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UBC Theses and Dissertations

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 S P I N T R A N S F E R M E A S U R E M E N T S O F T H E ird -» pp R E A C T I O N B y A n d r e w G . F e l t h a m B.Sc (Hons ) C a r l e t o n U n i v e r s i t y 1986 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D e p a r t m e n t of Phy s i c s W e accept th i s thes is as c o n f o r m i n g to the r equ i r ed s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1988 © A n d r e w F e l t h a m In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract A p r o t o n p o l a r i m e t e r has been con s t r uc ted at T R I U M F , w i t h des i gn spec i f i cat ions i n t e n d e d t o measu re the p o l a r i z a t i o n of p r o t on s over a n energy range of 100 MeV t o 300 MeV. It was b u i l t as the p r i n c i p l e detecto r i n a n e xpe r imen t t o d e t e r m i n e th ree sp in - t rans fe r pa r amete r s o f the f u n d a m e n t a l ird —» pp r e a c t i o n . In th i s thes is , some t heo re t i c a l a n d e x p e r i m e n t a l des i gn aspects of the sp in - t rans fe r measu rement are d i scussed. T h e i n ten t of th i s thes i s is t o desc r ibe an e x p e r i m e n t 1 w h i c h measures the p o l a r i z a t i o n of p ro ton s e m i t t e d f r o m t he nd —• pp r e a c t i on , u s i n g a n unpolarized t a r g e t 2 . T h e sole pu rpo se of th i s e xpe r imen t is t o demon s t r a t e t h a t ou r p o l a r i m e t e r a n d genera l a p p a r a t u s are capab l e of i d e n t i f y i n g the nd —> pp events f r o m a la rge b a c k g r o u n d presence, a n d t ha t t he s y s t ema t i c er rors a s soc ia ted w i t h the p o l a r i z a t i o n e x t r a c t i o n have been i den t i f i ed . T o th i s ex tent , t he s y s t em is r eady to p r oduce the 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 fo r the sp in - t rans fe r measurements . 1This experiment is identical in all respects to the spin-transfer experiment, except that here, the target is unpolarized. 2The polarization of the protons is well know from the analyzing power, Ajvo, of the time reversed pp —• dir reaction. i i Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgements viii I Introduction 1 1.1 T h e o r e t i c a l B a c k g r o u n d 1 1.2 G o a l s of E 331 3 1.3 C h o i c e of Trd —* pp C h a n n e l 6 1.4 E x p e r i m e n t a l D e s i g n P h i l o s o p h y 8 1.5 The s i s S u m m a r y 10 II Polarimeter Theory and Calibration 11 II. 1 D e f i n i t i o n of A n a l y z i n g P o w e r a n d P o l a r i z a t i o n fo r S p i n | Pa r t i c l e s 11 11.2 H o w are A n a l y z i n g Powers Mea su red ? 14 11.3 ANO as a C a l i b r a t i o n Source o f P o l a r i z e d P r o t o n s 15 11.4 P o l a r i z a t i o n E s t i m a t o r s 18 II. 4.1 T h e A c c e p t a n c e Test 21 III Description of Apparatus 24 III. 1 O u t l i n e of the 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 Ta rge t 27 III.2.2 T h e C a r b o n A n a l y z e r 28 i i i 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 E331 E v e n t T r i g ge r 33 111.2.6 T h e J - l l P rep roce s so 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 38 IV . 1 G e o m e t r y o f the S y s t e m 38 IV.2 O u t l i n e o f the Software P h i l o s o p h y 40 IV.3 C a l i b r a t i o n o f the REPDISK Sof tware 41 IV.3.1 ONLINE w i r e chamber c a l i b r a t i o n 41 IV . 3.2 O f f l i ne p a r a m e t e r c a l i b r a t i o n 46 IV.4 P o l a r i z a t i o n So f tware 47 IV. 5 D i a gno s t i c s 48 V Experimental Results 51 V . l P e r f o r m a n c e of A p p a r a t u s 51 V . l . l T h e S c i n t i l l a t o r s 51 V . l . 2 T h e W i r e C h a m b e r s 53 V.1.3 Sof tware D e f i n i t i o n of a " G o o d " E v e n t 55 V . l . 4 K i n e m a t i c s C a l c u l a t i o n s 55 V .2 P o l a r i z a t i o n Re su l t s 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 d imens i on s 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 essent ia l e xpe r imen t angles 40 III A l i s t o f o u t p u t f i les u sed for d i agnos t i c s 50 I V A t ab l e o f the po l a r i z a t i on s o b t a i n e d i n the p o l a r i m e t e r 60 V A c o m p a r i s o n p r e d i c t e d s p i n precessed p o l a r i z a t i o n s w i t h those mea su red 64 V I C o u p l i n g constants 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 represent ing e la s t i c a n d i ne l a s t i c NN p roces s ' . . 2 2 S p i n - c o r r e l a t i o n observables fo r the pp —> dix r e a c t i o n 4 3 Sp in - t r an s fe r observab le f i t s fo r the pp —> dw r e a c t i o n 5 4 S i m p l e m o d e l of sp i n -o rb i t c o u p l i n g 12 5 A n a l y z i n g power geomet r y 14 6 A n a l y z i n g power p lo t s t a k e n f r o m the fits of 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 de f i n i t i on s of Ayvo a n ( l PN 17 8 A n g u l a r dependence of pp —> drr a n a l y z i n g power 18 9 D i a g r a m of acceptance test. F i g u r e a) conta in s a c ros s - sect iona l v i e w of the acceptance test, b ) con ta i n s a n end - v i ew of the acceptance test. 23 10 T h e l a you t o f the M - l l E x p e r i m e n t a l A r e a fo r E331 25 11 S c h e m a t i c l a you t of the p o l a r i m e t e r a r m 26 12 L a y o u t of a Multi-wire Drift chamber p l a ne 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 t r i gger 34 14 T h e effect of the J l l o n the d a t a w r i t t e n (accepted) t o t ape 35 15 T h e geomet r y w h i c h defines the s ca t t e r i n g angles 39 16 T h e REPDISK flow char t 42 17 A flow char t of the s ub rou t i ne PION 43 18 C h e c k ca l i b r a t i on s 45 19 E x a m p l e o f the w i r e chamber combs 46 20 D e p e n d a n c e of the o p e n i n g ang le (8a + 6^) o n 8a 48 21 F l o w char t of the software r o u t i n e POLAR 49 22 a n d TOF d i s t r i b u t i o n s fo r the s c i n t i l l a t o r s 52 ax 23 A t y p i c a l r e so l u t i on p l o t 54 24 T h e x, y a n d z prof i les a n d RDS of the t race back t o the c a r b o n ana l yze r . 56 25 T h e x, y a n d z prof i les a n d RDS of the t race back t o the deu te r a t ed ta rget 57 26 T h e 1 -d imens iona l prof i les of the o p e n i n g ang le a n d c o p l a n a r i t y d i s -t r i b u t i o n s 58 27 T w o d i m e n s i o n a l k i n e m a t i c s d i s t r i b u t i o n 61 28 P o l a r i z a t i o n dependance 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 dependance o n 9A a n d 63 30 G e o m e t r y fo r target t r aceback 70 31 P o l a r i z a t i o n r o t a t i o n due t o r e l a t i v i s t i c boos t . 74 v i i Acknowledgements I w o u l d l i k e t o a cknow ledge the m a n y m e m b e r s o f the E 3 3 1 g r o u p , w i t h o u t w h o m a n e x p e r i m e n t o f t h i s n a t u r e w o u l d no t be pos s ib le . I n a d d i t i o n I w o u l d l i ke t o t h a n k P e t e r T r e l l e , M a r c e l l o P a v a n a n d P e t e r W e b e r fo r t h e i r v a l u a b l e c o n t r i b u t i o n s t o t he p ro jec t a n d i n p a r t i c u l a r G a r t h Jone s , w h o g u i d e d m e over m a n y h u r d l e s w i t h t he p ro jec t , e spec i a l l y i n t he d y i n g days o f t h i s thes is . F i n a l l y , I w o u l d l i ke t o acknowledge m y f e l l ow phy s i c s g roup ie s ; M a r k , C h r i s , 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 3 , whose ant i c s d u r i n g the last two years, have m a d e u n i v e r s i t y l i fe fun, at any h o u r o f the day. I r e a l l y have t o t h a n k m y r o o m m a t e H a r v , w h o has p u t u p w i t h a lo t as I p l o w e d t h r o u g h th i s thesis. 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, phys i c i s t s 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 reac t i on s i n a n effort to c o m p r e h e n d the m a n y c o n t r i b u t i n g processes w i t h i n t he nuc leus . T h e i m p o r t a n c e of the p i on ' s ro le t o t he u n d e r s t a n d i n g of nuc le i is e x e m p l i f i e d by the i t s s ign i f icant c o n t r i b u t i o n to the nuc lea r forces [1]. T h i s research has been a c c o m p a n i e d by extens i ve work s o n nuc l eon -nuc l eon (NN) a n d p i on - nuc l eon (nN) s c a t te r i ng i n o rde r t o p r o v i d e theor i s t s w i t h adequa te i n f o r m a t i o n o n w h i c h t o base a comp le te nuc lea r m o d e l . I n recent years , s i gn i f i cant progress has been ach ieved i n p a r a m e t e r i z i n g NN e la s t i c s c a t t e r i n g react ions u s i n g a One Boson Exchange ( O B E ) m o d e l [2]. T h i s e l emen ta r y process, s hown i n f i gure l a , occurs v i a the t rans fe r of a v i r t u a l bo son f r o m one nuc l eon t o another . U n f o r t u n a t e l y , a s im i l a r s i m p l e m o d e l does not ex i s t fo r 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 b y nuc le i . A p i o n i n t e r a c t i n g w i t h a nuc l eon w i t h i n a nuc leus , w i l l f r equen t l y f o r m a n exc i t ed i sobar such as a N* o r A (see figures l b , l c ) ) . T h e c o u p l i n g of these i sobars w i t h the secondary nuc leons open m a n y c o m p l e x pa th s by w h i c h a r e a c t i o n c an fo l low. In o rde r to f u l l y u n d e r s t a n d the nuc leus , i t is i m p o r t a n t to u n d e r s t a n d how the N* a n d A coup le w i t h nuc leons a n d p ions . T h i s c o m p l e x nuc lea r env i r onment c an be g rea t l y s imp l i f i e d t h r o u g h the s t u d y of 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 by few nuc l eon sys tems. T h e mos t f u n d a m e n t a l of the NNTT ^ NN react ions is ird ^ pp, because o f the two b o d y 1 A/N* \ A/N' N v> a) b) Figure 1: a) Feynman diagram for N — N elastic scattering; b), c) product ion of A or TV* through p ion absorption. nature of its i n i t i a l and final states. This p a t h also profits by involving only charged particles i n both entrance and exit channels, thus permit t ing easy detection experimentally. A t intermediate energies, the pp —> dir reaction forms predominantly an NA intermediate state [3]. Information on how the ird ^  NA ^ pp reaction develops is revealed i n terms of a part ia l wave expansion of the process. Such an expansion i n terms of angular momentum and spin components, provides a very useful interface between experiment and theory. The various contributions of the part ia l waves provide theorists w i t h knowledge of the quantum numbers which characterize the important intermediate states [4]. Because of their angular momentum basis, the part ia 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 t ha t e xpe r imen ta l i s t s have been ab le t o measure such s p i n r e l a t ed pa ramete r s . T h e lack o f ea r l y progress was p r i m a r i l y due t o t echno l og i c a l cons t ra in t s , such as l o w b e a m cur rent s a n d d i f f i cu l t ie 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 ex i s t ed a suff ic ient d a t a base f r o m w h i c h one c o u l d b e g i n the d e t e r m i n a t i o n of the var ious p a r t i a l wave a m p l i t u d e s [6] [7]. T o date , t he vast m a j o r i t y o f d a t a is ba sed o n s p i n observables a s soc ia ted w i t h t he p r o t o n channe l s . The se p r i m a r i l y cons i s t of the s p i n c o r r e l a t i o n pa ramete r s , ANN, ^-LL-I ASS, etc. The se observables cha rac te r i ze the sp in -dependence o f the pp —• dix c ross -sect ion fo r va r i ou s con f i gu ra t i on s of 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 gure 2, success fu l p a r t i a l wave a m p l i t u d e f i ts to th i s d a t a c an be 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 , there is no t a u n i q u e s o l u t i on , as there is mo re t h a n one set of p a r t i a l wave a m p l i t u d e s w h i c h f i t t he d a t a equa l l y we l l . T h e s e degenerate so lu t ions l e ad to s ign i f i cant amb i gu i t i e s i n t he p r e d i c t i o n s of t he deu te r on sp in -dependent observables , as shown i n figure 3. It is therefore c lear t h a t t he 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 deu te ron sp in -dependent p a r a m e t e r s is necessary to p rov i de i m p o r t a n t con s t ra i n t s o n the pos s ib le so lu t ions of t he p a r t i a l wave amp l i t ude s . 1.2 Goals of E331 T h e o n l y s i gn i f i cant deu te ron sp in -dependent d a t a , p u b l i s h e d to date , is the measu rement of iTu at S I N [8] 1. In o rde r to fill th i s v o i d , E331 at T R I U M F was des i gned t o measu re the sp in - t rans fe r observables Kis, KSS a n d KNN of the f u n d a m e n t a l ltd —* pp r e ac t i on . The se pa ramete r s eva luate the exchange of 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 tate p a r t i c l e t o one i n the final s tate. In E 3 3 1 , t he 1The 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. 4 0.40 0.30 -0.30 • i i i i i i ii< I I • / \ 0.20 0.10 0.00 -0.10 1 -0.20 -0.30 -0.40 I ' 1 1 1 I M 1 1 I 45. 90. 135. 180. 9" 0.30 K s s - o . i o i 1 1 1 1 i 1 1 1 1 i 1 1 1 45. 90. 135. 180. Figure 3: Spin transfer observable fits for the pp —> dw reaction (Tp = 500 MeV). Note that this corresponds to Tv = 105 MeV of the wd —> pp reaction. 5 degree of s p i n t rans fe r f r o m the p o l a r i z e d deu te r on i n the i n i t i a l s tate, t o one of the f i n a l s tate p r o t on s was measured . T o mea su re these pa ramete r s , th ree d i s t i n c t p o l a r i z a t i o n con f i gu ra t i on s were cons idered. KT,S refers t o the c o n t r i b u t i o n of l o n g i t u d i n a l l y p o l a r i z e d deuterons to the s ideways p o l a r i z a t i o n of the f i n a l s tate p r o t o n . S im i l a r l y , Kss, {KNN) refers to the in f luence of s ideways ( n o rma l ) p o l a r i z a t i o n o f the deu te r on , o n the s ideways ( n o r m a l ) p o l a r i z a t i o n of the p r o t o n . In th i s a p p l i c a t i o n , longitudinal p o l a r i z a t i o n o f 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 the center of mass s y s t em a n d normal is denned by the n o r m a l t o the s c a t t e r i n g p l ane . Sideways p o l a r i z a t i o n is t he componen t o r t h o g o n a l t o the 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 the M a d i s o n C o n v e n t i o n [9]. S ince i t has been c l a i m e d t ha t a measu rement of these observab les w i l l resolve the e x i s t i n g amb i gu i t i e s o f the p a r t i a l wave a m p l i t u d e s [10], t he goa l of E331 is t o f i n a l l y p i n d o w n a l l t he i m p o r t a n t a m p l i t u d e s of the red ^  pp r e ac t i on . T h i s d a t a w i l l a lso he lp t o c l a r i f y the c o n t r i b u t i o n s of the var ious i n t e r m e d i a t e states of t he irNN ^ NN r e a c t i on , a l l ow i n g theor i s t s t o deve lop a m o r e accu ra te m o d e l 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 nuc le i . 1.3 Choice of nd —+ pp Channel T h e r e are two channe l s ava i l ab le t o a n expe r imen te r i n t e n d i n g t o s t u d y the Tvd ^  pp r e ac t i on . T i m e rever sa l i n va r i ance suggests 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 the same react ions . In o the r words , one c an o b t a i n t he same bas i c i n f o r m a t i o n f r o m ab so rp t i o n as p r o d u c t i o n . The re f o re b o t h op t i on s mus t be we ighed w h e n des i gn ing an expe r imen t . T r a d i t i o n a l l y , mos t e xpe r imen t s have been p e r f o r m e d i n the pp —> dn d i r e c t i o n , because fo r m a n y years the q u a l i t y o f p r o t o n beams were b y fa r supe r i o r to p i o n beams. A s w e l l , p r o ton s have a lways been m u c h eas ier t o po l a r i z e t h a n 6 have deute rons , i n b o t h beams a n d targets . Hence the advantage of exper ience a n d c onven t i o n f avou red th i s channe l . In fact , a n e xpe r imen t i n t e n d i n g to measu re the s p i n t r an s fe r p a r a m e t e r ^ 5 5 is c u r r en 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 deu te r on p o l a r i m e t e r . A l t h o u g h t h i s e xpe r imen t benef i ts f r o m the m u c h h i ghe r s t a t i s t i c s , r e s u l t i n g f r o m the s i gn i f i c an t l y h i ghe r cur rent 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 d i sadvantages . F i r s t o f a l l , t he deu te ron p o l a r i z a t i o n con ta in s b o t h vec to r a n d tenso r t e r m s 2 . B o t h te rms w i l l c o n t r i b u t e t o the s p i n t rans fe r , yet t he i r effects are 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 the p o l a r i m e t e r is concerned . T h i s leads to a s i gn i f i c an t l y m o r e c o m p l i c a t e d ana l y s i s 3 . Secondly, a n d m o r e s i gn i f i cant ly , there ex ists l i t t l e o r 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 deu te r on p o l a r i m e t e r at the energies i n vo l ved . Converse ly , fo r the ltd —> pp r e ac t i on , there ex ists w e l l e s t ab l i s hed p r o t o n p o l a r i m e t e r techn iques . A s w e l l , successfu l t echno logy to p r o d u c e a n d measure p o l a r i z e d deu te r on targets has recent l y been deve loped at T R I U M F [12]. D u e t o the d i f ference i n the center of mass energies of the two reac t i on s , the -nd —> pp d i r e c t i o n a l lows us to inves t i ga te the r e a c t i o n over a m u c h la rger k i n e m a t i c range t h a n w o u l d be pos s ib le w i t h ava i l ab le T R I U M F energies i n the reverse sense. F o r e xamp le , expe r iment s c an be c a r r i e d ou t over a p i o n k i n e t i c energy range of 105 to 255 M e V . T h i s co r responds to a n equ iva lent p r o t o n energy range of 500 to 800 M e V . Such a range a l lows measu rement of the s p i n t rans fe r observables over the reg ion of the S-wave NA resonance, w h i c h peaks at Tv— 140 M e V (Tp « 565 M e V ) [1]. T h e p r i n c i p l e d i sadvantages of the ird —> pp c hanne l are l ow p i o n f luxes a n d 2Tensor terms are a result of the fact that the deuteron has a spin of lh. 3The 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. 7 o n l y b a r e l y adequa te deu te ron target p o l a r i z a t i o n s ( < 40 % ) . A s w e l l there is a cons ide rab le presence of b a c k g r o u n d due t o quas i - free deu te r on a b s o r p t i o n o f the p ions b y o the r m a t e r i a l s i n the reg ion of the target . F o r t u n a t e l y , b a c k g r o u n d r e m o v a l techn iques do ex i s t , as w i l l be d i scussed i n m o r e d e t a i l i n the nex t sect ion. 1.4 Experimental Design Philosophy T h e fac to r s w h i c h i n f l uenced the des ign of th i s e x p e r i m e n t were, p r i m a r i l y t he presence o f c o m p e t i n g b a c k g r o u n d reac t ions , t he la rge m a g n e t i c field i n the reg ion of t he ta rget a n d the requ i rement 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 he p o l a r i m e t e r fo r the e x t r a c t i o n of i t s observables. M u l t i - w i r e i o n i z a t i o n chambers are w i d e l y u sed fo r cha rged p a r t i c l e p o s i t i o n i d e n t i f i c a t i o n . T h e y are m o r e effect ive t h a n o the r types of detector s i n p r o v i d i n g a c cu r a te p o s i t i o n i n f o r m a t i o n over la rge d i s tances, as is r equ i r ed by the p o l a r i m e t e r . In o rde r t o separate the t rue ird —* pp events f r o m the b a c k g r o u n d , i t s two b o d y i n i t i a l a n d final s tate feature was used. A p i o n i n c i den t u p o n a deu te r on at rest w i l l p r o d u c e two p ro ton 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 on sh i p . If one ident i f ies t he energy o r ang le of one of the final s tate p ro tons , t he c o m p l i m e n t a r y p r o t o n mus t t r a v e l at a w e l l de f ined angle a n d energy w i t h respect t o i t . A l s o , con se r va t i on o f m o m e n t u m requ i res a l l pa r t i c l e s to m o v e i n the same p lane . O n t he cont ra ry , i f t he p i o n is ab so rbed b y a " qua s i - f r ee " deu te r on w i t h i n a la rger nuc leus , t he amoun t of energy ava i l ab le to the p r o t on s 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 nuc leus. T h i s w i l l l e ad to a s i gn i f i cant b r o a d e n i n g i n the energy a n d angle co r re l a t i on s be tween the pro tons . A n eff ic ient de tec t i on s y s t em mus t be ab le to a c cu r a t e l y i den t i f y p r o t o n pa i r s h a v i n g the a p p r o p r i a t e k i n e m a t i c r e l a t i on sh ip . T h r e e s t a n d a r d techn iques are ava i l ab le fo r th i s pu rpose . Time of flight 8 (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 of the two p ro tons . T h e s u m o f t he i r energies is a cons tant fo r a t r ue nd —• pp r e a c t i o n . T h e use of TEA was r u l e d ou t , s i m p l y because any p r o t o n w h i c h was s ca t te red b y the c a r b o n ana l yze r , w o u l d lose m u c h of i t s energy before en te r i n g the a b s o r p t i o n detector , therefore deg rad i ng the s y s tem ' s a b i l i t y t o resolve energy. It also w o u l d have r equ i r ed a n exo rb i t an t th i cknes s o f s c i n t i l l a t o r t o s top these h i g h energy p ro tons . T h e TOF t echn ique w o u l d requ i re the detector s t o be p l a c e d at a la rge d i s tance f r o m the target i n o rde r t o o p t i m i z e t he energy r e s o l u t i on . T h i s , however, w o u l d reduce the ove ra l l acceptance o f the p o l a r i m e t e r a n d t he reby reduce s ta t i s t i c s . S uch systems, however, w o u l d r e t a i n t he i r energy r e s o l u t i o n desp i te the t r a j e c t o r y d i s to r t i on s i n t r o d u c e d by the t a r ge t ' s m a g n e t i c f i e l d , s ince the p r o ton ' s energy is not af fected b y th i s f i e ld . W i t h the t h i r d m e t h o d , one ident i f ies the t ra jec to r i e s of the two p ro ton s , u s i n g w i r e chamber s , a n d checks the k i n e m a t i c a n gu l a r co r re l a t i on s be tween t h e m . T h i s p r o cedu re also ensures t h a t the pa r t i c l e s are co -p lanar . S u ch 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 the d i s t o r t i o n o f the t r a jec to r i e s due to the la rge m a g n e t i c f i e ld i n the reg ion of the target. However , s ince th i s f i e ld was w e l l k n o w n , i t is poss ib le to cor rect fo r i t s affect i n the subsequent ana lys i s . T h e de tec to r f a c i l i t y at T R I U M F was ab le t o p r o d u c e h i g h l y accu ra te , fast M u l t i - W i r e D r i f t C h a m b e r s at a reasonab le cost. S ince po l a r ime te r s r equ i re p o s i t i o n i n f o r m a t i o n anyway, i t was dec i ded to use the t r a j e c t o r y techn ique. A s a resu l t , two a r m s c o n t a i n i n g w i r e chamber s were con s t r uc ted . T h i s cho ice a l l owed us t o p ro f i t f r o m the la rger acceptance w h i c h c o u l d be o b t a i n e d b y l o c a t i n g the p o l a r i m e t e r c loser to the target . In 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 to have s i gn i f i cant va lue i n the ana ly s i s , as severa l p a r amete r s were f o u n d to have a t r a j e c t o r y dependence, (see chapte r 5) 9 1.5 T h e s i s S u m m a r y In the p r e c e d i n g sect ions, the need for f u r t h e r deu te r on sp in -dependent p a r a m e t e r s o f the ird —> pp r e ac t i on , has been d i scussed a n d a n e x p e r i m e n t ( E331 ) , i n t e n d e d t o measure th ree o f t h e m was desc r ibed . It is t he pu rpo se of th i s thes i s t o desc r ibe the te s t i ng a n d c a l i b r a t i o n o f the E331 s y s t em, b o t h w i t h respect t o d e m o n s t r a t i n g i r s a b i l i t y to reject b a c k g r o u n d reac t i on s a n d t o c o n f i r m success fu l o p e r a t i o n o f the p o l a r i m e t e r itself. In th i s r e ga rd , th i s thes is descr ibes a n e xpe r imen 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 of p ro ton s p r o d u c e d f r o m the ird —> pp r e a c t i on , a process i n v o l v i n g a n u n p o l a r i z e d deu te r a t ed t a r g e t 4 . S uch a r e a c t i o n is u se fu l because t he p o l a r i z a t i o n of the 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 the s c a t t e r i n g p l a n e a n d equa l to the w e l l - k n o w n a n a l y z i n g power, A^o, of the t i m e reversed pp —> dn channe l . Hence , one is p r o v i d e d w i t h a source o f p ro ton 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 source c an t hu s be used to check for s y s t ema t i c effects w h i c h i n f l uence 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 th i s thes is i n t roduces the de f i n i t i on s a n d t heo r y r equ i r ed for a n u n d e r s t a n d i n g o f the po l a r ime te r . In chapte r 3, a d e s c r i p t i o n o f the e x p e r i m e n t a l a p p a r a t u s is p resented. T h i s e x p e r i m e n t a l setup is i d e n t i c a l t o t h a t i n t e n d e d for the sp in - t ran s fe r measurements . C h a p t e r 4 conta in s a d e s c r i p t i o n of the sof tware ana ly s i s r ou t ine s . F i n a l l y , chapte r 5 s ummar i z e s the pe r f o rmance o f the appa ra tu s , presents the p o l a r i z a t i o n resu l t s a n d deals w i t h the s y s temat i c s of the expe r imen t . 4 The 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 In th i s s ec t i on , t he concepts o f polarization a n d analyzing power for s p i n | pa r t i c l e s are d i scussed. A de s c r i p t i on of p o l a r i z a t i o n is best p receded by i t s de f i n i t i on . A n ensemble o f pa r t i c l e s whose sp ins are a r r anged i n a n o n - r a n d o m f a sh i on is s a i d t o be polarized. F o r s p i n | pa r t i c l e s , p o l a r i z a t i o n is d e t e r m i n e d f r o m the d i f ference between the n u m b e r of pa r t i c l e s a l i gned p a r a l l e l t o a pre fe r red ax i s (N+) a n d those d i r e c ted a n t i - p a r a l l e l (iV_). T h i s d i f ference is n o r m a l i z e d by the t o t a l n u m b e r of pa r t i c l e s i n the ensemble a n d expres sed as a percentage. p=^rfixl00% (1) T h e i d e a o f analyzing power is usefu l i n de s c r i b i n g the s p i n dependence of nuc lea r cross-sect ions. It is a w e l l - k n o w n p h e n o m e n o n [13] tha t a p o l a r i z e d b e a m of pa r t i c l e s i m p i n g i n g o n a nuc lea r target , w i l l p r o d u c e a s c a t t e r i n g d i s t r i b u t i o n w h i c h is a z i m u t h a l l y a s y m m e t r i c . S im i l a r l y , a n u n p o l a r i z e d b e a m o f pa r t i c l e s s ca t te red off nuc le i w i l l exper ience v a r y i n g degrees o f p o l a r i z a t i o n , i n the f i na l s tate, w h i c h is a f u n c t i o n o f the s ca t t e r i n g ang le a n d i n c i den t p a r t i c l e m o m e n t u m . B y u t i l i z i n g these effects, one has a m e t h o d fo 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 o f p o l a r i z e d pa r t i c l e s . T h e s e p h e n o m e n o n are a re su l t o f a n o n - c e n t r a l tenso r fo rce w i t h i n t he nuc leus , w h i c h p re fe ren t i a l l y def lects pa r t i c l e s of 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 ch a force c an be s i m p l y i l l u s t r a t e d as b e i n g a consequence of the 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 sp in -dependent effects. T h u s the analyzing power of a r e a c t i o n is a n u m b e r w h i c h re lates the s ize o f the s p i n dependent c oup l i n g t o a l l o t he r forces o f the i n t e r a c t i o n . W h e n u sed i n the context o f a p o l a r i m e t e r , t he concept o f analyzing power is used t o desc r ibe t he p o l a r i z a t i o n measu rement ef f ic iency o f a p a r t i c u l a r r eac t i on . Analyzing power fo r s p i n | pa r t i c l e s is de f ined m a t h e m a t i c a l l y as the r a t i o o f the 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 igure 5) t o the i n c i den 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 is averaged over a l l the co l l i s i on processes, c, t he a s ymmet r y , is an eas i l y mea su r ab l e q u a n t i t y de f ined by e — NL a n d NR represent the n u m b e r s ca t te red t o the left a n d right of the i nc iden t p r o t o n d i r e c t i o n i n the r eac t i o n p lane . Left is i n the d i r e c t i o n of the c ros s -p roduct of the i n c i den t u n s c a t t e r e d vec to r a n d the n o r m a l to the s ca t t e r i n g p lane. E q u a t i o n 2 f o l l ows d i r e c t l y f r o m the m o r e genera l expres s ion for the sp in -dependent cross - sect ion o f p o l a r i z e d s p i n | pa r t i c l e s i nc iden t o n an u n p o l a r i z e d ta rget (or v ice-versa) . dap(6,<f>) _ do-0{6,<t>) -dn dn dn do0(6, <f>) dn [1 + PA(9)} (3) where n is de f i ned i n e q u a t i o n 6. F r o m th i s express ion, one c a n o b t a i n the a n a l y z i n g power , dat (6,4.) MO) = (4) Here , t he 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-sect ion w i t h respect t o the t o t a l cross - sect ion. 13 Polarization G Proton Analyzer. Counters F i g u r e 5: T h e analyzing power is mea su red by c o m p a r i n g the n u m b e r of p ro ton s w h i c h s ca t te r left a n d r i gh t , fo r a g i ven i n c i den 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 he effect ive nuc leon -nuc leus a n a l y z i n g p o w e r is s o m e w h a t m o r e c o m p l e x t h a n the n i ave m o d e l d i scussed i n the p rev i ou s sec t ion . T h e s c a t t e r i n g is a m i x t u r e o f severa l t ypes o f e las t i c a n d i ne l a s t i c i n te rac t i on s . E a c h process has i t s o w n a n a l y z i n g powe r due t o the different coup l i n g s i nvo l ved . T h e t o t a l a n a l y z i n g power is a we ighted average o f a l l of these effects a n d is p r i m a r i l y dependent o n i n c i den t b e a m energy s ca t te r i ng ang le a n d the energy a ccep tance o f the detecto r . Becau se of the m a n y c o m p l e x c on t r i bu t i o n s , the m o s t 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 measu re i t e xpe r imen ta l l y . T h i s is done by s c a t t e r i n g a b e a m o f nuc leons , w i t h k n o w n p o l a r i z a t i o n o n a de s i r ed target. G i v e n a monoene rge t i c b e a m , the a n a l y z i n g power fo 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 he s c a t t e r i n g a s y m m e t r y e at t ha t angle, as expressed b y e q u a t i o n 2 a n d de sc r i bed i n figure 5. 14 An appropriate target for the measurement of proton polarization is carbon because its principle component, 1 2 C , has zero spin 1 and the nuclear scattering 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]2 was chosen to 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. On 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 straggling3 and dispersion of the 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 1 A spin zero nucleus is free from the extra complexity of spin-spin coupling. 2It should be noted that there are some errors in this paper [17] which have been identified after communications with the authors. 30ur 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. 15 -0.1 H 1 1 1 1 1 h 0 100 200 300 400 500 600 Proton Energy (MeV) 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) Ana-lyzing power distribution for 7 cm thick carbon, where the average proton energy at the analyzer center is 200 MeV. 16 p cm P TT pp -* dn P P 6 cm 7Td -> pp F i g u r e 7: A n g u l a r de f in i t i ons fo r ANO a n d PN of the —• dn a n d 7rd —• p p react ions . sec t i on II. 1, is a p p l i e d i n the reverse sense. P a r i t y a r gument s , ensure t h a t o n l y n o r m a l l y p o l a r i z e d p r o t o n s c an be p r o d u c e d b y t he ltd —• pp r e a c t i o n i f an u n p o l a r i z e d target is e m p l o y e d . T i m e rever sa l i n va r i ance i nd i ca te s t h a t t he i r p o l a r i z a t i o n , PN, is e qua l t o the we l l - known a n a l y z i n g power , ANO for pp —* dn [18]. In the u sua l ang le conven t i on ANO is expres sed w i t h respect t o t he ou t - go i ng p i o n , 8„ [19] a n d PN w i t h respect the f o r w a r d p r o t o n 8j. T h i s conven t i on is de f ined i n figure 7, w i t h |#* | = i n the center o f mass . ANO c a n be represented by t he r a t i o be tween t he p o l a r i z e d a n d u n p o l a r i z e d d i f f e ren t i a l cross-sect ions as de s c r i bed be e q u a t i o n 4. T h e s e cross sect ions m a y be fitted t o a s soc ia ted Legend re a n d Legend re p o l y n o m i a l s respect ive ly . U s i n g the m o s t recent energy -dependent coeff ic ients ( p r o v i ded b y W a l d e n [20]) a s imp le c o m p u t e r p r o g r a m y i e l d e d the 8* dependent ANO d i s t r i b u t i o n at the p i o n energies t y p i c a l l y s t u d i e d i n E 331 . T h e d i s t r i b u t i o n co r r e spond i ng to t he p i o n energy used 17 .0 0 50 150 200 Figure 8: Plot of Analyzing Power A^0 for pp —• dn with Tp = 700 MeV corre-sponding to TTT = 205 MeV in the time reversed reaction. 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]; where n is normal to the p - C scattering plane, defined by equation 6, PB is the polarization of the incident proton beam, A(9) is the p - C analyzing power, I(e,<f>) = I0(6)[l + A(8)PB-h] (5) n = Pi x Pf P, XP> (6) and Jo(0) is the unpolarized cross-section. The geometry of this system is defined 18 i n figure 15. T h e C a r t e s i a n componen t s o f the p o l a r i z a t i o n a re PB = Pxx + Pyy + PJ (7) Pz is i gno red s ince componen t s i n the d i r e c t i o n o f m o m e n t u m do not affect t he 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> are the p o l a r a n d a z i m u t h a l angles o f the s ca t te r i ng . In t e rms o f component s , F r o m e q u a t i o n 2, one c an w r i t e e^(8) = A(8)Py a n d es(8) = A(8)PX, whe re e# is the 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 onto the p — C s c a t t e r i n g p l a ne a n d es is the a s y m m e t r y p r o j e c t e d on to the o r t h o g o n a l p l ane def ined b y the i n i t i a l p r o t o n m o m e n t u m pi a n d the n o r m a l t o the r eac t i on p lane. B o t h a s ymmet r i e s are mea su red abou t the ax i s de f ined by p~. T h e t e r m Ac(8, (f>) is a d d e d t o ref lect t he acceptance o f the sy s tem. T h i s f a c to r account s fo r the i n s t r u m e n t a l con s t r a i n t s o n the de tec t i on o f t he s ca t t e r i n g pa r t i c l e s , w h i c h , i f i gno red , c o u l d p r o d u c e false a s ymmet r i e s . F o r e x a m p l e , pa r t i c l e s w h i c h t r a ve l near the edge of the detecto r are sub jec ted t o b ias s ince those w h i c h scat ter t owa rd s the center o f the appa r a tu s w i l l be de tec ted whereas those w h i c h deflect ou twa rd s m a y no t . T o s i m p l i f y t he f o l l ow i ng d i scus s ion , t he de r i va t i on s a re p e r f o r m e d for t he subset o f a l l events h a v i n g a c o m m o n s ca t t e r i n g angle, 8k. E x p a n s i o n t o the c o m p l e t e d a t a set w i l l be de sc r i bed subsequent ly. O n e c a n compensa te f o r the effect of the acceptance 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 are met . F o r a g i ven 8k, Besset et a l . [21] have shown t h a t i f the accep tance f u n c t i o n Ac(4>) is s y m m e t r i c w i t h a p e r i o d of 7r 4 4This is referred to in this thesis as azimuthal symmetry. 1(8, <f>) = I0(8)[l + A(8)Py cos(<j>) + A(6)PXsin(<£)] x Ac(6, <f>) (8) 19 (Ac(<f>) = Ac(<f> + TT)) 5, the following relations are true r2ir t7n r7* ( * f(<f>) cos(<f>)d<f> = eN j * f(<f>) cos\<f>)d<f> + es I * f(<t>) sm(<f>) cos(<t>)d4> Jo Jo Jo F f(<f>) sm(<t>)d<f> = eN F f{4>) sin(<£) cos(<£)<ty + es F f(<f>) sm2(<f>)d<j> (9) Jo Jo Jo 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 o f((j>)cos(<j>)d<t> — £ s i n ( < ^ ) « Jo f(<t>)s\n(4>)d<t> 1 £ cos2(<M * jf* fit) c o s 2( W ^ £ s i n 2 ( t f , ) « f(<f>)sm\<f>)d4> 1 f2w — y^sin(^/)cos(^/) « / f(4>)sm(<t>)cos(4>)d<f> iv , JO Using an obvious matrix notation, equations 9 can be written as; / £ , c o s ( < ^ ) \ / £ , c o s 2 ( < ^ ) £ , s in (^ , ) cos (<^) W \ V Eisin(^/) / ~ \ E;sin(^j)cos(^/) £ /s in 2 (<£, ) J \ e s / nn) from which one may solve ek = Fk Bk. For each 6k, the polarization may be calculated using The overall polarization, P , can be estimated by weighting each 6k contribution by 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. Vk(e) « Fl1 (12) 5This is ensured by the subroutine AC.TST described in section II.4.1 20 and V(Pk) = A(eky (13) So (14) V / i t h Bp and Fp now denned as £/ A(8k)cos(<t>i) E , A ( ^ ) s i n ( ^ , ) (15) k D A ^ c o s 2 ^ , ) £ ; A 2 ( ^ ) s i n ( ^ , ) c o s ( ^ ) E i A2(f?*)sin(<£,)cos(<^) ZiA2(6k) sin2(4>,) (16) Now we can write P = Fp BP and V ( P ) = F p 1 It should be noted that the acceptance function Ac(8,<f>) does not appear in 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 in the detectors and apply an "acceptance test" to insure the azimuthal symmetry of the apparatus. II.4.1 The Acceptance Test A s indicated in 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 and 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 in figure 9, the proton vector incident upon the carbon, is extrapolated to the last wire chamber (wc 6) of the polarimeter a r m 6 . In order to 6This 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 symmetrically accepted event scottered 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 e x p e r i m e n t was p e r f o r m e d i n the M - l l e x p e r i m e n t a l a r ea at T R I U M F (see f i gu re 10). T h e M - l l beam- l i ne is capab le of p r o d u c i n g la rge p i o n f luxes over a great range o f energies. A b e a m prof i le , at the ta rget , of 1.5 c m ( F W H M ) i n the h o r i z o n t a l a n d 1 c m ( F W H M ) i n the ve 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 ivergence o f Ad = ± 0 . 6 7 ° (horz.) a n d A</> = ± 3 . 2 ° (vert. ) [22] are t y p i c a l l y o b t a i n e d by M - l l . S u ch spec i f i cat ions c o m m i t t e d a l l b u t the ta i l s of the b e a m to h i t t he s m a l l ta rget a rea (see sect ion III.2.1). T h e deu te r a t ed b u t a n o l ta rget was m o u n t e d i n the same c ryogen ic s y s t em used fo r the p o l a r i z e d target ( i n o rde r t o test the b a c k g r o u n d re jec t ion ) a n d was l o c a t e d at the focus of the p i o n b e a m . T h e p ro ton s p r o d u c e d i n the r e a c t i o n were i den t i f i ed 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) con f i gu red i n two a rms . T h e f o r w a r d a r m , ( a r m A ) def ined the s o l i d ang le of the d e t e c t o r 1 . T h e re l a t i v e p o s i t i o n of the two a rms was governed by the a p p r o p r i a t e 2 -body k i n e m a t i c r e l a t i on sh i p between the p ro tons . T h e p o l a r i m e t e r was c o n t a i n e d i n a r m A . It cons i s ted of a set o f th ree M W D C ' s o n e i ther s ide of a c a r b o n ana l yze r , as s hown i n figure 11. T h e three chambe r s d o w n s t r e a m f r o m the c a r b o n were s i gn i f i c an t l y l a rger t h a n the f o r w a r d ones, i n o rde r t o t r a c k la rge ang le scat ter ings f r o m the ana l yze r . S c i n t i l l a t o r s were 1Wire chamber 3 defined the solid angle (about 40 mSter.) since it was fully illuminated by the outgoing protons. 24 F i g u r e 10: T h e l ayou t 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 . p l a c e d at detection points at b o t h the en t rance a n d ex i t of the la rge c h a m b e r set i n o rde r to i den t i f y pa r t i c l e s w h i c h h a d passed f r o m the ana l y ze r t h r o u g h the l a t t e r h a l f o f the po l a r ime te r . A r m B was u sed t o i den t i f y the t r a j e c t o r y of the b a c k w a r d p r o t o n . It con t a i ned th ree s m a l l M W D C ' s fo r p o s i t i o n a l i n f o r m a t i o n . A g a i n , s c i n t i l l a t o r s were s i t u a t e d at detection points o n b o t h ends of th i s a r m t o q u i c k l y s i gna l the passage of a cha rged pa r t i c l e . Severa l p e r i p h e r a l detectors were present i n the a rea, whose p r i n c i p l e f u n c t i o n was to p r o v i de b e a m m o n i t o r i n g a n d d iagnost ic s . A s m a l l finger s c i n t i l l a t o r ( S i ) p l a c e d i n the p r i m a r y b e a m , p r o v i d e d a measu re of the i n c i den t p i o n b e a m ra te . 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 ro f i l e of the b e a m i nc i den t o n the ta rget , a n d a l l owed one to check for sh i f t s o r b r o a d e n i n g of the p i o n b e a m . D i s p l a c e m e n t o f the b e a m was also m o n i t o r e d b y a d o w n s t r e a m hodoscope. T h i s 25 63 cm 21 cm 111 cm to tgt. < \J/ 63-cfrJ ZU 21 cm — > 54" to hgt. 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 f ou r p a d d l e dev ice w h i c h was cen t r a l l y l o ca ted i n the b e a m , such t ha t any sh i f t w o u l d create a re la t i ve increase of the count r a te i n one of the panels . F i n a l l y , t w o sets of counters were u sed t o m o n i t o r f l u c t u a t i o n s i n the p i o n b e a m ' s in tens i ty . A m u o n counter (pi • fa) w a s p l a c e d bes ide the M - l l b e a m p i pe to count decay m u o n s f r o m the p i o n b e a m . T h e m o n i t o r te lescope ( M T 1 • MT2 • M T 3 ) was p l a c e d away f r o m the b e a m , a i m e d at the target . It c oun ted a l l cha rged pa r t i c l e s r a d i a t i n g f r o m the target . B o t h o f these techn iques 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 i n c i den t p i o n b e a m rate. A l l s ignals p r o d u c e d by the detectors were sent v i a fast, l ow a t t e n u a t i o n c o a x i a l cables t o the M - l l c o u n t i n g r o o m where the event l og i c was processed, a n d EVENT i n f o r m a t i o n was s to red on m a g n e t i c tape. T h e d a t a a c q u i s i t i o n was 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 f u n c t i o n was to w r i t e t he event d a t a onto m a g n e t i c tape, b u t w h i c h a l so 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 h i s tog rams f r o m these fo r on l i ne d iagnost i c s . I n s t r u m e n t a l i n f o r m a t i o n was pas sed to t he c o m p u t e r v i a Analog to Digital Converters (ADC) w h i c h mea su r ed the e l ec t r i c a l charge of the s i gna l pulses f r o m each event a n d Time to Digital Converters (TDC), w h i c h mea su red t i m e in te rva l s be tween pulses. T h e s e u n i t s were s i t u a t e d i n a CAMAC c ra te , w h i c h was i n te r f aced to the PDP-11 t h r o u g h a C A E Starburst J - l l preprocessor . III.2 D e s c r i p t i o n of E x p e r i m e n t a l Components III.2.1 Deuterium Target T h e ta rget u sed i n t h i s e xpe r imen t cons i s ted of 1 m m d i a m e t e r beads of deu te r a t ed 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 1 9 m o l e c u l e s / m L o f deu te r a t ed EHBA - Crv c omp lex . T h e beads were h e l d i n a te f l on ( F E P ) con ta i ne r whose d imens ions were 18 x 22 x 6 mm? (2.4 c m 3 ) d u r i n g the J a n u a r y 27 1987 r u n a n d 20 x 20 x 10 m m 3 (4.0 c m 3 ) fo r t he M a y / J u n e 1987 r u n [25]. T h e ta rget was s u r r o u n d e d by l i q u i d 3He a n d 4He w h i c h was coo led b y a d i l u t i o n r e f r i ge ra to r t o a few 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 co 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 to def ine the ax i s of p o l a r i z a t i o n fo r E 3 3 1 , the sp in - t rans fe r e xpe r imen t . T h e f i e l d was m a i n t a i n e d d u r i n g the coarse o f th i s e x p e r i m e n t i n o rde r t o s imu la te , as c lose ly as poss ib le , t he cond i t i o n s of E 3 3 1 . 111.2.2 The Carbon Analyzer T h e c a r b o n ana l y ze r cons i s ted of layered g r aph i t e of d imens i on s 7 x 30 x 30 c m 3 a n d dens i t y 1.7 g/cm3. T h e layers of v a r y i n g th icknesses p r o v i d e the f r e e d o m to change the o ve r a l l ana l y ze r d e p t h , a l t h o u g h th i s feature was not e m p l o y e d i n ou r e xpe r imen t . T h e ana l y ze r was l a t e r a l l y p o s i t i o n e d o n the a r m A ca r t , such t ha t i t s center was co l l i nea r w i t h the centers of the w i r e chamber s o n b o t h sides. A l o n g t he c e n t r a l ax i s of the po l a r ime te r , the c a r b o n was l o c a t e d equ i d i s t an t be tween t he t h i r d a n d f o u r t h chambers . 111.2.3 The Scintillators T h e c o m m o n p l a s t i c s c i n t i l l a t o r s u sed i n the expe r imen t , p r o v i d e d fast log ic s ignals fo r the mas te r t r i gger , as we l l as energy depo s i t i o n (dE/dx) a n d t i m e of flight (TOF) i n f o r m a t i o n needed for p a r t i c l e i d en t i f i c a t i on . T h e la rge s c i n t i l l a t o r s 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 t he i r u p p e r a n d lower ends. T h i s inc reased the i r l i ght co l l e c t i on eff ic iency, a n d r educed po s i t i on -dependen t t i m i n g er rors b y ave rag ing b o t h t op a n d b o t t o m t ime s w i t h a m e a n t i m e r . T h e s c i n t i l l a t o r s of a r m B were m u c h sma l l e r a n d hence o n l y r equ i r ed s ing le ended l i ght co l l ec t i on . T h e d imens i on s of the s c i n t i l l a t o r s are l i s t ed 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 imens i on s D e t e c t o r A r m P o s i t i o n Width x Height x thickness cm3 A f o r w a r d 18 x 34 x 0.4 b a c k w a r d 36 x 70 x 0.7 B f o r w a r d 18 x 35 x 0.4 b a c k w a r d 18 x 35 x 0.4 P a i r s of s c i n t i l l a t o r s were p l a c e d at each de tec t i on po i n t . B y r e d u c i n g t he i r i n d i v i d u a l s izes, one c o u l d increase detecto r eff ic iency. A r e d u n d a n t s c i n t i l l a t o r (S5A) was p o s i t i o n e d cen t r a l l y after the last s c i n t i l l a t o r p a i r of a r m A . It p r o v i d e d a u se fu l check o f the re la t i ve eff ic iencies of the p o l a r i m e t e r a r m s c i n t i l l a t o r pa i r s , s ince a di f ference i n eff ic iency at th i s p o i n t c o u l d i n t r o d u c e a r t i f i c i a l s c a t t e r i n g a symmet r i e s . III .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 Multi-Wire Drift Chambers ( M W D C ' s ) were used to p r o v i de p o s i t i o n a l i n f o r m a t i o n fo r charged pa r t i c l e s pas s ing t h r o u g h the appa ra tu s . The se detectors use i n f o r m a t i o n f r o m b o t h the anodes a n d cathodes t o d e t e r m i n e the coo rd ina te s of a t r ack . D i s c r e te nearest w i r e po s i t i on s are o b t a i n e d d i r e c t l y f r o m the anodes w h i c h are s i t u a t e d 8 m m apar t . In a d d i t i o n , the e lectrons p r o d u c e d by the i o n i z i n g pa r t i c l e s have a h i g h l y p o s i t i o n sens i t ive d r i f t t ime . T h e c o m b i n a t i o n of the two pieces o f i n f o r m a t i o n p rov i de a p o s i t i o n a c cu r a c y of the o rde r of 500 m i c r o n s [26]. F u r t h e r deta i l s of t he i r o p e r a t i n g cha rac te r i s t i c s w i l l be con ta i ned i n the thes is o f P a v a n [26]. E l e v e n chamber s were c on s t r u c t ed b y the T R I U M F detec to r f a c i l i t y for the E331 g roup . D u r i n g the e xpe r imen t s i x s m a l l chamber s of a c t i v e a rea 30 x 30 c m 2 , a n d th ree la rge chamber s of ac t i ve a rea 60 x 60 c m 2 were e m p l o y e d . A s ing le spare c h a m b e r of each s ize was ava i l ab le i n case of b r e a k d o w n . E a c h c h a m b e r cons i s ted 29 Cathode Bus X EVEN RIGHT Anode Delay Line F i g u r e 12: T h e l a you t o f a s ingle p l ane of a M W D C . T h e ca thodes are r e ad out o n one s ide t h r o u g h the ODD/EVEN bus , a n d the anodes f r o m the RIGHT a n d LEFT sides o f the anode de lay l ine. o f two i ndependen t p lanes of w i res , o r i en ted o r thogona l l y , t o 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 of a t y p i c a l p l a ne is p r o v i d e d i n f igure 12. A p l ane c on t a i ned a l t e r n a t i n g anode a n d c a thode wi res , whose cent ro id s were spaced 4 mm apa r t . T h e anode w i re s were a l l c onnec ted at a c o m m o n e n d to a de lay l i ne , w h i c h was r e ad out at b o t h ends. T h e c a t h o d e w i re s were a l t e r na t e l y a t t a c h e d t o ODD a n d EVEN busses. A s i gna l i n d u c e d o n a n anode w i r e 2 w i l l sp l i t a n d t r a ve l t owa rd s b o t h ends of t he de lay l i ne . T h e t imes of a r r i v a l of these s ignals at the r i gh t a n d left TDC channe l s 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 + nda + tkR 2The reader is referred to Sauli [27] for more information on the detection mechanism of gas chambers. 3The START being provided from the master trigger. 30 tL = td + (N -n)da + tkL (17) W h e r e td is the c o m m o n d r i f t t i m e i n the chambers , n is the anode w i r e numbe r , N is the t o t a l n u m b e r of w i res , da is the a m o u n t of de lay per w i r e p r o v i d e d by the anode de lay l i ne , a n d tkR(tkl) is the respect i ve de lay cons tant a s soc ia ted w i t h each l i ne a n d a s soc ia ted e lect ron ic s . T h e w i r e p o s i t i o n c an be o b t a i n e d b y c a l c u l a t i n g the d i f ference be tween i # a n d t R - t L = U + nda + tkR - t d - ( N - n)da - tkh = 2dan + (tkR+tkL) + Nda (18) s/ constant T h i s q u a n t i t y is a l i n ea r f u n c t i o n of the w i r e n u m b e r n. T h e d r i f t t i m e is c a l c u l a t ed f r o m the s u m of a n d ti-tfi + tL = td + nda + tkR + td + (N - n)da + tkL = 2td + (tkR+tkL) + Nda (19) N v ' constant=kanoC[e T h i s q u a n t i t y ( tw i ce the d r i f t t i m e p lu s a cons tan t ) is i ndependen t of the w i r e numbe 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 a s soc ia ted w i t h the d r i f t t i m e i n f o r m a t i o n , as i t is not k n o w n on w h i c h s ide of the anode w i r e the event h a d passed. T o resolve th i s amb igu i t y , the a m p l i t u d e s of the ca thode ODD a n d EVEN ana l og s ignals are measu red . It has been s hown [28] t h a t the s i gna l i n d u c e d o n the c a thode w h i c h is closest to the event w i l l be 10 — 15 % l a r ge r t h a n t ha t i n d u c e d on the nex t nearest c a t hode (on the o the r s ide of the anode) . Hence the cor rect s ide is d e t e r m i n e d by c a l c u l a t i n g the d i f ference i n a m p l i t u d e of the ODD a n d EVEN s igna l s , (ODD - g • EVEN) (ODD + g • EVEN) K ' 31 W h e r e g is t he r e l a t i v e g a i n m a t c h i n g cons tant fo r the ODD a n d E V E N channels . T h e d i f ference is n o r m a l i z e d by the s u m to remove energy dependence. F i n a l l y , t he checksum is a p a r amete r w h i c h c an be u sed to i d e n t i f y events 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 or m u l t i p l e h i t s . E s sent i a l l y , one is c o m p a r i n g the d r i f t t i m e measu rement s o b t a i n e d f r o m the cathodes a n d anodes. T h e c a thode d r i f t t i m e is mea su red by the t i m e of a r r i v a l o f the ODD s i gna 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 the ODD t i m e are, whe re td is a ga i n the d r i f t t ime , dc is the de lay pe r u n i t d i s t ance of the ca thode bus , x is t he p o s i t i o n a l ong the anode bus, a n d tkQ is t he de lay a s soc ia ted w i t h the ODD s i gna l cables a n d as soc ia ted e lect ron ics . T h e xdc c o m b i n a t i o n is a non -neg l i g i b l e de lay - l i ne effect i n t r o d u c e d by the ca thode bu s c o n s t r u c t i o n . U s i n g t he k n o w n p o s i t i o n i n f o r m a t i o n f r o m e q u a t i o n 18, one c an co r rec t fo r th i s effect. T h e va lue r e m a i n i n g is s i m p l y the d r i f t t i m e p lu s a cons tant . A " g o o d " c h e c k s u m w i l l be cha rac te r i z ed by a cons tant va lue. E r r oneou s i n f o r m a t i o n i n c l u d i n g m u l t i p l e h i t s w i l l l e a d t o i n c o m p a t i b l e d r i f t t i m e measu rement s , such t ha t e q u a t i o n 22 w i l l dev i a te f r o m th i s cons tant . to = tj. + xdc + tko (21) checksum — £R + ti — C(to — (tp, — ti)dc) — 2 t j - j - kanode Ctd C'kcath,ode (22) where C is the re l a t i ve c a l i b r a t i o n of the ca thode d r i f t scale to the anode scale, a n d kcathode is t he cons tant t i m e de lays a s soc ia ted w i t h the ca thode channe l . 32 111.2.5 The E331 Event Trigger T h e m a s t e r event t r i gge r ' s p r i m a r y pu rpo se is t o i den t i f y those events f o r w h i c h a p a r t i c l e has passed t h r o u g h b o t h detecto r a rms , i n co inc idence . T h i s p rov ide s a n effect ive f i r s t -o rde r b a c k g r o u n d remova l . It thus defines t he c o m m o n START fo 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 ompa red . I n o rde r t o def ine a n EVENT, co inc idences mu s t have been obse rved by one of t he s c i n t i l l a t o r s at a l l f ou r de tec t i on po i n t s o n the detec to r car t s . START = (SIA + S2A) • (S3A + SAA + S5A) • (SIB + S2B) • (S3B + SAB) v v < v ^ . A B (23) whe re t he a r m A t i m i n g s igna l s are genera ted b y m e a n t i m e r s c onnec ted t o the t o p a n d b o t t o m p h o t o m u l t i p l i e r s of the re spect i ve s c i n t i l l a t o r s . T h e o ve r a l l t i m i n g was def ined by the (SIA + 5 2 A ) s i g na l s ince the p ro t on s t r a ve r s i ng a r m A were of h i ghe r energy a n d therefore were less a f fected b y the t i m e v a r i a t i o n s due t o energy loss. T h e re la t i ve t i m i n g o f the pulses a n d the log ic de f i n i t i on s are g i ven i n figure 13. 111.2.6 The J - l l Preprocessor A ve r y i m p o r t a n t componen t of ou r d a t a a c q u i s i t i o n s y s t em was the C r e a t i v e E l e c t r o n i c s S y s tems ( C E S ) Starburst Fas t P rocessor . T h i s dev ice was a s m a l l c o m p u t e r w h i c h sat as a m o d u l e i n the C A M A C crate. T h e J - l l C P U of the s t a rbu r s t h a d a n a r ch i t e c tu re very s i m i l a r t o tha t of a D i g i t a l P D P - 1 1 . It was capab le of u s ing t he C A M A C bus as an ex ten s i on of i t s m e m o r y a n d the reby h a d easy access t o d a t a channe l s of the o the r modu l e s i n the crate. T h i s f ea tu re a l l owed i t t o p e r f o r m r a p i d ca l cu l a t i on s o n the d a t a , a n d make fast dec i s ions fo r each event. S ince the p — C a n a l y z i n g power, r e su l t i n g f r o m the nuc lea r i n t e r a c t i o n , 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 lc — 420 mm, which is the distance between the outer chambers of each set. In this expression, the small-angle approximation, 6 « tan 8 was used. 6X — (wc6x — wcAx — {wclx — wc3x))/lc 8y = (wc6y — wcAy — [wcly — wc3y))/lc (24) 34 J—11 Filter OFF QJN o-l. -200-::BS: 200 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 was t h e n r equ i r ed t h a t the x o r y s lope changed b y at least 6° i n o rde r to accept the event. F o r such a " g o o d event , " t he J - l l w o u l d i n d i c a t e t o the P D P - 1 1 t ha t the event was t o be read . T h e effect of the J - l l f i l t e r o n the a ccep ted d a t a is d e m o n s t r a t e d i n f i gu re 14. I I I . 2 . 7 O n l i n e D a t a A c q u i s i t i o n T h e STAR system of G . S m i t h [29] was chosen as the d a t a a c q u i s i t i o n r ou t i ne . It is a P D P - 1 1 sof tware package w h i c h does severa l tasks. The se i n c l u d e r ead i n g the modu l e s i n the C A M A C crate, p e r f o r m i n g ca l cu l a t i on s , p r o d u c i n g se lected h i s t og r ams a n d w r i t i n g the r a w d a t a to m a g n e t i c tape. T h i s s y s t e m benef i t s f r o m i t s s i m p l i c i t y a n d flexibility. It is eas i l y a d a p t a b l e to va r ious e x p e r i m e n t a l con f i gu ra t i on s . A m i n i m u m of " o n l i n e " c a l cu l a t i on s were p e r f o r m e d o n the d a t a , d u r i n g the a c q u i s i t i o n , as i t was i m p o r t a n t to reduce c o m p u t e r dead t ime . T h e p h i l o s o p h y a d o p t e d was to ensure t h a t the bas i c component s of the a p p a r a t u s were f u n c t i o n i n g p rope r l y , a n d save h i ghe r l eve l c a l cu l a t i on s fo r the of f l ine ana ly s i s . T h e essent ia l c o m p u t a t i o n s p e r f o r m e d on l i ne a l l owed one t o m o n i t o r the progress of the r u n a n d i den t i f y s y s t emat i c effects a n d fa i l i ng s as t h e y o c cu r r ed . I m p o r t a n t c a l cu l a t i on s i n c l u d e d ; • w i r e p o s i t i o n (combs) , a n d checksums, to i n d i c a t e the p e r f o r m a n c e q u a l i t y of the chambers . • t he J - l l x a n d y s c a t t e r i n g angles, t o ensure t ha t the cuts i n the J - l l h a d no t d r i f t e d s i gn i f i can t l y f r o m the center (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 the p i o n b e a m h a d no t s h i f t ed 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 bas i c g eomet r y u sed to desc r ibe the react ions i n the off l ine ana l y s i s f o l l ows the conven t i on of severa l au tho r s [30,31,32,21]. It is I l l u s t r a t e d i n f igure 15. T w o angles are u sed t o desc r ibe the re l a t i on sh i p be tween the s ca t te red a n d i nc i den t p r o t o n vectors . The se angles are def ined w i t h respect t o a reference s y s t em, i n w h i c h the i n c i den t p a r t i c l e m o m e n t u m defines t he z -ax i s , a n d the o r t h o g o n a l y -ax i s is fixed. S i nce th i s e xpe r imen t as w e l l as E 331 were d u a l s c a t t e r i n g e xpe r imen t s , i t was necessary t o def ine two such geometr ies . In the i n i t i a l nd —> pp r e a c t i on , t he y -ax i s (y ) was chosen t o be v e r t i c a l i n the l a b f r ame, w h i c h was a cen t r a l va lue f o r the n o r m a l t o the s ca t t e r i n g p lane. T h i s ax i s was we l l de f i ned a n d o r t h o g o n a l t o the i n c i den t p i o n b e a m . F o r the p — C r e a c t i on , the y -ax i s was chosen to be n o r m a l to the s c a t t e r i n g p l ane of the first r e a c t i o n ( equa t i on 6) (nd), a n d was cons i s tent w i t h the 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 ang le QQ represents t he ang le be tween the z ax i s a n d the m o m e n t u m vec to r of the o u t g o i n g pa r t i c l e . T h e a z i m u t h a l ang le <f>c denotes t he angle be tween the p r e v i ou s l y de f ined y -ax i s a n d the n o r m a l to the s c a t t e r i n g p l ane (n). T o a v o i d u s i n g a bas is t h a t 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 de ta i l s of each event cons idered, the componen t s of a l l vec to r s i n the off l ine ana l y s i s were expres sed i n t e rms of a coo rd i na te s y s tem, fixed t o t he l abo ra to 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 1 , and the 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 This 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 of essent ia l e xpe r imen t angles. od c-(^ ) - es Oc <f>d <f>c c o s ( ^ c ) - ( i ^ x f t , ; , ) s i gn of <f>d s i gn of <f>c H d X (lfi..-xft,,|) •PP. the f u n d a m e n t a l va r i ab le s u t i l i z e d i n the software. IV.2 O u t l i n e of the Software Philosophy T h e so f tware i n h e r i t e d b y the E331 g r o u p was a n ensemble of s ub rou t i ne s w r i t t e n by severa l au tho r s . T h e u l t i m a t e goa l was t o p r o d u c e a coherent ana l y s i s package des i gned spec i f i ca l l y fo r the needs of E 3 3 1 , yet l eav i ng i t f l e x i b l e enough t ha t the r ou t i ne s m a y be eas i ly u t i l i z e d b y other e xpe r imen t s u t i l i z i n g the p o l a r i m e t e r . O t h e r r equ i rement s were: user f r iend ly , c o m p u t e r eff ic iency, a n d p r o v i s i o n of a w i d e range of d iagnos t i c s ava i l ab le fo r t r o u b l e shoot ing . I n o rde r t o s i m p l i f y the ana lys i s , t he process was d i v i d e d i n t o two separate p rog rams . T h e f i r s t (REPDISK) r e ad a l l t he events f r o m tape a n d , by a p p l y i n g a series o f c a l cu l a t i on s a n d cuts , r educed the d a t a to a subset of " g o o d " events, whe re a " g o o d " event is def ined i n sect ion V . l . 3 . The se events were a n a l y z e d b y the second p r o g r a m (POLAR), w h i c h c a l c u l a t e d the p r o t o n p o l a r i z a t i o n f r o m the s c a t t e r i n g ang le i n f o r m a t i o n , repdisk is de sc r i bed i n the r e m a i n d e r o f th i s sec t ion a n d the nex t , a n d POLAR is de sc r i bed i n sect ion IV.4. REPDISK was a ve r y genera l r o u t i n e w h i c h c o u l d be 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 ua t i on s . It served to i n i t i a l i z e r u n pa ramete r s , def ine h i s tog rams , test files, t ake care of book - keep i ng a n d o u t p u t var ious o u t p u t d i agnos t i c f i les (see sec t i on IV.5) . It o r che s t r a ted the r ead i n g o f the event buffers f r o m tape or d i sk 40 a n d t r e a t e d each a c co rd i n g t o buf fer t y p e 2 . D i f fe rent sub rou t i ne s were ca l l ed fo r sca ler, header , t r ue event, e n d o f f i le etc., buffers. F o r each r e a l event buf fer , REPDISK w o u l d c a l l t he s ub r ou t i ne PION, w h i c h o r g a n i z e d a l l t he ca l cu l a t i on s p e r f o r m e d o n the da t a . T h e f i n a l se lected d a t a were t h e n b i n n e d a c c o r d i n g to the h i s tog rams d e n n e d d u r i n g the i n i t i a l i z a t i o n . T h e flow of REPDISK is o u t l i n e d i n f i gure 16. T h e log ic a n d flow o f PION is i l l u s t r a t e d i n f i gure 17. F o r those w i s h i n g t o use the p o l a r i m e t e r i n o the r e x p e r i m e n t a l con f i gu ra t i on s , t he p o l a r i m e t e r rou t i ne s are m o s t l y sel f -suff ic ient a n d i ndependen t of o t he r a p p a r a t u s 3 . Neces sa ry changes t o the sof tware f o r a di f ferent 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 genera l l y be m a d e i n the s ub r ou t i ne PION, s ince each e x p e r i m e n t w i l l i n genera l r equ i re some different ana ly s i s r ou t ine s . D u e to the 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 necessary t o c a l l o r r emove di f ferent c a l c u l a t i o n sub rou t i ne s where r equ i r ed . I V . 3 C a l i b r a t i o n o f t h e R E P D I S K S o f t w a r e IV.3.1 ONLINE wire chamber calibration In th i s sec t i on , a s u m m a r y of the on l i ne w i r e c h a m b e r c a l i b r a t i o n p rocedure s are presented. F o r exped iency , o n l y coarse w i r e r e s o l u t i o n ( ± 4 mm) was r e q u i r e d by the on l i ne rou t ine s . T h e c a l i b r a t i o n used fo r the chamber s d u r i n g of f l ine ana lys i s , a l t h o u g h s i m i l a r was m u c h mo re precise, o f fer ing a r e s o l u t i o n of a n o rde r of m a g n i t u d e be t t e r t h a n the on l ine s y s tem. T h e off l ine techn ique w i l l be d i scussed i n t he thes i s of P a v a n [26]. T h e r e are severa l steps i n vo l v ed i n the c a l i b r a t i o n sof tware. T h e o rde r i n 2 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. 3These routines are available from the author upon request. 41 In i t i a l i ze Repdisk Read in ONLIN.SET Drift table Define histos/dot plots Read Scalers 7Tv Read Event Buffer (from disk) Scaler Buffery 3= 67 Normal End of Run \ / Save scaler info \ / Save test resul t s \ / Save h is to info \ / Write good event pa ramete r s to AMAT###.DAT Parity Error Pass info to POLAR ->(END Pro ton? Scint. Cuts \ , Yes / c a l l PION \ / B i n Histos \ / B i n Dot Plots No Figure 16: The REPDISK flow chart 42 c a l l CBLK2 : calculates combs, checksums and ODD-EVENs \ / c a l l TSTBLK(2): applies cuts to checksums \ / c a l l WCPOS: calculates drift position and adds to wire position \ / c a l l CODE_CHAM: suff. good WC combinations \ / Yes c a l l TRAJECTORY: fits slopes and intercepts from WC positions \ / c a l l XYZTRBK trace back to deuteron tgt. \ / c a l l TRAJ FIT: refits trajectories to include target as origin \ / c a l l COPL: kinematics angles of calculates and scat. 1st react. No No t-H Q CU W cc; o d u 0 +-> CU u c CD > T l CQ No No (^RETIJRNJ^' c a l l BOXSTUFE-appties 2-d cuts to data c a l l TSTBLK(3):checfcs if event came from target  Fes c a l l AC_TST: checks scattering symetry in detector c a l l CARBONTRBK: traces back to origin of scat, in carbon c a l l TSTBLK(4):cheefcs if good traceback to carbon, Yes c a l l C_ANG calculates scattering angles in analyzer c a l l TSTBLK(5): checks if event scat'd more than req'd minimum Yes c a l l AMAT : writes good evnt param. to disk  Figure 17: A flow chart of the subroutine PION. 43 which they are performed is important , since many of the calibrations depend on earlier results. Initially, one must calibrate the indiv idual chamber planes, and 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 tem for cal ibration 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 cal ibrat ion process. T h e constant C from equation 22 relates the T D C scales of the ODD cathode to the anode drift t ime measurements. C was obtained by p lot t ing 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, dc, (see equation 22) was included. T h i s constant was obtained by plott ing the checksum vs uncalibrated difftime (t^ — tn). T h i s procedure is demonstrated i n figure 18 b). To obtain 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 known, so the relation between the data bins and the distance is given as 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 obtain a relative posit ion cal ibration of the chambers. A straight line from the target origin, through the middle of chamber three also passes through 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 around the central two wires of chamber three 4 . T h e required offset for each chamber was then obtained from the 4Chamber three was chosen because it was the only chamber which was fully illuminated. scale factor = (data bin of right wire — data bin of left wire) (n — 1) • 8mm 44 -1500 -1000 -500 0 500 1000 1500 Chamber Position Figure 18: Calculation of relative TDC and the position dependent calibration con-stants for the checksum. 45 1000 u i i i r r i i -150 -100 - 5 0 0 50 100 150 Difftime (mm) (wire 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 arm B before its components were transformed into the arm A frame. T h e second parameter involved the correction necessary for the nd —> pp opening angle dependence on 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 8a + 6b ys 6a, as in figure 20. A s a result, the opening angle was redefined as 6a + 80 = 8a + 80 + 6a • corr . Such a correction 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 and the normal to the scattering plane of the first reaction (defined by p^ x pj). It was found that its width did not significantly depend on any kinematic quantities and hence did not require any calibrations. IV.4 Polarization Software T h e program 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, 6C, <j)c • the kinematic parameters, 6a + 6b and coplanarity In POLAR the carbon scattering angles were binned in multi-dimensional arrays with respect to the variables contained in A M A T # # # . D A T , with the intention of enabling investigation of the dependence of the polarization on these 47 200 216 232 248 264 280 296 312 328 344 360 6 (tenths of degrees) Figure 20: Dependance of the opening angle (8a + 8^) on 8a. This slope is obtained simply by hand. parameters. For example, for each bin of the <f>d distribution, there corresponds a 2-dimensional 8C, <f>c 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 and last run numbers —minimum scattering angle in carbon —2—D position of peak in 0„+0b vs Copl. dist. — r a d i u s of foreground/background separation i Zero all arrays > V Read in Event get the next event 1 Distinguish the foreground from the background V Bin arrays according to; f(run no.,SC,0C,status) f(0 d,e c,0 c,status) f(0 d,0 e,0 c,status) etc. Status is flag for background \U no more events to read 8C and 0O are Calculate polarization as a function of the parameters binned above call SUMS(k,eo,0o,status) required for all polarizatior. calculations k is a fixed parameter which represents the p a r t i c u l a r bin for which the polarization is being evaluated Figure 21: Flow chart of the software routine P O L A R . 49 Table III: A list of output files used for diagnostics header files RUN### .REP logs a series of run queues from the offline analysis, errors due to opening a file etc. are written in here. test files T S T # # # . R E P 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. scalar files SCL### .REP contains a list of all the final hardware scalar totals for each run. histogram files R ^ ^ ^ . H S T contains the histogram information used by the STAR system histogram routine. The following histograms are stored for each run: 6a + 9b, coplanarity, Be, carbon traceback, carbon RDS distribution, reso-lution 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. scatter plots R U N # # # . D O T contains the dot plot information used by DOT. The following dot plots are stored for each run: kinematics and target trace-back scatter plots. scalar files DIA###.DAT contains the incremental scalar values as a function of time. The following scalar information is stored for each run: count-ing rates of the beam, monitor tele-scope, and muon counters, the up/down and left/right hodoscopes and the clock counter. The scalers were read every 5 minutes. 50 Chapter V Experimental Results V . l Performance of Apparatus V . l . l The Scintillators T h e s c i n t i l l a t o r s ' p e r f o rmance was eva l ua ted by how we l l t he p r o t o n s of the t r ue ird —> pp events c o u l d be sepa ra ted f r o m o the r react ions . A s w e l l as the qua s i - ird —• pp r eac t ions , s i gn i f i cant b a c k g r o u n d was c o n t r i b u t e d b y quas i - free TTN s c a t t e r i n g whe re the r e s u l t i n g p ro ton s h a d a w i d e range of genera l l y lower energies. F i g u r e 22 a) shows d i s t r i b u t i o n s of energy depo s i t ed i n t y p i c a l s c i n t i l l a t o r s of a r m s A a n d B. T h e la rger peak is due to p ro ton s . T o i t s left is t he re s i dua l 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 th i s peak has been r e m o v e d by the ha rdwa re 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 to the r i gh t of the p r o t o n peaks are i n genera l the quas i - f ree TTN p ro ton s . T h e ^ d i s t r i b u t i o n s i n the s c i n t i l l a to r s p r o v i d e d a f i r s t l eve l b a c k g r o u n d sepa ra t i on . T h e mono-energe t i c p ro ton s p r o d u c e d a w e l l de f ined peak w h i c h p e r m i t t e d a s imp le , yet ef fect ive one d i m e n s i o n a l cut . T h e TOF d i s t r i b u t i o n s were not as u se fu l for b a c k g r o u n d r e m o v a l as the ^ i n f o r m a t i o n . T h i s was a resu l t of the short flight p a t h s be tween the target a n d sc i n t i l l a t o r s . However , s ince the b a c k g r o u n d s epa ra t i on was not c r i t i c a l l y dependent o n the TOF i n f o r m a t i o n , these resu l t s were qu i t e acceptab le . O n l y loose cut s were a p p l i e d to th i s d i s t r i b u t i o n , a n e x a m p l e of w h i c h is s hown i n figure 22 b ) , because of the p o o r re so lu t i on . TOF i n f o r m a t i o n was o n l y ava i l ab le 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 sed to def ine the S T A R T . V.1.2 The Wi re Chambers I n the ana ly s i s , the w i r e chamber s were s t r i c t l y u sed for vec to r c o n s t r u c t i o n . Hence i t was i m p o r t a n t to k n o w how we l l the s lope a n d i n te r cep t o f a t r a j e c t o r y c o u l d be de te rm ined . T h e se l f -cons i s tency of the s lope 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, the r e s o l u t i on c a l c u l a t i o n . T h i s expres s ion e n a b l e d c o m p a r i s o n of the e x p e c t e d p o s i t i o n o n a c h a m b e r ( c a l cu l a ted o n the bas i s o f t he p o s i t i o n i n the ad jacent chamber s ) w i t h the a c t u a l p o s i t i o n i n t h a t chamber . T h e r e s o l u t i o n p l o t , s h o w n i n f i gu re 23 was p r o d u c e d u s i n g events fo r w h i c h a l l chamber s h a d success fu l check sums a n d w i t h t he c e n t r a l c h a m b e r h a v i n g the e x t r a c o n d i t i o n of g o o d O D D / E V E N i n f o r m a t i o n WC3X/y - WClx/y resolution = — — — wc2x/y (26) T h i s d i s t r i b u t i o n was a sens i t ive i n d i c a t o r o f m i s c a l i b r a t i o n s of the software. If the re l a t i ve offset o r ga i n of one of the chamber s h a d changed, a sh i f t or b r o a d e n i n g i n the r e so l u t i on peak w o u l d re su l t , F o r a w e l l c a l i b r a t e d s y s t em, the w i d t h s of such r e s o l u t i on peak were t y p i c a l l y 700 microns F W H M , w i t h a m e a n va lue o f less t h a n 300 microns. T h e t racebacks to the s ca t t e r i n g o r i g i n at the c a r b o n ( d e u t e r i u m ) were a re f l ec t i on of the cons i s tency between the sets ( a rms) of chamber s . S ince, i n genera l , two a r b i t r a r y vector s do not intersect , the p o s i t i o n of m i n i m u m d i s t ance be tween t h e m def ined the " o r i g i n " of the t raceback. T h e m a g n i t u d e of th i s d i s t ance i n d i c a t e d how we l l the vector s agreed w i t h each o ther . T h i s d i s t ance 1For 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). 53 2500 Front of arm A, y-planes 2000-1500 c D O o 1000 500--60 -40 -20 0 20 40 60 Resolution Plot (tenths of millimeters) F i g u r e 23: A t y p i c a l r e s o l u t i on p l o t . d e n o t e d by Root Difference Squared (RDS) was c a l c u l a t e d i n the s t a n d a r d t r aceback r o u t i n e w h i c h is de sc r i bed i n a p p e n d i x A . O f the two target t r aceback s , t he deute ron ' s R D S was s i gn i f i c an t l y b roade r , s ince i t was e x t r a p o l a t e d over a m u c h g reater d i s tance , a n d was d i s t o r t e d by the m a g n e t i c f i e l d s u r r o u n d i n g the target . A cut o n th i s d i s t ance o f 50 mm fo r the d e u t e r o n a n d 10 m m for the c a r b o n was f o u n d to be a p p r o p r i a t e va lues at w h i c h t o reject p o o r d a t a , as seen by f igures 24 a n d 25. B o t h target t racebacks p r o d u c e d good d i s t r i b u t i o n s (see figures 24 a n d 25) abou t the reg ions where the targets were expec ted . Hence a cut o n the d a t a whose t r aceback i n d i c a t e d t ha t the events h a d o c c u r r e d ou t s i de the ta rget r eg i on 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 aceback 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 s ca t te red by a n e x t e r n a l target. T h e large ang le be tween a r m s A a n d B (about 165°), r e su l t ed i n the p o o r r e s o l u t i on of the t a r ge t ' s z - component . T h i s w i d t h was g rea t l y i m p r o v e d by con s t r a i n i n g the o r i g i n to l i e close t o the ax i s of 54 t he p i o n b e a m . In s u m m a r y , the tests a p p l i e d to the t r a jec to r i e s en su red t h a t the t racebacks were k n o w n to a r e s o l u t i on o f abou t 2 cm. W i t h these cuts abou t | o f the d a t a was r e m o v e d at the deu te ron t r aceback a n d abou t h a l f o f t he d a t a te s ted i n the c a r b o n t r aceback was re jected. V . 1 . 3 S o f t w a r e D e f i n i t i o n o f a " G o o d " E v e n t A " g o o d " event h a d t o sat i s fy the f o l l o w i n g c r i t e r i a i n o rde r t o be r e t a i n e d for p o l a r i z a t i o n ana l y s i s i n POLAR. 1. p r o v i d e co inc idence event i n b o t h a rms , i n d i c a t i n g a 2 -body final state. 2. l i e w i t h i n the ^ a n d TOF cu t s i m p o s e d by the s c i n t i l l a t o r s , t hu s b e i n g i den t i f i ed as p r o t on s i n each a r m . 3. have at least 2 of 3 w i r e chamber s p r o v i d e g o o d events fo r b o t h x a n d y p lanes i n each of the 3 sets. 4. t r a j ec to r i e s mu s t enab le g o o d q u a l i t y t r acebacks t o the s c a t t e r i n g o r i g i n at the deu te r a ted target a n d c a r b o n ana l yze r . 5. mu s t pass the acceptance test de sc r i bed i n sec t i on II.4.1. 6. mu s t s ca t te r i n the c a r b o n by 9 ° 2 , o r more . A b o u t 5 % o f a l l events w r i t t e n to t a p e qua l i f i ed as " g o o d " events. V . 1 . 4 K i n e m a t i c s C a l c u l a t i o n s T h e quas i - f ree nd —» pp b a c k g r o u n d was the mos t d i f f i cu l t t o d i s t i n gu i s h f r o m the f o reg round s ince the final s tate pa r t i c l e s were the same. However , figure 26 shows t h a t the t r u e nd —> pp s i gna l was we l l s epa ra ted f r o m the m u c h flatter b a c k g r o u n d . 2 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). 55 n 1 r -200-160-120-80 -40 0 40 80 120 160 200 Carbon Traceback; x-component (mm) i 1 r 200-160-120-80 -40 0 40 80 120 160 200 Carbon Traceback; y-component (mm) -200-160-120-80 -40 0 40 80 120 160 200 Carbon Traceback; z-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 4000 ,3000 J2000H 1000 8000 6000 54000 2000 -100-80 -60 -40 -20 0 20 40 60 80 100 Target Traceback; x-component (mm) -100-80 -60 - 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 1600-1400 -1200 -1425 1475 1525 1575 1625 1675 1725 1775 1825 c?o + 8b (tenths of degrees) 1000 Coplanarity (tenths of degrees) Figure 26: The 1-dimensional profiles of the opening angle and coplanarity distri-butions. 58 S ince t he o p e n i n g ang le of the two f i n a l s tate p r o t on s was 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 by the 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 cop l ana r i t y , t o f i r s t o rder , was i ndependent of energy, s ince a l l ird —> pp reac t i on s m u s t be cop l ana r , regardless o f the ava i l ab le energy. It was t he u n c e r t a i n t y of t r a jec to r i e s w h i c h b roadened th i s f u n c t i o n . T h e p i o n b e a m , whose d ivergence was not m o n i t o r e d , c o n t r i b u t e d mos t s i gn i f i c an t l y 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 . Its d ivergence i n the y - d i r e c t i o n at t he ta rget [22] is abou t 5° as c o m p a r e d w i t h o n l y 1° i n the x - d i r e c t i o n . S e p a r a t i o n o f the b a c k g r o u n d f r o m the f o reg round is o b t a i n e d b y p l a c i n g a two d i m e n s i o n a l cu t a r o u n d the k i n e m a t i c s peak, as seen i n f i gu re 27. 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 the m a i n peak is abou t 10 % 3 . S ince the r e m a i n i n g b a c k g r o u n d has sat i s f ied the p r o t o n ^ a n d TOF cu t s o f the s c i n t i l l a t o r s of b o t h a rms , a n d are b r o a d l y accepted b y the d u a l a r m co inc idence , these events are a t t r i b u t e d to the quas i - ltd —• pp r eac t i on . F o r the purposes of th i s thes i s , no f u r t h e r effort was m a d e to remove the 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 the rea l vd —> pp p o l a r i z a t i o n s i gna l was d i s rega rded at th i s p o i n t , b u t fo r E 3 3 1 , i t is a n t i c i p a t e d t ha t co r rec t i on s w i l l be a p p l i e d t o e l i m i n a t e the b a c k g r o u n d ' s i n f luence. V.2 Polarization Results T h e p o l a r i z a t i o n resu l t s p resented fo r th i s thes is were mea su red w i t h the deute rons 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 i n c i den t p i on s was 205 MeV a n d the p ro t on s were detected at a cen t r a l ang le o f 27° ( l ab ) 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 f o r e g r ound a n d b a c k g r o u n d are presented i n t a b l e IV . 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 by m e a s u r i n g the 3In 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. 59 T a b l e IV : P r o t o n p o l a r i z a t i o n s as mea su red b y the p o l a r i m e t e r . M e a s u r e d P r e d i c t e d P o l a r i z a t i o n s (%) P o l a r i z a t i o n s (%) ( f r o m A N 0) Fo reg round ; N o r m a l c o m p o n e n t 3 6 . 5 ± 2 4 39 S ideways c o m p o n e n t 1 0 . 8 ± 2 5 0 B a c k g r o u n d ; N o r m a l c o m p o n e n t 2 5 . 2 ± 3 6 -S ideways c o m p o n e n t n 0 ± 3 . 5 -p o l a r i z a t i o n of the events ou t s i de o f the cut s hown i n figure 27. T h e e r ro r s quo ted are p u r e l y s t a t i s t i c a l , a n d are c a l c u l a t e d u s i n g the f o rmu l ae o f sec t i on II.4. A s is seen i n t ab l e IV , the n o r m a l p o l a r i z a t i o n is cons i s tent w i t h t h a t p r e d i c t e d by the ANQ fits ( a l t h o u g h s l i gh t l y low) . M o r e s ign i f icant is the cons i s tent non - ze ro s ideways p o l a r i z a t i o n for b o t h the f o r eg r ound a n d b a c k g r o u n d events. A s was d i scussed i n sec t i on II.3, p r o t on 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), s h o u l d c o n t a i n o n l y a n o r m a l p o l a r i z a t i o n c omponen t . Searches fo r s y s t emat i c effects, p resented i n figures 28 a n d 29, ve r i f i ed t h a t t he p o l a r i m e t e r was p e r f o r m i n g as expec ted . T h e i ndependence o f p o l a r i z a t i o n as a f u n c t i o n of c a r b o n s ca t te r i ng ang le (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 ra te s the dependence of the p o l a r i z a t i o n o n 9d a n d (f>d. S ince ANO var ies r a p i d l y w i t h 9d, such a dependence of PN, is not u n e x p e c t e d . However , because o f the la rge e r ro r bar s w i t h th i s d a t a , no f u r t h e r ana ly s i s was a t t e m p t e d . T h e s y s t ema t i c effect w h i c h c o u l d e x p l a i n the s ign i f i cant s ideways p o l a r i z a t i o n observed by the po l a r ime te r , was the precess ion of the p r o t o n ' s s p i n due to the la rge 2.5 T m a g n e t i c field i n the reg ion of the target . W i t h some r o u g h ca l cu l a t i on s , i t was qu i c k l y s hown t ha t th i s effect was s i gn i f i cant . A c o m p u t e r m o d e l was deve loped to p r ed i c t the n a t u r e of the s p i n precess ion. 60 105 100-95-CL 0 u O c D 90-85-80-75 150 155 160 165 170 175 F i g u r e 27: 2 - d imen s i ona l d i s t r i b u t i o n of the k i n ema t i c s . T h e c i r c l e represents the de s i gna ted cut w h i c h d i s t i n gu i s hed the f o reg round f r o m the b a c k g r o u n d . Re su l t s f r o m th i s c o m p u t e r m o d e l , w h i c h is de s c r i bed i n a p p e n d i x C , showed t ha t there was a s i gn i f i cant dependance of the s p i n p recess ion o n the p a r t i c l e ' s t r a j ec to r y . 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 ve d i r e c t i o n o f the f i e l d e xpe r i enced b y the s p i n component s over di f ferent t r a jec to r i e s . 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 xpec ted to be a f u n c t i o n o f s c a t t e r i n g ang le 9a ( the f o r w a r d s c a t t e r i n g ang le of the p r o t on ) . The re f o re i t 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 the final p o l a r i z a t i o n for a l l the events i n the p o l a r i m e t e r ' s acceptance. T h e p roceedu re deve loped to account fo r th i s was the f o l l ow ing . T h e t o t a l p recess ion was c a l c u l a t e d for a set o f even l y spaced t r a jec to r i e s , w h i c h s p a n n e d the acceptance o f the detector . The se were we igh ted by the r e l a t i v e n u m b e r of events w h i c h f o l l owed each p a t h f r o m the e x p e r i m e n t a l d a t a set. T h e we i gh ted s u m t h e n represented the average p o l a r i z a t i o n mea su red b y t he p o l a r i m e t e r . 61 100 -75" 50-25 o--25" -50 -75 - P = 0.365 Foreground Normal Polarization P t I I I I 6 8 10 12 14 16 18 20 100 75 50" 25 44-0 -25--50 H P_= 0.108 -75 Foreground Sideways Polarization 10 12 14 16 18 20 e 100" 75" 50" 25" 0 -25--50 -75 -100 I I I I I I i i i t 0.024 P = 0.365 N Foreground Normal Polarization —i—i— i—i—i—i—i—T 295 296 297 298 299 300 301302 303304305 Run Number 0 -25 -50 -75 -100 [ i i i t i 1 1 1 ± 0.025 P = 0.108 s Foreground Sideways Polarization 295 296297 298 299300 301302303304305 Run Number Figure 28: Polarization dependance on 9c and run number. 62 100 75 50 25 0 -25 -50 -75 -100 I I I I I I I I I I I I I I I I I I 100 75" 'o 50" i i 25' f f i - o- 1 t 1 c ± 0.024 rizatic -25-± 0.025 P = 0.365 - o o CL -50- P = 0.108 s Foreground Normal Polarization - oton -75" Foreground Sideways Polarization CL I I I I 1 I I I I -100 I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 e 7 8 9 10 100 75" 50 25 o--25" -50 -75" - i o o i— r i i i i i i i i i P = 0.365 N Foreground Normal Polarization -i—i—i—i—i—i—r 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 100 75 50 25 0 -25 -50 -75 -100 _ i I I I L_ Foreground Sideways Polarization i i i i I i i r 5 -4 -3 -2 -1 0 1 2 3 4 5 ure 29: Polarization dependance on and <j>d. Note the units are arbitrary. 6 3 T a b l e V : A comparison of polarizations of those predicted b y the pro-gram FINDANG and those measured b y the polarimeter. M e a s u r e d P o l a r i z a t i o n s (%) P r e d i c t e d P o l a r i z a t i o n s (%) N o r m a l p o l a r i z a t i o n 3 6 . 5 ± 2 4 36.5 S ideways p o l a r i z a t i o n 1 0 . 8 ± 2 " 5 12.6 In o u r case where the 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 is qu i t e s t ra i ght f o r w a r d to ca l cu l a te the s p i n precess ion o f the p r o t o n t o the ana l y ze r . A l t h o u g h the p o l a r i m e t e r o n l y mea su red the two componen t s o r t h o g o n a l t o the p a r t i c l e ' s d i r e c t i o n of m o t i o n , one c o u l d u n a m b i g u o u s l y c o m p a r e t he p o l a r i z a t i o n mea su r ed e x p e r i m e n t a l l y w i t h t h a t p r e d i c t e d b y t he preces s ion 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 , t ab l e V compares the p r ed i c t ed p o l a r i z a t i o n c omponen t s w i t h those mea su r ed by the po l a r ime te r . It is seen tha t the s p i n precess ion success fu l l y account s fo r the p o l a r i z a t i o n seen b y the p o l a r i m e t e r to we l l w i t h i n e x p e r i m e n t a l e r ro r . 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 pion events from 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 and acts as a check of the software posit ion cal ibration. T h e Root Difference Squared (RDS) is a check of consistence amongst the sets of wire chambers. The 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 common origin. W i t h respect to background separation, it has been demonstrated that our system can eliminate a l l but about 10 % of the background which lies under the true event kinematics peak. P r o t o n spin 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 providing an in i t ia l proton polarizat ion equal to A^0 of the identical t ime 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 polarizat ion extraction are appropriate for our system, i n particular for Tp = 200 MeV (the typical proton energy obtained i n this experiment). In this regard, it would be useful to repeat 65 th i s e xpe r imen t for a l l energies at w h i c h the sp in - t rans fe r mea su remen t w i l l be p e r f o r m e d , i n o rde r to ve r i f y the usefulness of these a n a l y z i n g powers over the en t i re range. F i n a l l y , i t has now been e s tab l i s hed t ha t a v a l i d d a t a ana l y s i s package ex ists fo r use i n f u r t h e r p o l a r i m e t e r expe r iment s . 66 Bibliography [1] D. A s h e r y a n d J .P . Schiffer. Ann. Rev. Nucl. Part. Sci., 36:207, 1986. [2] R. M a c h e i d t . T h e meson t heo r y of nuc lea r forces a n d nuc lea r m a t t e r . 1985. Lec t u r e s p resented at the 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 Q u a r k - N u c l e a r Phy s i c s . [3] D.V. B u g g . Ann. Rev. Nucl. Part. Sci., 35:295-320, 1985. [4] M . B e t z et a l . Theo r i e s of p i o n p r o d u c t i o n i n nuc l eon -nuc l eon co l l i s ions . I n v i t e d pape r p resented at the W o r k s h o p 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 i n N u c l e i . [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, 1985. [6] D.V. B u g g . Nucl. Phys., A416:227, 1984. [7] D.V. B u g g . Nucl. Phys., A437:534, 1985. [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 P res s , M a d i s o n W i s c o n s i n , 1971. [10] G . Jones, p r i v a te 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 te 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., A254:263-269, 1987. [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. Add i s on -Wes l e y , R e a d i n g Mas sachuset t s , 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 Sons, 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, 1983. [16] M . W . M c N a u g h t o n et a l . Nucl. Inst. Meth., A241:435-440, 1985. [17] P. Webe r , p r i v a te c o m m u n i c a t i o n . [18] F . Sper i sen , W . G rueb l e r , a n d V . K o n i g . Nucl. Instr. Meth., 204:491-503, 1983. 67 [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 te c o m m u n i c a t i o n . [21] D. Besset , Q . H . D o , B. Fav i e r , L . G . G reen i au s , R. Hess, 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, 1979. [22] T r i u m f user ' s h andbook . Second E d i t i o n , u n p u b l i s h e d . [23] R. H e n d e r s o n et a l . IEEE Trans. Nucl. Sci, NS-34:528, 1987. [24] R. H e n d e r s o n et a l . IEEE Trans. Nucl. Sci., NS-35-.477, 1988. [25] G . D . W a i t a n d D.C. Healey. 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 te c o m m u n i c a t i o n . [27] F. S a u l i . Principles of Operation on Multiwire Proportional and Drift Chambers. 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 epo 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 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. u n p u b l i s h e d . [30] D. Besset, B. Fav i e r , L . G . G reen i au s , R. Hess, C . Lechano i ne , D. R a p i n , a n d D. W . W e r r e n . Nucl. Instr. Meth., 166:515, 1978. [31] R .D. R a n s o m et a l . Nucl. Instr. Meth., 201:309-313, 1982. [32] G . W a t e r s et a l . Nucl. Inst. Meth., 153:401-408, 1978. [33] M . Sev ior . p r i v a te c o m m u n i c a t i o n . 68 Appendix A Traceback Algebra T h e t r a ceback r ou t i ne s were u sed by the software to dec ide whe re the s ca t t e r i n g o r i g i n was s i t u a t ed . In o rde r t o d e t e r m i n e the s c a t t e r i n g o r i g i n , t he pos i t i on s o n the i n i t i a l a n d final s ca t te r i ng t r a jec to r i e s c o r r e s pond i n g t o t he closest a p p r o a c h was first d e t e rm ined . T h e " o r i g i n " was t h e n def ined as the m i d p o i n t of the l i ne j o i n i n g these two po in t s . F o r th i s pu rpose , the geomet r y de f ined i n figure 30 was used. RA a n d RB are vecto r s w h i c h e x t e n d toward s a k n o w n p o i n t i n space f r o m —* —* the c oo rd i n a t e o r i g i n . A a n d B are vector s de f ined by the 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 w i t h eA a n d eg as the i r u n i t d i r e c t i o n a l vectors . T h e vector s tA a n d tB are de f ined by the f o l l ow i ng re la t ions . tA = RA - kAeA (27) f f l = R B - kBeB (28) where kA a n d KB a r e va r i ab l e pa ramete r s . S ince tA a n d tB i n d i c a t e po i n t s on vector s A a n d B re spect ive ly , t he a i m is to m i n i m i z e t he di f ference of these vector s . T h e m a g n i t u d e of th i s d i f ference is expres sed as \tA — tB\2, where \tA-tB\2 = \RA-RB\2+kA+kl+2kB(RA-RB)-eB-2kA(RA (29) T h i s exp re s s i on is a m i n i m u m w h e n b o t h p a r t i a l der i va t i ves w i t h respect to the 1For example, the coordinate from one of the wire chambers. 69 apparatus origin to 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 fo r target t raceback . p a r a m e t e r s kA a n d kB are equa l t o zero. d\tA-tB\* d\tA-tB\2 dk, dk B u n d e r these cons t ra in t s ; kA = kBeA • eB + (RA - RB) • eA —* —* kB = KAeA • eB - (RA - RB) • eB A f t e r d e c o u p l i n g these equat ions , one ob t a i n s the f o l l o w i n g express ions; k B = ( i _ ( g * . g B ) 2 ) [{{^A ~ ' ^ A ' ^ ~ i ^ A ~ ' a n d , kA — kB (eA • eB) • eB The se p a r a m e t e r s were s u b s t i t u t e d back i n t o equat i on s 27, 28 to o b t a i n the des i red p o i n t s o n the vectors . T h e o r i g i n is t h e n (tA + tB) 0(x,y,z) 70 B y evaluating the m i n i m u m distance of \tA — ts\, one can evaluate the quality of the traceback. If this distance is large, it is an indicat ion that the trajectory information was poor, or the two vectors d i d not share a common i n reality. \tA — 2B| is denoted as the Root Difference Squared ( R D S ) . 71 Appendix B Spin Precession Unfortunately, it was not possible to correct for the spin precession based solely on information recorded by 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. The approach henceforth taken, was to calculate the effect of precession on a known polarization (from Ay0) and compare its final components outside the effective magnetic field w i t h those measured by the polarimeter. The obscure geometry of the system d i d not allow for a simple analytic solution, so it was necessary to perform the calculations numerically. T h e program F I N D A N G [33] had previously been wri t ten to calculate energy losses, kinematic angles, and trajectories of charged particles passing i n and out of the magnetic field of the polarized deuterium target. FINDANG allows one to determine the magnetic field and particle velocity components at any point along a particle's path. B y choosing an appropriate step size, the spin precession could be calculated at each interval, and integrated over the entire path 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 which describes the time rate of change of the spin of a particle i n its rest frame w i t h respect to the electric field, magnetic field and velocity components expressed i n the laboratory frame. dsR e _ •s x dt mc 'f-^)*-(5-0^(W + electric field term (34) Since no electric fields were present i n the lab system, the last term was ignored. 72 T h i s e q u a t i o n was t h bas i s fo r the s p i n p reces s ion c a l c u l a t i o n . A t th i s p o i n t i t is u se fu l t o ou t l i n e the f r ames of reference u sed by FINDANG t o desc r ibe the p o l a r i z a t i o n , a n d e x p l a i n the sub t l e d i f ferences be tween the s p i n a n d p o l a r i z a t i o n t r an s f o rma t i on s l i n k i n g these f rames . A l t h o u g h s p i n is a n inva r i an t quant i t y , the m a g n i t u d e of i t s c o m p o n e n t s are f r a m e dependent . T h i s is de sc r i bed b y equat ion s 37, 38. Hence the s p i n c o m p o n e n t p a r a l l e l to t he 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 increase for a n observer m o v i n g at 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 observer i n the p a r t i c l e ' s rest f r ame. T h i s c an also l e ad to a n apparent r o t a t i o n o f the s p i n w h e n i t is obse rved f r o m di f ferent f rames. V e c t o r p o l a r i z a t i o n o n the o the r 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 de s c r i bed by e q u a t i o n 1. Its d i r e c t i o n a l c omponen t s descr ibe the m e a n d i r e c t i o n of a l l t he sp ins i n a b e a m o f pa r t i c l e s . W h e n th i s m e a n is boo s t ed i n t o ano the r f r ame , i t w i l l a lso r o t a t e as de s c r i bed above, b u t i t is not cor rect to 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 a l so change. T h e n u m b e r s a l i gned 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 he m e a n d i r e c t i o n w i l l not change. T h r e e f r ames of reference were u sed 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 were expressed i n the l a b o r a t o r y f r a m e as de sc r i bed i n sec t ion IV . 1, s ince th i s was the f r a m e re levant to the po l a r ime te r . T h e s p i n p reces s ion c a l cu l a t i on s ( equa t i on 34) r equ i r ed the s p i n componen t s to be de f ined i n the p a r t i c l e ' s rest f r ame , whereas the i n i t i a l p o l a r i z a t i o n was de sc r i bed i n the center of mass f r a m e as is the conven t i on for s p i n t rans fe r observables [5]. A s is seen i n f igure 31, one c an re la te the center o f mas s p o l a r i z a t i o n to t ha t seen i n t he l a b f r a m e by a s imp le 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 angle u. cos u = cos 9* cos 91 + 7CM s i n 9* s i n 9L (35) 73 F i g u r e 31: R o t a t i o n due t o r e l a t i v i s t i c boos t f r o m center of mass to l a b f r ame , for the genera l r e a c t i o n c(a,c)d. TYl s i n a; = — (s in#* cos 6L — *)CM c o s 9* s m ®L) (36) hi* W h e r e 9*, a n d 6^ are the angles be tween the o u t g o i n g p r o t o n a n d the i n c i den t p i o n i n t he center o f mass a n d l ab f rames respect ive ly , a n d to = 9cm — 9T,. T h i s r o t a t i o n leaves any n o r m a l c omponen t unaf fected. T o boo s t be tween the rest a n d l a b o r a t o r y sys tems, equa t i on s 37, 38 were used. SR = SL ~ T~T fe • P) P (37) 2 sL = ?R + - 1 — (sR • 0) 0 (38) 7 + 1 v ' W h e r e 7, a n d ft desc r ibe the m o t i o n of the rest f r a m e (p ro ton ) i n the l a b s y s tem. T h i s boos t is t r a j e c t o r y dependent . In s u m m a r y , 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 l a b f r a m e 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 va lue i n the center o f mass f r ame. T h i s va lue is 74 boosted to the rest frame as a spin whose magnetic precession is integrated over the entire p a t h length through the magnetic field. The f inal spin is boosted back to the lab where it can again be renormalized to a polarizat ion, 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. 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 Pk's are the various components of 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. INPN + gNPs + hNPL (39) P's = fsPN + gsPs + hsPL (40) Ps<x Ps- KSs + Pi • KLS + tensor terms PN OC ANO + PN ' KNN + tensor terms (41) 76 PL oc Pg • KSL + PL ' KLL + tensor terms A s d e m o n s t r a t e d b y equat i on s 41 t he size a n d s i gn of the p r o t o n ' s s ideways 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 l i nea r f u n c t i o n o f the s ideways a n d l o n g i t u d i n a l vec to r p o l a r i z a t i o n of the deu te ron ( together w i t h tensor t e rms ) . T h e n o r m a l p o l a r i z a t i o n of the deu te ron , o n the o the r h a n d , is i ndependen t of t he deu te r on p o l a r i z a t i o n i n the s ca t t e r i n g p lane. U s i n g these features , one c an c o m p a r e resu l t s fo r r u n s whe re the deu te r on p o l a r i z a t i o n i n the s c a t t e r i n g p l a n e is o f oppo s i t e s ign, a n d s i m p l i f y t he above equat ions . PN+) = JNPN + RgNPs + RhNPL -{PN\ = INPN - gNPs - hNPL) (42) PN -PN' = {\ + R)gNPs + (l + R)hNPL a n d P ^ = fsPN + RgsPs + RhsPL -(Ps~\ = ISPN - gsPs - hsPL) (43) P£ -Ps^ = (i + R)gsPs + (i + R)hsPL set KN - P*+) - P^ a n d Ks = P*+) - P^ ( _ ) . N o w we have two equa t i on s a n d two u n k n o w n s , whe re R is t he r a t i o of the m a g n i t u d e s of the p o l a r i z a t i o n c o r r e spond i n g t o the p o s i t i v e l y a n d nega t i v e l y p o l a r i z e d d e u t e r i u m targets . R = l ^ t l i m \P*(+)\ { ] N o w s o l v i n g fo r P$ a n d PT, one ob ta i n s , {Ks - ^KN) l P S ~ (1 + *) ^ (Ks - ^ K N ) i In o rde r t o o b t a i n t he c o u p l i n g constants , one c an use the s p i n precess ion p r o g r a m d i scussed i n a p p e n d i x B. B y se t t i ng the i n i t i a l center o f mas s p o l a r i z a t i o n to p u r e n o r m a l , s ideways o r l o n g i t u d i n a l , one c an measu re 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 PN fs p' PN gN P1 LM. P ? gs P' Is. p ? hN P ' LM. Pf, hs 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 = ^- (47) JN or PN = T- (48) Js This technique is simple and unambiguous since no components of P 5 or Pi can contribute. 78 

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