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Calibration of a polarized proton target Shypit, Rickey Lee 1981

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CALIBRATION OF A POLARIZED PROTON TARGET by RICKEY LEE SHYPIT B.Sc., The U n i v e r s i t y of Winnipeg, 1979 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE •in . The F a c u l t y o f . G r a d u a t e .Studies Department of P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d The U n i v e r s i t y o f B r i t i s h Columbia A p r i l , 1981 R i c k e y Lee S h y p i t , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 ABSTRACT Proton p o l a r i z a t i o n of a b u t a n a l 5% H 20-EHBA-C r(V) p o l a r i z e d t a r g e t have been measured by e l a s t i c s c a t t e r i n g t o an accuracy o f ± 2%. The four c u b i c centimeter t a r g e t sample, suspended i n a 25 KG f i e l d , was c o o l e d by a 3 pumped He r e f r i g e r a t o r . L e f t r i g h t asymmetry of s c a t t e r e d p a r t i c l e s were d e t e c t e d by a two arm t e l e s c o p e , each arm c o n s i s t i n g of a MWPC and a p l a s t i c s c i n t i l l a t o r . Separa-t i o n o f e l a s t i c two body events i n hydrogen from non-e l a s t i c two body events i n nonhydrogenous m a t e r i a l was achieved u s i n g a c o p l a n a r i t y t e s t . - i i -T A B L E O F C O N T E N T S page I. INTRODUCTION . 1 I I . FORMALISM 6 2.1 The S c a t t e r i n g M a t r i x 6 2.2 Dynamic Nuclear O r i e n t a t i o n 12 2.3 Q-meter D e t e c t i o n 17 I I I . APPARATUS 20 3.1 P o l a r i m e t e r 2 0 3.2 Super Conducting S o l e n o i d 23 3.3 C o l l i m a t o r 24 3.4 4B3 Bending Magnet 24 3.5 Quadrupole and S t e e r i n g Magnets 26 3.6 Monitor 4CM8 27 3.7 P o l a r i z e d T a r g e t 27 3.8 NMR System 30 IV. THE POLARIZATION MONITOR 3 2 V. DATA ACQUISITION 3 9 VI. DATA ANALYSIS 4 5 6.1 R e c o n s t r u c t i o n of Proton Tracks 4 5 6.2 Background S u b t r a c t i o n s 51 6.3 The F i t t i n g E quation 56 6.4 E r r o r A n a l y s i s 60 V I I . RESULTS AND CONCLUSION 6 3 7.1 R e s u l t s 63 7.2 C o n c l u s i o n 69 - i i i -LIST OF TABLES page 3.1 S o l e n o i d C u r r e n t s C a l c u l a t e d to e f f e c t a 90° P r e c e s s i o n 25 3.2 C u r r e n t s i n the Bending Magnet C a l c u l a t e d f o r a 35° Bend 25 5.1 P r e c i s e Beam En e r g i e s i n MeV (at e x t r a c -t i o n , a t the p b l a r i m e t e r , a f t e r the p o l a r i -meter, and a t the t a r g e t 4 4 6.1 Run Averaged Hydrogen to Background R a t i o s 6.2 P r e c e s s i o n Angle i n the Bending Magnet 5 5 6.3 Values of the P o l a r i z a t i o n Parameter a t 26° LAB 59 7.1 F i n a l Values of A, W and SA(eT) 64 7.2 F i n a l Values of C 65 7.3 NMR P o l a r i z a t i o n I n t e g r a l s and A b s o l u t e T a r g e t P o l a r i z a t i o n s 66 - i v -LIST OF FIGURES page 2.1 Kinematic Diagram 8 2.2 Energy L e v e l Diagram 16 3.1 Beamline 4C 21 3.2 Beamline P o l a r i m e t e r 22 3.3 NMR F u n c t i o n a l Diagram 28 3.4 Schematic of the C r y o s t a t 29 4.1 P o l a r i z a t i o n Monitor 33 4.2 Three Beam D e f i n i n g S c i n t i l l a t o r s 35 4.3 E l e c t r o n i c L o g i c Diagram 36 4.4 Diagram of a De l a y - L i n e M u l t i - W i r e P r o p o r t i o n a l Chamber 37 5.1 Scope Trace Showing NMR S i g n a l (Re-produced from P o l a r o i d Photograph) 41 5.2 A t t e n u a t i o n In Beam Tra n s m i t t e d Through The Target ..... 4 2 6.1 TDC D i f f e r e n c e Histogram 47 6.2 L i n e I n t e g r a l o f the B F i e l d 50 6.3 Coplanar Histogram 53 -v-ACKNOWLEDGEMENT I would l i k e to thank my c o l l e a g u e s i n t h i s experiment who f i r s t i n t r o d u c e d me to experimental n u c l e a r p h y s i c s - D. Axen, D.V. Bugg, M. Comyn, J . Edgington, D. Healey, G. Ludgate, J . S t a n l e y , N. Steven-son, and N. Stewart. I a l s o thank R. Dubois, P. Bennett and A. Haynes f o r t h e i r a s s i s t a n c e w i t h many computing problems. I would e s p e c i a l l y l i k e to thank my s u p e r v i s o r Dave Axen f o r h i s con s t a n t encouragement and advi c e d u r i n g the experiment and throughout the w r i t i n g of t h i s t h e s i s . - v i -1 Chapter I INTRODUCTION The c a l i b r a t i o n of a p o l a r i z e d p roton t a r g e t by-n u c l e a r s c a t t e r i n g techniques i s d e s c r i b e d i n t h i s t h e s i s . The c a l i b r a t i o n i n v o l v e d a measurement of the l e f t r i g h t asymmetry i n proton proton s c a t t e r i n g w i t h the beam and t a r g e t a l i g n e d l o n g i t u d i n a l l y . The measured asymmetry i s g i v e n by a product of the beam p o l a r i z a t i o n , the t a r g e t p o l a r i z a t i o n and the C L L parameter. The C L E parameter was computed from a phase s h i f t a n a l y s i s of the world data and the beam p o l a r i z a t i o n was determined u s i n g a p r e v i o u s l y c a l i b r a t e d p o l a r i m e t e r . The t a r g e t p o l a r i z a t i o n e x t r a c t e d from t h i s asymmetry measurement was then compared to the value o f t a r g e t p o l a r i z a t i o n determined from an NMR measurement. Measurements were made a t beam e n e r g i e s of 210, 325, 425, 460, 500 and 520 MeV. The energy r e g i o n covered by these measurements extends from the minimum to the maximum beam energy a v a i l a b l e a t the TRIUMF c y c l o t r o n . 2 The measurement of t a r g e t p o l a r i z a t i o n by the nu c l e a r s c a t t e r i n g method was motivated by the need f o r an a c c u r a t e d e t e r m i n a t i o n o f the t a r g e t p o l a r i z a t i o n which was r e q u i r e d i n the measurement of Ao T• Aa i s the J-J j - i d i f f e r e n c e i n proton proton t o t a l c r o s s e c t i o n s f o r s p i n s a l i g n e d p a r a l l e l and a n t i - p a r a l l e l along the l o n g i t u d i n a l 1 2 3 4 d i r e c t i o n . P r e v i o u s measurements of Ao^ ' ' ' have r e -l i e d o n l y on n u c l e a r magnetic resonance techniques to determine the t a r g e t p o l a r i z a t i o n . I n c o n s i s t e n c i e s i n p r e v i o u s l y p u b l i s h e d data might be e x p l a i n e d by syst e m a t i c e r r o r s i n the n o r m a l i z a t i o n of the t a r g e t p o l a r i z a t i o n . T h i s experiment i s e s s e n t i a l l y a t e s t o f t h a t h y p o t h e s i s . In an NMR measurement the t a r g e t p o l a r i z a t i o n i s determined from the f o l l o w i n g r e l a t i o n p = [NMR enhanced s i g n a l p T ~~ jNMR thermal s i g n a l " thermal where p t h e r m a l the a b s o l u t e t a r g e t p o l a r i z a t i o n a t thermal e q u i l i b r i u m . Thermal e q u i l i b r i u m r e f e r s to the s t a t e i n which the n u c l e a r s p i n system i s i n e q u i l i b r i u m w i t h i t s surroundings. P., -, i s c a l c u l a t e d from the thermal fundamental Boltzman equation EXP (yB/kT) - EXP (-yB/kT) thermal EXP (yB/kT) + EXP (-yB/kT) where y i s the proton magnetic moment, B i s the s t r e n g t h of the s t a t i c e x t e r n a l f i e l d , k i s the Boltzman co n s t a n t and T i s the e q u i l i b r i u m temperature. p t h e r m a l t h u s 3 depends c r i t i c a l l y on an accurate knowledge of the temperature which appears i n v e r s e l y i n the e x p o n e n t i a l f a c t o r . Measurement of the e q u i l i b r i u m temperature and i n p a r t i c u l a r the e l e c t r o n s p i n temperature i s d i f f i c u l t as the l a t t i c e temperature must be determined i n d i r e c t l y by m o n i t o r i n g vapor p r e s s u r e i n the c r y o s t a t . ~* Un-c e r t a i n t y i n the temperature c a l c u l a t i o n i n t r o d u c e s a sys t e m a t i c e r r o r i n t o the o v e r a l l NMR measurement. A second experimental d i f f i c u l t y i s a s s o c i a t e d with the weak thermal e q u i l i b r i u m s i g n a l which was the order of 0.1% of the enhanced s i g n a l . D i s t o r t i o n s a r i s i n g from e l e c t r o n i c a m p l i f i c a t i o n and n o i s e are much more s e r i o u s f o r the sma l l thermal s i g n a l , p a r t i c u l a r l y as the n u c l e a r s i g n a l appears as a s m a l l a.e. modulation on a l a r g e d.c. o f f s e t . A f u r t h e r l i m i t i n g a s p e c t of the NMR measurement i s t h a t a l l p a r t s of the sample c o n t r i b u t e e q u a l l y to the measurement. Inhomogenieties i n the magnetic f i e l d and s u r f a c e e f f e c t s i n the sample cause l o c a l v a r i a t i o n s i n the Larmor p r e c e s s i o n a l frequency of the protons. The s i g n a l i s an average over the e n t i r e volume of the sample. In comparison the n u c l e a r s c a t t e r i n g measurement samples o n l y those protons i n the t a r g e t t h a t c o n t r i b u t e to s c a t t e r i n g . Hence t h i s method i s more d i r e c t i n g i v i n g i n f o r m a t i o n on the i n i t i a l s p i n s t a t e of those protons which i n t e r a c t v i a a s p i n dependent p o t e n t i a l i n the fundamental process being s t u d i e d ( i . e . A a T ) . Li Consequently a l l systematic e r r o r s are estimated to l i m i t the o v e r a l l accuracy of the NMR measurement to the order of 6%. The measurement of t a r g e t p o l a r i z a t i o n by the n u c l e a r s c a t t e r i n g method i s l i m i t e d by s t a t i s t i c a l e r r o r s s i n c e the data can be accumulated o n l y f o r a f i n i t e time. T h i s time was estimated such t h a t the s t a t i s t i c a l e r r o r s were l e s s than u n c e r t a i n t i e s a r i s i n g from the phase s h i f t p r e d i c t i o n s f o r C ^ L and the other components of the s c a t t e r i n g matrix which enter i n the c a l c u l a t i o n . These e r r o r s were added i n quadrature w i t h the s t a t i s t i c a l e r r o r s . To these was added a small u n c e r t a i n t y i n the angular d i s t r i b u t i o n of the s c a t t e r e d events. The over-a l l u n c e r t a i n t y o b t a i n e d i n t h i s measurement was of the order of ± 2%. T h i s measurement thus p r o v i d e d a c a l i -b r a t i o n of the NMR system. The formalism d e s c r i b i n g the n u c l e a r s c a t t e r i n g measurement i s summarized i n Chapter 2. The d e s c r i p t i o n i s based on the d e n s i t y matrix formalism which i s p a r t i -c u l a r l y w e l l s u i t e d to t h i s a p p l i c a t i o n . A b r i e f d i s -c u s s i o n of dynamic n u c l e a r o r i e n t a t i o n i s g i v e n i n the same ch a p t e r . Included i s a d e s c r i p t i o n of the method used to prepare the i n i t i a l p o l a r i z a t i o n s t a t e f o r the s c a t t e r i n g experiment. F i n a l l y Q meter d e t e c t i o n which c o n s t i t u t e d the d e t e c t i o n system of the NMR measuring apparatus employed i n t h i s experiment i s d e s c r i b e d . A g e n e r a l d e s c r i p t i o n of beam l i n e apparatus and e s s e n t i a l equipment i n c l u d i n g the p o l a r i z e d t a r g e t and NMR apparatus i s g i v e n i n Chapter 3. Chapter 4 i s de-voted e n t i r e l y to a d e s c r i p t i o n of the p o l a r i z a t i o n monitor which was used to observe the n u c l e a r s c a t t e r i n g . In Chapter 5 a s h o r t account o f the procedure f o r re-, c o r d i n g experimental data i s g i v e n . The d u r a t i o n of the experiment was one week hence o n l y some g e n e r a l aspects of the data a c q u i s i t i o n program are i n c l u d e d . D e t a i l s of the a n a l y s i s used to o b t a i n the t a r g e t p o l a r i z a t i o n are summarized i n Chapter 6 . Track r e c o n s t r u c t i o n of protons s c a t t e r e d i n t o the monitor i s d e s c r i b e d . The r e c o n s t r u c t e d t r a j e c t o r i e s were used t o compute the copl a n a r and opening angle f o r each s c a t t e r i n g event. Use of a c o p l a n a r i t y t e s t to separate e l a s t i c two body events i n hydrogen from background s c a t t e r i n g i s d e s c r i b e d . The f i t t i n g e quation used to o b t a i n the t a r g e t p o l a r i z a t i o n are presented and d i s c u s s e d . F i n a l l y the c a l c u l a t i o n of an o v e r a l l u n c e r t a i n t y i n the measurement i s d e s c r i b e d . The r e s u l t s o f the n u c l e a r s c a t t e r i n g measurement wit h e r r o r a n a l y s i s and c o n c l u s i o n are presented i n the f i n a l Chapter. 6 Chapter I I FORMALISM T h i s Chapter g i v e s a d e s c r i p t i o n of the d e n s i t y m atrix formalism i n nucleon-nucleon s c a t t e r i n g , the methods of dynamic n u c l e a r o r i e n t a t i o n and the technique of Q-meter d e t e c t i o n 2.1 The S c a t t e r i n g M a t r i x The k i n e m a t i c s of e l a s t i c proton proton s c a t t e r i n g are completely determined by two parameters, which may be chosen as the c e n t r e of mass momentum, k, of the i n c i d e n t proton and the c e n t r e of mass s c a t t e r i n g angle 0 . v.* • ILL • The s p i n dependence of the i n t e r a c t i o n has been conven- .. i e n t l y formulated i n a matrix r e p r e s e n t a t i o n . In t h i s formalism the s c a t t e r i n g amplitudes, f ( k , 0 c m ), f o r each p o s s i b l e s p i n c o n f i g u r a t i o n are the components of a m a t r i x . The i n i t i a l s p i n s t a t e n i s s p e c i f i e d by a (n) fo u r component v e c t o r X . In the asymptotic l i m i t as r becomes much l a r g e r than the range of the i n t e r a c t i o n the wave f u n c t i o n of the s c a t t e r e d p a r t i c l e i s g i v e n by 7 .(n) i k - r (n) , _(n) i k r i p = e — — X + r e where f ( n ) = MX ( n ) (2.1) M i s a 4x4 matrix which i s a f u n c t i o n o f k, 6 _ _ and the c .m. P a u l i s p i n o p e r a t o r s , a ^ and a ^  , f o r the two p a r t i c l e s The i n c i d e n t momentum v e c t o r k and the s c a t t e r e d momentum v e c t o r k' are used t o d e f i n e t h r e e orthonormal c a r t e s i a n v e c t o r s i n the c m . frame gi v e n by k+k* ^ kxk' „ k-k 1 N = i , „ . , . , i , K = I.. .., I = N x P (2.2) The most g e n e r a l form of M t h a t p r e s e r v e s i n v a r i a n c e under space r o t a t i o n s , r e f l e c t i o n s and time r e v e r s a l i s (2.3) where the s u b s c r i p t s k,N, and P i n d i c a t e the d i r e c t i o n of the s p i n v e c t o r w i t h r e s p e c t t o the d e f i n i t i o n s i n 2.2. The c o e f f i c i e n t s a,c,m, g and h are known as the amplitude of the s c a t t e r i n g m a t r i x . The s p i n s c a t t e r i n g matrix M i s r e l a t e d to the s c a t t e r i n g matrix S by M = if < 0 f * f I s - l | e i * i > (2.4) C M . FRAME FIGURE 2.1 KINEMATIC DIAGRAM 9 where the b r a and k e t v e c t o r s s p e c i f y the d i r e c t i o n o f motion of the outgoing and incoming nucleons whose r e -l a t i v e momentum i s d e s c r i b e d by wave number p. The matrix M i s . u s e d i n the framework of a d e n s i t y 7 matrix formalism to r e l a t e the i n i t i a l s p i n s t a t e of two s c a t t e r i n g nucleons t o the e x p e c t a t i o n value of the s p i n s i n the f i n a l s t a t e g i v e n by < a ( 1 ) a D ( 2 ) ( a , g = 0,1,2,3) (2.5) a p t where = 1 corresponds t o a measurement where the s p i n component i s not determined and 1,2 and 3 correspond to " " " (n) the d i r e c t i o n P,N and k. For a pure s p i n s t a t e x / the d e n s i t y matrix i s d e f i n e d as p i j = X i " * * j n ) + ( 2 - 6 ) where x ^  i s the i t h component of the column v e c t o r x ^ (n) ^  and x i s the a d j o i n t row v e c t o r . For an i n c o h e r e n t mixture of pure s p i n s t a t e s as i s a beam produced by an a c c e l e r a t o r , the d e n s i t y m a t r i x i s d e f i n e d v (n) (n) + ,~ n, P • • =') P X• X • (2.7) in L ^n A i A i n J where p n i s the p r o b a b i l i t y of f i n d i n g the system i n s t a t e x ^  • The e x p e c t a t i o n v a l u e < s ^ > of any s p i n (u) • ope r a t o r s i s E P ( <n> S ( y ) y ( n ) ) S E P x . ( n ) t S ( y ) y ( n ) (y) = h n U / b x ; = n i j n x i b i j x j I P n ( x ( n ) , x ( n ) ) " z ? P n . x l n ) t x ' n ) n n n 2. n l l T ( P f S ( y ) ) = — (2.8) T r ( p f ) 10 Hence ( 1 ) ( 2 ) <«<» .' 2>> f - ^ ' ^ °° ° 6 ' (2.9, " T r ( l > f ' The f i n a l s t a t e d e n s i t y matrix p f i s expressed i n terms of the matrix M and the d e n s i t y matrix p^, which d e s c r i b e s the p r e p a r a t i o n of the i n i t i a l s t a t e . From 2.1 p. = Z P M X ( n ) x(N)V'' = M p.M+ (2.10) K f n n A I and the f i n a l s t a t e e x p e c t a t i o n v a l u e of s p i n i s (1) (2) T r ( M P . J M V 1 ) a g 2 ) ) { 2 m l l ) a P 1 T (Mp.M1^) r I D e f i n i n g S ( y ) = ^ ^ ° ^ r v = 1,2, ... 16, the r e l a t i o n between i n i t i a l and f i n a l p o l a r i z a t i o n s t a t e s i s 16 I v=l K S ( ^ > = i I <S ( V )>. T ( M S ( v ) M + S ( l ° ) (2.12) f 4 L , l r where I i s the d i f f e r e n t i a l s c a t t e r i n g c r o s s e c t i o n from the i n i t i a l s t a t e |e^ <J>^ > i n t o the f i n a l s t a t e < Q ^ ^ \ . For experimental convenience the r e f e r e n c e frame i s r e - d e f i n e d to be the l a b o r a t o r y c o o r d i n a t e system. The c r o s s e c t i o n , I , c o r r e s p o n d i n g t o a measurement where a ot p p o l a r i z e d beam wit h p o l a r i z a t i o n along a s c a t t e r s from a p o l a r i z e d t a r g e t w i t h p o l a r i z a t i o n along 3, and the f i n a l p o l a r i z a t i o n s are not observed, i s w r i t t e n Xa, = I T r ( M M + ) + i T ( M a ( 1 V ) < a ( 1 ) > . -+ * T ( M a R ( 2 V ) <a f t ( 2 ) > . 4 r a ' a l 4 r 8 B l + ( M C ( 1 ^ 2 V ) < C ( 1 > > . < C r ( 2 ) > . (2.13) 4 r a 3 a l 3 l 11 The f i r s t term i s I Q , the d i f f e r e n t i a l c r o s s e c t i o n f o r s c a t t e r i n g o f an u n p o l a r i z e d beam from an u n p o l a r i z e d t a r g e t and the next two terms are the p o l a r i z a t i o n IQP-By d e f i n i n g a ten s o r C _ such t h a t a p zo c a e = 1 V ^ a ^ °l2) M + ) ( 2 ' 1 4 ) equation 2.13 takes on the form I a = I n { l + P ( 6 <a ( 1 )>. + 6 O M<a 0 ( 2 )>.) aB 0 l aN a 1 BN B 1 + C . <a ( 1 )>.<a D ( 2 )>.} (2.15) aB a l B l where <S i s the Kronecker d e l t a . <a ^  > . i s the expecta-a l c t i o n v alue o f the beam s p i n along a i n the i n i t i a l s t a t e (2) and <a^ >^ i s the e x p e c t a t i o n v a l u e of the t a r g e t s p i n along B i n the i n i t i a l s t a t e ( i . e . the components of beam and t a r g e t p o l a r i z a t i o n ) . The dependence o f the s c a t t e r i n g on the two kinematic v a r i a b l e s i s i m p l i c i t i n the de-f i n i t i o n o f P and C „ through t h e i r r e l a t i o n to the matrix a B M, whose components are the s c a t t e r i n g amplitudes. Numerous experiments have been done to determine the s c a t t e r i n g m a t r i x . The pr e s e n t work was not designed to c o n t r i b u t e to t h a t d e t e r m i n a t i o n but r a t h e r made use of e x i s t i n g i n f o r m a t i o n on M to determine the t a r g e t p o l a r i z a t i o n . E q u a t i o n 2.15, expressed i n Aa T c o n f i g u r a -t i o n (eq. 6 . 4 ) , was p r e c i s e l y the equation used to determine the t a r g e t p o l a r i z a t i o n . In t h i s c o n f i g u r a t i o n both beam and t a r g e t had zero component o f p o l a r i z a t i o n 12 i n the normal d i r e c t i o n . Hence the second term of 2.15 d i d not c o n t r i b u t e to s c a t t e r i n g and i s absent i n 6.4. The components of the tensor which d i d c o n t r i b u t e were C T , C o c and C c . The c o n t r i b u t i o n o f the C c c and C „ T terms r e s u l t e d because both the beam and the t a r g e t s p i n s had a non zero component i n the sideways d i r e c t i o n . The o r i g i n o f these n o n - l o n g i t u d i n a l components i s ex-p l a i n e d i n the d i s c u s s i o n on apparatus g i v e n i n Chapters three and f o u r . 2.2 Dynamic Nuclear O r i e n t a t i o n Some of the b a s i c concepts i n v o l v e d i n the p r e p a r a t i o n of an i n i t i a l p o l a r i z a t i o n s t a t e are d e s c r i b e d i n t h i s s e c t i o n , b e g i n n i n g w i t h a d e s c r i p t i o n o f the funda-mental p r i n c i p l e s o f n u c l e a r o r i e n t a t i o n . or atom i s r e l a t e d to i t s angular momentum v e c t o r Ih by the e x p r e s s i o n where 8 i s the Bohr magneton and g i s the g - f a c t o r a p p r o p r i a t e t o the system being c o n s i d e r e d . The i n t e r -a c t i o n energy o f the d i p o l e moment w i t h a magnetic f i e l d The magnetic d i p o l e moment v e c t o r y_, of a p a r t i c l e y_ = q $ I (2.16) i s E = - y_-H0 (2.17) 13 The d i p o l e e x i s t s i n d i s c r e t e energy l e v e l s g i v e n by E m = g3mH 0 (2.18) Each l e v e l i s c h a r a c t e r i z e d by m, the e x p e c t a t i o n value of the s p i n component i n the d i r e c t i o n of the a p p l i e d f i e l d . For an assembly of i d e n t i c a l d i p o l e s i n e q u i l i b r i u m a t a temperature T(°K) , the r e l a t i v e number, N m, of the whole assembly i n s t a t e m i s giv e n by the Boltzman equation N = EXP(-E /kT) (2.19) m m ' When the temperature i s s u f f i c i e n t l y low t h a t the magnetic energy i s l a r g e r than the random thermal energy, the number of d i p o l e s w i t h s p i n s a l i g n e d p a r a l l e l to the f i e l d , N +, exceeds the number wit h s p i n s a l i g n e d a n t i -p a r a l l e l N_. The s p i n s are p r e f e r e n t i a l l y o r i e n t e d and a s u i t a b l e measure of t h i s p o l a r i z a t i o n P, i s g i v e n by I . ^ T mN P = 1 _EZZ| 51 (2.20) The magnetic p r o p e r t i e s of the assembly can be d e s c r i b e d by the m a g n e t i s a t i o n M, d e f i n e d by I l m EXP(g3mH 0/kT) „ m=-T M = N <?3 — (2.21) £ EXP(gBmH Q/kT m=-I where N i s the t o t a l number of d i p o l e s i n the assembly 14 The e x p o n e n t i a l f a c t o r s appearing i n the d e f i n i t i o n of M are N m as g i v e n by 2.19. T h e r e f o r e the m a g n e t i s a t i o n i s simply r e l a t e d to the p o l a r i z a t i o n by M = NgglP (2.22) At a temperature o f approximately 1°K and a f i e l d o f 25 KG the e l e c t r o n s i n the paramagnetic ions are e s s e n t i a l l y 100% p o l a r i z e d but the n u c l e a r s p i n s t a t e s are e s s e n t i a l l y e q u a l l y populated except f o r the small Boltzman f a c t o r which lea d s t o pro t o n p o l a r i z a t i o n s o f the order o f 0.5%. The proton p o l a r i z a t i o n can be enhanced by dynamic methods. The important f e a t u r e s o f dynamic p o l a r i z a t i o n can be i l l u s t r a t e d i n a simple model. In t h i s model the sample i s an assembly of paramagnetic ions each i n a h y p e r f i n e c o u p l i n g w i t h the f r e e protons. The sample i s p l a c e d i n a microwave c a v i t y a t l i q u i d He temperature and i n a st r o n g e x t e r n a l f i e l d . Denoting the e l e c t r o n magnetic quantum number S and n u c l e a r s p i n p r o j e c t i o n : m, the energy l e v e l s o f the system are E(s,m) = gBH QS + Asm (2.23) Here g i s the e l e c t r o n i c s p e c t r o s c o p i c s p l i t t i n g f a c t o r and A takes on a value such t h a t the h f s i n t e r a c t i o n i s secondary but s t i l l much l a r g e r than the i n t e r a c t i o n , o f the proton s p i n d i r e c t l y w i t h the e x t e r n a l f i e l d , which i s n e g l e c t e d . The s p l i t t i n g s of the l e v e l s i n u n i t s of kT 15 (Figure 2.2) are 6 = A/2-kT and A g&H Q/kT (2.24) The l a b e l l i n g of s t a t e s i n F i g u r e 2.2 corresponds to the e l e c t r o n i c and n u c l e a r s p i n p r o j e c t i o n s . For ex-ample, (++) means (S=+%, m-+h) e t c . Dynamic n u c l e a r p o l a r i z a t i o n i s achieved by connecting an r f o s c i l l a t o r to the c a v i t y c o n t a i n i n g the sample. With the o s c i l l a t o r frequency s e t such t h a t hv = g3H Q, t r a n s i t i o n between the s t a t e s (+-) and (-+) are induced. T h i s t r a n s i t i o n corresponds to f l i p p i n g the e l e c t r o n s p i n s up and the n u c l e a r s p i n s down. The p o p u l a t i o n s of these two l e v e l s w i l l e v e n t u a l l y become equal s i n c e r f induced emission and a b s o r p t i o n r a t e s are e q u a l . Assuming N +_ = N_ + = 1, the r e l a t i v e p o p u l a t i o n of the other two l e v e l s w i l l be determined by the r e -l e v a n t r e l a x a t i o n mechanism. As the dominant mechanism f a v o r s t r a n s i t i o n s where As=l and Am=0, the steady s t a t e p o p u l a t i o n s tend to ++ -A-o , +- 1 , n = e and — = -j^- (2.25) -+ — e r e s u l t i n g i n dynamic n u c l e a r p o l a r i z a t i o n of the system approximately equal to -A/2. Now i f the o s c i l l a t o r i s s e t to induce t r a n s i t i o n s between the s t a t e s (—) and (++) the r e l a x a t i o n mechanism w i l l e s t a b l i s h a steady s t a t e p o l a r i z a t i o n of +A/2. 16 (a) (b) s m ( - + ) F IGURE 2.2 E N E R G Y L E V E L D I A G R A M column (a) gives the relative populations at thermal equilibrium; column (b) gives the dynamic equilibrium populations achieved by r.f. saturation o f the transition shown 17 2.3 Q-Meter D e t e c t i o n The NMR measuring apparatus used i n t h i s experiment d e t e c t e d n u c l e a r o r i e n t a t i o n s by a simple method known as Q-meter d e t e c t i o n . T h i s method e x p l o i t s some of the w e l l known p r o p e r t i e s of r f c o i l s which w i l l be reviewed b r i e f l y i n t h i s s e c t i o n . A c u r r e n t i c i r c u l a t i n g i n the c o i l produces a magnetic f i e l d H l i n e a r l y p o l a r i z e d along the a x i s of the c o i l . The magnetic f l u x a c r o s s the c o i l due to t h i s . . . c u r r e n t i s $ = L i (2.26) where L i s the c o i l i nductance. For a c u r r e n t w i t h time dependence given by i = Re { l e j a 3 t J (2.27) The q u a l i t y f a c t o r Q, of the c o i l i s d e f i n e d P = Ww/Q (2.28) 2 where P = h I r i s the power d i s s i p a t e d i n the c o i l of 2 r e s i s t a n c e r , and W = h L I i s the maximum energy stored in the c o i l . E q u i v a l e n t l y Q can be w r i t t e n Q = Lu/r (2.29) The c o i l i s arranged such t h a t the r f f i e l d i s p e r p e n d i c u l a r to a uniform e x t e r n a l f i e l d H . Thus the 18 n u c l e a r m a g n e t i s a t i o n i s f o r c e d t o precess about by the d r i v i n g r f f i e l d . The ma g n e t i s a t i o n has a component, M.// , along the r f f i e l d d i r e c t i o n equal to M / / = Re {2xHe :' a , t} (2.30) i n s i d e the sample and zero o u t s i d e . The complex r f s u s c e p t i b i l i t y , x , can be w r i t t e n as X - ' - J x " and i s v a n s i h l y small except a t resonance. The f l u x a t the i n d u c t i o n v e c t o r , B = H + 4TTM, a cross the c o i l i s B = Re {L(1+4TT X TI) I e j c o t } (2.31) where TI i s the sample f i l l i n g f a c t o r . T h i s i s e q u i v a l e n t to s a y i n g t h a t i n the presence of n u c l e a r m a g n e t i s a t i o n the inductance takes on the complex v a l u e s L ( l + 4 T r x n ) . When a c a p a c i t a n c e C i s added i n s e r i e s w i t h the c o i l c o n t a i n i n g the sample and the c i r c u i t i s d r i v e n by a c o n s t a n t c u r r e n t generator i t develops a v o l t a g e across i t s t e r m i n a l s p r o p o r t i o n a l to i t s s e r i e s impedance Z, g i v e n by Z = r + j (OJL (1+4TTXTI) - - ) (2.32) coc When the r f f i e l d i s swept through the resonant frequency, e x c i t a t i o n o f d i p o l e t r a n s i t i o n s i n the sample causes power t o be absorbed from the c o i l . T h i s a b s o r p t i o n changes the impedance of the c i r c u i t and. r e s u l t s i n a f r a c t i o n a l change i n v o l t a g e a c r o s s the c i r c u i t p r o p o r t i o n a l t o 19 4TT r, Q X " (2.33) X " i s r e l a t e d to XQ> the s t a t i c C u r i e s u s c e p t i b i l i t y , g v i a d i s p e r s i o n r e l a t i o n s and s i n c e the n u c l e a r magnetisa-9 -t i o n i s p r o p o r t i o n a l to XQ-7 the recorded s i g n a l amplitude i s p r o p o r t i o n a l to the n u c l e a r p o l a r i z a t i o n . For p o s i t i v e p o l a r i z a t i o n power i s t r a n s m i t t e d to the c o i l by the n u c l e a r s p i n s r e s u l t i n g i n a f r a c t i o n a l change i n v o l t a g e of the o p p o s i t e s i g n . 20 Chapter I I I APPARATUS Beam l i n e 4C a t TRIUMF was designed s p e c i f i c a l l y f o r the t r a n s m i s s i o n of a p o l a r i z e d proton beam to the p o l a r i z e d proton t a r g e t . T h i s d e s c r i p t i o n of apparatus begins w i t h v a r i o u s components t h a t can be termed beam l i n e apparatus and i n c l u d e s the p o l a r i z e d t a r g e t and NMR system. 3.1 P o l a r i m e t e r A p o l a r i m e t e r , p r e v i o u s l y c a l i b r a t e d t o an accuracy of ± 1.5%"^, was used t o measure beam p o l a r i z a t i o n and monitor beam i n t e n s i t y . T h i s d e v i c e i s a f o u r arm t e l e s -cope which measured the l e f t - r i g h t asymmetry i n beam s c a t t e r e d from a 50 ym t h i c k CH,, t a r g e t suspended i n the beam p i p e . Each arm c o n s i s t e d o f two p l a s t i c s c i n t i l l a t o r s a c c u r a c t e l y a l i g n e d . The forward arms were p o s i t i o n e d at 26° w i t h r e s p e c t to the beam and the r e c o i l arms were at the conjugate angle o f 60°. For proton proton e l a s t i c s c a t t e r i n g w i t h the i n c i d e n t beam p o l a r i z e d v e r t i c a l l y and the t a r g e t u n p o l a r i z e d , the p o l a r i z a t i o n , ' d e n o t e d P, , 21 BEAMLINE POLARIMETER PRECESSING SOLENOID COPPER COLLIMATOR NEUTRON COLLIMATOR B3 ^(BENDING MAGNET) SM4 (VERTICAL STEERING) ro LU c r o LU SM5 (HORIZONTAL STEERING) M8 POL. PROTON TARGET ^2 < L L L J C D 22 FORWARD FORWARD FIGURE 3 . 2 BEAMLINE POLARIMETER 23 i s r e l a t e d to the s c a t t e r i n g asymmetry e by P b = e/P where e i s d e f i n e d #Left - # • Right  E #Left + # Right The p o l a r i z a t i o n parameter P has been a c c u r a t e l y measured i n a p r e v i o u s experiment."^ The c o r r e c t i o n r a t e s f o r the carbon content o f the t a r g e t have a l s o been determined i n the same experiment. 3.2 Super Conducting S o l e n o i d The TRIUMF c y c l o t r o n produces beam p o l a r i z e d i n the v e r t i c a l d i r e c t i o n . In t h i s experiment the proton s p i n s were precessed i n two stages to the l o n g i t u d i n a l . The f i r s t p r e c e s s i o n through 90° i n t o the sideways d i r e c t i o n was performed by a super conducting s o l e n o i d . The s o l e n o i d used was lm long w i t h a 10 cm warm bore. I t i s capable o f producing a uniform 6 t e s l a magnetic f i e l d when operated a t the maximum r a t e d c u r r e n t of 212A. The l i n e i n t e g r a l o f the magnetic f i e l d along the a x i s of the s o l e n o i d w a s . c a l c u l a t e d to be 0.028187 T-M-A-1. The p r e c e s s i o n angle o f the s p i n v e c t o r i n a uniform magnetic f i e l d B was c a l c u l a t e d u s i n g the 24 where e i s the proton charge, P i s i t s momentum and u i s the proton magnetic moment. The s o l e n o i d c u r r e n t s c a l c u l a t e d t o e f f e c t a 90° p r e c e s s i o n are gi v e n i n Table 3.1. 3.3 C o l l i m a t o r A c o l l i m a t o r o f approximately 30 cm t h i c k n e s s and 3.5 m i n l e n g t h f o l l o w e d the s o l e n o i d . The c o l l i m a t o r was c o n s t r u c t e d from 2 s t e e l p l a t e s 50 mm t h i c k and f i l l e d with..lead. I t was o r i g i n a l l y designed f o r use i n neutron experiments. There are 11 e q u a l l y spaced p o r t s from -3° to + 27°, each p o r t having an up stream diameter of 10 cm and a down stream diameter of 12.5 cm. T h i s experiment u t i l i z e d o n l y the 0° p o r t . A 20 cm. long copper p l u g w i t h 1mm bore was i n s e r t e d i n the 0° p o r t . The p l u g was used to reduce the beam i n t e n s i t y to the 5 6 l e v e l o f 10 to 10 r e q u i r e d f o r t h i s experiment. 3.4 4B3 Bending Magnet T h i s d i p o l e magnet performed two f u n c t i o n s . F i r s t i t was used t o d e f l e c t the c o l l i m a t e d beam through a 35° bend i n the beam l i n e on i t s way to the experimental area. T h i s angle i s approximately optimum f o r p r e c e s s i n g proton s p i n s from the sideways t o the l o n g i t u d i n a l d i r e c t i o n a t TRIUMF e n e r g i e s . The amount of d e f l e c t i o n was c a l c u l a t e d from the e x p r e s s i o n Table 3.1 S o l e n o i d C u r r e n t s C a l c u l a t e d to E f f e c t a 90° P r e c e s s i o n ENERGY. : CURRENT 519."9 74.25 500.7 72.50 459.4 68.95 423.3- 65.78 329.6 56.26 208. 2 44.03 (A) Table 3.2 Currents i n the Bending Magnet C a l c u l a t e d f o r a 35° Bend ENERGY CURRENT (A) 519.9 2295.0 500.7 2205.0 459.4 1995.0 423.3 1833 .0 329 .6 1500.0 208. 2 1131.0. 26 8 = - B x d& P J " where 9 i s i n r a d i a n s . The c u r r e n t s needed to d e f l e c t the i n c i d e n t protons through 35° were c a l c u l a t e d f o r each of the experimental e n e r g i e s and are t a b u l a t e d i n Table 3.2, Secondly, protons which had s u f f e r e d energy l o s s i n the c o l l i m a t o r were d e f l e c t e d a t angles g r e a t e r than 35°. They were e f f e c t i v e l y swept out of the beam by the f i e l d . T h i s r e s u l t e d i n the s e l e c t i o n o f a narrow momentum b i t e . In coming through the bending magnet the proton s p i n s were precessed not e x a c t l y to the l o n g i t u d i n a l , but by an amount w.r.t. the sideways d i r e c t i o n g i v e n by f = M 8 where = (-) Y M protons The r e s u l t i n g sideways component o f t h e • s p i n v e c t o r , given by C O S ( ! | J ) i s r e s p o n s i b l e i n p a r t f o r the c o n t r i b u t i o n o f the C v component of the tensor C . to the c r o s s e c t i o n sL a 3 i n 2.15. 3.5 Quadrupole and S t e e r i n g Magnets A v e r t i c a l s t e e r i n g magnet, two quadrupole f o c u s s i n g magnets and a h o r i z o n t a l s t e e r i n g magnet were pl a c e d i n s u c c e s s i o n down stream of the bending magnet. In t h i s experiment the quadrupole magnets were not used as the beam p r o p e r t i e s were found t o be.acceptable without 27 them. The s t e e r i n g magnets each have a po l e gap of 100 mm and a maximum r a t e d c u r r e n t o f ± 7A. They were used t o c e n t r e the beam on the t a r g e t . 3.6 Monitor 4CM8 Monitor 4CM8 was used to check beam focus and c e n t e r i n g approximately lm upstream from the t a r g e t . I t was permanently i n s t a l l e d i n the beam l i n e and evacuated by the beam l i n e vacuum system. The monitor c o n s i s t e d of an aluminum outer c a s i n g housing a m u l t i wire pro-p o r t i o n a l chamber. The monitor was swung i n t o the beam path to o b t a i n a beam p o f i l e , then swung a s i d e . 3.7 P o l a r i z e d T a r g e t C e l l The t a r g e t c e l l was a p o l y t e t r a f l u o r o e t h y l e n e ( t e f l o n ) c y l i n d e r measuring 2.4 cm (length) by 1,5 cm (diameter). The t a r g e t composition was, by weight, 95% C 4 H ^ Q O and 5% E^O doped with 2 - e t h y l - 2 h y d r o x y b u t y r i c a c i d - Cr(v) Complex ( i . e . EHBA-Cr(v)). The t a r g e t m a t e r i a l was f r o z e n i n t o beads 1.0 to 1.7 mm i n diameter. The d e n s i t y of f r e e hydrogen i n the t a r g e t , determined from three weighings o f the t a r g e t m a t e r i a l and measurements 3 of the c e l l volume, was 0.0765±.0003 gm/cm . On o p e r a t i n g , 3 temperature o f 500 m i l l i k e l v i n was produced by a He ev a p o r a t i o n r e f r i g e r a t o r , designed and p r e v i o u s l y used a t L i v e r p o o l U n i v e r s i t y . Proton s p i n alignment was achieved i n a f i e l d o f 25 KG produced by a p a i r o f superconducting EXTERNAL PHASE RF INPUT (lOOmvJ^ TO CRYOSTAT ft NMR COIL A ATTENUATOR S SPLITTER G RF GAIN LFG LOW FREQUENCY GAIN C I ID O d O m g > CD > O O F I L L I N G T U B E FIGURE 3.4 SCHEMATIC OF THE CRYOSTAT ,30 Helmholtz c o i l s . A K l y s t r o n operated i n the neighbourhood of 71 GHz enhanced t a r g e t p o l a r i z a t i o n w i t h s p i n f l i p s accomplished by a d j u s t i n g the frequency. 3.8 NMR System A saddle c o i l wrapped around a t e f l o n form c o n t a i n i n g the t a r g e t c e l l formed the i n d u c t i v e element of a s e r i e s tuned LRC c i r c u i t . The saddle c o i l was c o n s t r u c t e d from 0.05 mm t h i c k copper f o i l and s a t i s f i e d the requirement t h a t the r f f i e l d d i r e c t i o n should be p e r p e n d i c u l a r to the s t a t i c f i e l d d i r e c t i o n . Radio frequency power was s u p p l i e d by a Rockland model 5600 frequency s y n t h e s i z e r . The r f d r i v e r was programmed to sample 3 01 f r e q u e n c i e s c e n t e r e d on 108.8 MHz i n steps of 1 KHz. The main d e t e c t o r employed a balanced r i n g modulator (BRM) o p e r a t i n g as a f u l l wave synchronous r e c t i f i e r 1 1 to reduce measurement i n a c c u r a c i e s but a c o n v e n t i o n a l magnitude d e t e c t o r was i n c o r p o r a t e d to f a c i l i t a t e t u n i n g . Q u a l i t y f a c t o r curves sampled s l i g h t l y o f f resonance (by a d j u s t i n g the s t a t i c magnetic f i e l d ) were d i g i t i z e d and s t o r e d i n micro p r o c e s s o r memory. NMR s i g n a l r e c e i v e d d u r i n g the experiment was d i g i t i z e d and s u b t r a c t e d from the s t o r e d Q curve by the micro p r o c e s s o r , then i n t e g r a t e d and w r i t t e n to an output r e g i s t e r . The q u a n t i t y recorded onto magnetic tape was thus the p o l a r i z a t i o n i n t e g r a l i n u n i t s of v o l t s - KHz, where the p o l a r i z a t i o n i n t e g r a l i s the area between the two curves as shown i n F i g u r e 5.1 a f t e r ) the enhanced 31 s i g n a l has been normalized to the Q curve a t two s e l e c t e d p o i n t s as i l l u s t r a t e d i n 5.1. 32 Chapter IV THE POLARIZATION MONITOR The p o l a r i z a t i o n monitor was a two arm t e l e s c o p e . Each arm c o n s i s t e d of a m u l t i w i r e p r o p o r t i o n a l chamber and a p l a s t i c s c i n t i l l a t o r . The p o s i t i o n s o f the chambers and s c i n t i l l a t o r s were a c c u r a t e l y measured to ± 2 mm. The forward arm subtended angles of 21.3° to 40.9° a t the t a r g e t c e n t e r . The r e c o i l arm viewed the t a r g e t i n the angular range of 43.8° to 63.1°. The t a r g e t c a n n i s t e r i n c l u d i n g Helmholtz c o i l s and t a r g e t c e l l was p h y s i c a l l y r o t a t e d through 12° i n the h o r i z o n t a l plane. T h i s r o t a t i o n was necessary i n order to monitor C t- i n the r e g i o n of g r e a t e s t s e n s i t i v i t y (45° cm.) J _ l l j and where i t i s b e s t known from phase s h i f t a n a l y s i s . The p h y s i c a l c o n s t r u c t i o n of the apparatus allowed un-impeded beam e x i t o n l y to 50° on e i t h e r s i d e of an a x i s d e f i n e d by the c o i l s . D e t e c t i o n of r e c o i l events out p a s t 60° was accomplished by e f f e c t i v e l y o f s e t t i n g the i n c i d e n t beam d i r e c t i o n by the 12° r o t a t i o n of the t a r g e t assembly. 33 LL) SCINTILLATOR FIGURE 4.1 POLARIZATION MONITOR 34 Consequently the 12° r o t a t i o n of the t a r g e t i n t r o d u c e d the c o n t r i b u t i o n of an a d d i t i o n a l term i n C _ and one.in C to the c r o s s e c t i o n of 2.15. A l s o sL ss the r o t a t i o n l i m i t e d the experiment to a s i n g l e co-i n c i d e n c e t e l e s c o p e r a t h e r than a symmetric f o u r armed apparatus. The MWPC's had outer dimensions of 28.25 cm square. A imagic gas mixture of 69.7% argon, 30% isobutane and .3% f r e o n composition was c o n t i n u a l l y c i r c u l a t e d through each chamber. A plane of 1000 cathode wires spaced 2mm ap a r t were maintained a t a v o l t a g e of 4 kv. A g r i d o f h o r i z o n t a l and v e r t i c a l sense wires on e i t h e r s i d e of the cathode d e t e c t e d i o n t r a c k s produced when a charged p a r t i c l e passed through. Each sense plane determined two c o o r d i n a t e s . The p u l s e s a r r i v i n g a t both ends of a sense wire due to a p a r t i c l e t r a c k anywhere along the wire were recorded. The fou r p u l s e s from each chamber were de s c r i m i n a t e d i n an ECL to NIM u n i t and a d i g i t a l p u l s e t r a n s m i t t e d to a time to d i g i t a l c o n v e r t e r . Each chamber had a s c i n t i l l a t o r p l a c e d d i r e c t l y behind i t . The s c i n t i l l a t o r were i d e n t i c a l w i t h dimensions of 16.1 cm x 14.5 cm x 2 mm. The chambers were t r i g g e r e d on a c o i n c i d e n c e between the three beam d e f i n i n g s c i n t i l l a t o r s w ith the forward and r e c o i l s c i n t i l l a t o r s of the p o l a r i z a t i o n monitor. F o r m a l l y the t r i g g e r was c o n s t r u c t e d from SI.s2.S3.L.R 35 SI UP SI DOWN S2 LEFT S2 RIGHT THREE FIGURE 4 .2 BEAM DEFINING SCINTILLATORS 43 ns. TRIGGER 0 F I G U R E 4 . 3 E L E C T R O N I C L O G I C D I A G R A M 37 X 2 TO ECL TO NIM W RECEIVER HORIZONTAL AND VERTICAL SENSE WIRES AND CATHODE GAS H.V GAS FIGURE 4 .4 DIAGRAM OF A DELAY-LINE MULT I WIRE PROPORTIONAL CHAMBER 38 where s i i s an up down s p l i t s c i n t i l l a t o r , s2 i s a l e f t r i g h t s p l i t s c i n t i l l a t o r , s3 i s a 1 cm i n d i a m e t e r c i r c u l a r s c i n t i l l a t o r and L,R denote the s c i n t i l l a t o r s i n t h e l e f t and r i g h t arms o f t h e m o n i t o r . T h i s t r i g g e r p r o v i d e d a common s t a r t p u l s e f o r a l l t h e TDC*s w i t h t h e s t o p p u l s e s coming from each o f t h e chamber c o o r d i n a t e s d e s c r i b e d above. The absence o f a s t o p p u l s e from any-chamber c o o r d i n a t e prompted an o v e r f l o w b i t t o be s e t i n the a p p r o p r i a t e TDC. T h i s i n d i c a t o r was used t o dete r m i n e t h e d e t e c t i o n e f f i c i e n c y o f t h e chambers w h i c h was d e f i n e d w i t h r e s p e c t t o t h e t r i g g e r ( i . e . the t r i g g e r was assumed t o be 100% e f f i c i e n t ) . The d e t e c t i o n e f f i c i e n c y o f each chamber was t a k e n as t h e f r a c t i o n o f e v e n t s i n w h i c h p u l s e s a r r i v e d from a l l f o u r c o o r d i n a t e s o f t h a t chamber. The d e t e c t i o n e f f i c i e n c y o f t h e m o n i t o r was t a k e n t o be the f r a c t i o n o f ev e n t s i n w h i c h t h e two chambers d e t e c t e d i n c o i n c i d e n c e an ev e n t t h a t had produced a t r i g g e r . The s p l i t s c i n t i l l a t o r s p r o v i d e d a v e r y s e n s i t i v e m o n i t o r o f beam s t a b i l i t y . By m o n i t o r i n g t h e count r a t i o i n a l l f o u r q u a d r a n t s a s l i g h t s h i f t i n beam p o s i t i o n c o u l d be d e t e c t e d . As a p r e c a u t i o n t o guard a g a i n s t f a l s e asymmetries t h e beam s p i n was f l i p p e d r e g u l a r l y . The count r a t i o i n t h e s c i n t i l l a t o r s i n d i c a t e d no o b s e r v a b l e s h i f t i n t h e beam when t h e p o l a r i z a t i o n was f l i p p e d . 39 Chapter V DATA ACQUISITION A b r i e f d e s c r i p t i o n o f the procedure f o r r e c o r d i n g data and the p r e p a r a t i o n i n v o l v e d i s giv e n i n t h i s chapter. The d u r a t i o n of the experiment was one week. The p o l a r i z e d i o n souce s p i n c o n t r o l l e r was operated i n automatic s p i n mode g i v i n g f i v e minutes each- of s p i n up and s p i n down beam f o l l o w e d by one minute of u n p o l a r i z e d beam. The i n t e n s i t y o f the e x t r a c t e d beam was monitored by the up-stream p o l a r i m e t e r . A l l s i n g l e s r a t e s and u s e f u l c o i n c i d e n c e s were s c a l e d . S c a l a r i n f o r m a t i o n along w i t h TDC data was read through the CAMAC dataway by a PDP-11/34 computer and w r i t t e n onto magnetic tape. A computer generated "busy" or a " s p i n busy" from the s p i n c o n t r o l l e r formed a master i n h i b i t , t h a t i s the s c a l e r s and TDC's were d i s a b l e d w h i l e the computer was busy p r o c e s s i n g an event or when the beam p o l a r i z a t i o n was being r e v e r s e d . Histograms of each m u l t i w i r e p r o p o r t i o n a l chamber were c o n s t r u c t e d o n - l i n e and d i s p l a y e d on a CRT. Run times averaged 2 hours f o r a t y p i c a l accumulation o f 50,000 events. 40 The t a r g e t p o l a r i z a t i o n was r e v e r s e d p e r i o d i c a l l y by a d j u s t i n g the k l y s t r o n frequency. One h a l f hour was r e q u i r e d f o r the p o l a r i z a t i o n s i g n a l to grow to i t s maximum. P o l a r i o d photographs of the scope t r a c e showing the NMR en-hanced s i g n a l were made a f t e r each adjustment (see F i g u r e 5.1) . The monitor 4CM8 was used t o determine beam p r o f i l e s a f t e r each new beam tune. P o l a r o i d f i l m exposures of the beam spot s i z e a t t a r g e t entrance and e x i t were recorded. S t e e r i n g of the beam onto the t a r g e t was achieved u s i n g the v e r t i c a l and h o r i z o n t a l s t e e r i n g magnets. The beam was swept through the t a r g e t independently i n the v e r t i c a l and h o r i z o n t a l d i r e c t i o n s to ensure unbiased c e n t e r i n g . C e n t e r i n g o f the beam on the t a r g e t was observed as a two percent a t t e n u a t i o n i n the t r a n s m i t t e d beam. A p l o t of the a t t e n u a t i o n d u r i n g one such v e r t i c a l scan i s shown i n F i g u r e 5.2. Data were taken a t s i x e x t r a c t i o n e n e r g i e s . Energy l o s s e s i n t r a n s m i s s i o n down the beam l i n e were c a l c u l a t e d . The e n e r g i e s of i n t e r e s t were those a t the p o l a r i m e t e r and t a r g e t c e n t r e , which were needed f o r kinematic c a l c u l a t i o n s , and the energy of the beam a f t e r p a s s i n g the p o l a r i m e t e r , which was needed t o c a l c u l a t e p r e c e s s i o n and bend angles i n the s o l e n o i d and bending magnet. There were no beam l i n e elements d i r e c t l y i n the path of the beam i n the i n t e r v a l from the p o l a r i m e t e r t o the bending magnet, hence the bend 41 CO o > FREQUENCY (khz) FIGURE 5.1 S C O P E TRACE SHOWING NMR SIGNAL (REPRODUCED FROM POLAROID PHOTO) 42 FIGURE 5.2 ATTENUATION IN BEAM TRANSMITTED THROUGH THE TARGET 43 and p r e c e s s i o n angles were c a l c u l a t e d a t the same energy. These t h r e e beam e n e r g i e s along w i t h the energy of the beam at e x t r a c t i o n are giv e n i n Table 5.1. Table 5.1 P r e c i s e Beam Energies i n MeV (at e x t r a c t i o n , a t the p o l a r i m e t e r , a f t e r the p o l a r i m e t e r , and a t the t a r g e t centre) Energy a t E x t r a c t i o n Energy a t the P o l a r i m e t e r Energy A f t e r P o l a r i m e t e r Energy a t T a r g e t 520.7 520.3 519.9 516.5 501.4 501.0 500.7 497.1 460.2 459.8 ' 459.4 455.8 424.1 423.7 423.3 419.5 330.6 330.1 329.6 325.4 209.8 209.0 208.2 202.7 45 Chapter VI DATA ANALYSIS A d e s c r i p t i o n of t r a c k r e c o n s t r u c t i o n f o r protons s c a t t e r e d i n t o the t a r g e t p o l a r i z a t i o n monitor i s g i v e n a t the beginning of t h i s c h a p ter. T h i s chamber informa-t i o n was used to separate e l a s t i c hydrogen events from non e l a s t i c two body events i n non hydrogenous m a t e r i a l by a c o p l a n a r i t y t e s t . The a b s o l u t e t a r g e t p o l a r i z a -t i o n was obtained from the e x t r a c t e d e l a s t i c data u s i n g a non l i n e a r f i t t i n g r o u t i n e w i t h c h i - s q u a r e d minimiza-t i o n . 6.1 R e c o n s t r u c t i o n of Proton Tracks T r a j e c t o r i e s of s c a t t e r e d protons were obtained by f i r s t t r a n s l a t i n g the time to d i g i t a l c o n v e r t e r i n -formation i n t o d i s t a n c e s w i t h r e s p e c t to a c a r t e s i a n c oor-d i n a t e system. S c a t t e r i n g angles were then computed with r e s p e c t t o t h i s system, and c o r r e c t e d f o r d e v i a t i o n due to the magnetic f i e l d surrounding the p o l a r i z e d t a r g e t . The t r a n s l a t i o n of TDC i n f o r m a t i o n to p a r t i c l e t r a j e c t o r i e s , was done as f o l l o w s . The chambers were of 46 the d e l a y l i n e read out type. That i s the passage of a charged p a r t i c l e through the chamber induced a pu l s e on the de l a y l i n e sense wire which t r a v e l l e d outward i n both d i r e c t i o n s . The TDC d i g i t i z e d a time p r o p o r t i o n a l to the time i n t e r v a l between the a r r i v a l of these p u l s e s a t e i t h e r end and a common a r b i t r a r y s t a r t p u l s e taken to be the t r i g g e r s c i n t i l l a t o r p u l s e . I t i s the d i f f e r e n c e i n the a r r i v a l time between p u l s e s a r r i v i n g a t o p p o s i t e ends of a chamber t h a t determines the l o c a t i o n o f the t r a c k i n the chamber. Hence common o f f s e t s i n the time a t both ends c a n c e l i n the computa-t i o n . L o c a t i o n of the t r a c k w i t h r e s p e c t to the c a r t e s i a n c o o r d i n a t e system r e q u i r e s d e t e r m i n a t i o n of the correspondence between a f i x e d TDC channel and the co o r d i n a t e system as w e l l as the c a l i b r a t i o n of TDC channel numbers w i t h r e s p e c t to d i s t a n c e . Both the c a l i b r a t i o n and r e f e r e n c e p o i n t were obt a i n e d from histograms o f the time d i f f e r e n c e s mentioned above. The c a l i b r a t i o n i s determined from the d i g i t a l width o f the d i s t r i b u t i o n w h i c h . i s j u s t twice the p r o j e c t e d width o f the s c i n t i l l a t o r onto the chamber as seen a t the t a r g e t . The histogrammed d i s t r i b u t i o n i s cent e r e d on the median l i n e of the s c i n t i l l a t o r , whose c o o r d i n a t e s were measured. Histograms were c o n s t r u c t e d f o r both the h o r i z o n t a l and v e r t i c a l planes of each chamber. Thus from the chamber i n f o r m a t i o n the s i x FIGURE 6-1 TDC DIFFERENCE HISTOGRAM 0.0 • !« 2«o.o -2»«.o i ' 2??.0 I « I « 7 ?lt>.0 I <5 » S X I S X 210.0 I 2«« » 2 « « < « « « « 2,m.o i 2 « « x ^ » « 2xixxx < « »; 198.0 i «««» « « « l»»<«« « « 192.0 I 51 7 X X X X X X i i*itn »««««<« »»« ««« i a « . o i «« ««««««»« « «««»««« xxx xxx ix 180.0 - X X 3 X X X X X X I X 7 S KB x x x x x x i ? x x x x > i x ( xxxxxxx I I i 7 a . o i x x x x x x x x x x x n « x x >»<(<<<ii«<<(ii x x x x x x x < » ; 1*8.0 I S X X I X X X X X X X X X X X X X B X X X X X X X K X X X X X I X I I I X X X X X X IXX» 1*2.0 I X X X X X X X X X X X H X X X X K X X X X X I X X X X X X « X I X « X X X X X X X XXXI 1 S 6 . 0 I X X X X X X X X X X X X X X X X X X X X X X X X I X X X X X X X X X X X X X X X X X X X X X X • 1S0.0 I X X X X X X X X X X X X X X X X X I X X X X X X X X X X X X X X X I X X X X X X X X X X X X X I taa.o I x x x x x x x x x « x x x « x x x x x x x x x x x x x x x x x x x x x x « x x x x x x « x x x N 1JB.0 I x x x x x x « x x x x x x x x x x « x x x x x x x x x x x x x x x x x « x « x x x x x « x x « U2.0 I xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx C l i f t . 0 I X X X X X X X X X X I I X X X X X X X X X X X I X I X X X X X « « « » X X X « X X X I X X X X 0 120.0 - X X X X X X X X X X X X X X I X X I X I X X X X X X X X X X X X I X X X X X X X X X X X X K X l U 114.0 I X X X X X X X X X X X X X X X X X x X X X X X X X X X X X X X X X X I I X X X X X X X X X X X X N I0S.O I X X X X X X X X X X X I X X X X X X X X X X X X X X X X I I X I X X X X X X X X X X X X X X X X f 10?.0 I X X X X X X X X X X I X X X X X X X X X X X X X X X X X X X X X I X X X X X X X X I X X X X X X S 9 6 . 0 I X X X X I X I X X X X X I X I X I X X X X X X X I X X X I I X I X X X I X I X X X X X X X X X X 90.0 I X X X X X X I X X X X X X I X X I X X X U X U X X X X X X X X X X X X X X X I U X X X X X a«.o I xxxxxxxxxxxxxxxxxxxxxxixxxxixxxxxxxmxxxxxxxxn 78.o I xxxxxixxixxxxixxxxxixxxiixxxxxxixxixxaxxxxxxxxxx 72.0 I X X X X X X X X X X X X X X X X X K X X X X I X I X X X X X X X X X X X X X X I I X X I X X X I kb.O I I X X X X X X X X X X I X X X X X X X X X X X X X I X X I X X X X X X X X I I I I X X I X I X X kO.0 • I X I X X X I X X X X X X X X X X X X X X X X X X I I X X X I X X I I X X X X X X X X I X I X I M.O I X X X X X X X X I I I I X I X X I X X I X I X X I I X X X X I X I X X X I X X X X X X X X X X I *8.0 I X X X X I I X X X X X X X X X X X I X X I X X X X X I X X X I I X X X X X X X X X X X I X I I I • 2.0 I I X X X X X I X X X X X X X X X X X X I I X X I X X I I X X X X I I X X I X X X X X X I X X X X 16.0 I X X X X X I X I X X X X X X X X X X X X X X X X X I I X X I X X I C X X X I X I X X I X X X X X 10.0 I X X X X X X X X X X X I X X X X X X X X X X X X X I X X X X I X X X X I X X X I X X X X X X X I 2 a . o • (ixxxxxixxxxxxxixixxxixxxxxxxxxxxixxxxiixxxiixxx 18.0 I S X K X X X X X X I X X I X X X X I X X X X I X X X X X X I X X X X X X X X X X X X X X I I I X X 12.0 I X X X « X X X X X X X X X I X I < X X X I X X X X X X X X I X X X X X X X X X X X X < X I I X X X 6.0 I 2 2 I I I X X X X X X X X I X X X X X X X X X X X I X X X X X X X I X X X X X I I I X X X I 2 2 2 TDC SPECTRUM (DIFFERENCE) —1 48 c a r t e s i a n coordinates, of each proton p a i r t r i g g e r i n g the monitor were c a l c u l a t e d . The proton t r a c k s were d e f l e c t e d up to 6° by the 25 KG f i e l d r e q u i r e d f o r the p o l a r i z e d t a r g e t . The amount of d e f l e c t i o n f o r each t r a j e c t o r y was c a l c u l a t e d by an i t e r a t i v e procedure and the c o r r e c t i o n s a p p l i e d to o b t a i n the t r u e s c a t t e r i n g angle. C o r r e c t i o n s were a p p l i e d s i m i l a r l y to the i n c d i e n t t r a j e c t o r i e s . In the i t e r a t i v e procedure, as a f i r s t a p p r o x i -mation, the s c a t t e r i n g angle was chosen to be the angle a t which the p a r t i c l e had been d e t e c t e d . The momentum P of the s c a t t e r e d protons a t t h i s angle was then c a l c u -l a t e d u s i n g the f o l l o w i n g e q u a t i o n ( d e r i v e d i n Appendix A) which a p p l i e s t o the case of a t a r g e t p a r t i c l e a t r e s t . P(E 2+m 2+2E ¥m -P 2 c o s 2 9 ) + (-2PTm2 cosO-2E Tm P T cose) = 0 I p l p l I p I p l (6.1) where E J ' P J are the i n c i d e n t proton energy and momentum r e s p e c t i v e l y and m^ i s the proton mass. In each i t e r a t i o n the c a l c u l a t e d momentum was c o r r e c t e d f o r magnetic de-f l e c t i o n , u s i n g the procedure d e s c r i b e d i n the f o l l o w i n g , and the computed angle compared w i t h the observed angle. The t r i a l s c a t t e r i n g angle was then v a r i e d and the. i t e r a t i o n continued t i l l the d i f f e r e n c e between the c a l c u l a t e d and observed angle was l e s s than 5 m i l l i r a d i a n s . The r e q u i r e d p r o p e r t i e s o f the magnetic f i e l d were c a l c u l a t e d u s i n g the computer s i m u l a t i o n r o u t i n e SOL RAY, a v a i l a b l e a t TRIUMF. SOL RAY simulates the magnet f i e l d produced by the Helmholtz c o i l s of the p o l a r i z e d t a r g e t . The f i e l d i s c a l c u l a t e d u s i n g the B i o t - S a v a r t law f o r a c u r r e n t loop, which i s a p p r o p r i a t e s i n c e the f i e l d o f the Helmholtz c o i l s i s the super-p o s i t i o n o f f i e l d s from two such c u r r e n t l o o p s . In the s i m u l a t i o n , protons of d e f i n i t e momentum were generated a t the t a r g e t c e n t r e . The t r a j e c t o r y o f each proton was f o l l o w e d through the magnetic f i e l d w i t h the c o o r d i n a t e s of the proton being output i n one cen t i m e t e r i n t e r v a l s over the f l i g h t path. From these c o o r d i n a t e s the l i n e i n t e g r a l of the t r a n s v e r s e component of the magnetic f i e l d was computed over the angular range of the monitor. The angular dependence of the i n t e g r a l was found to be a l i n e a r f u n c t i o n of the angle i n the r e g i o n of the monitor. As shown i n F i g u r e 6.2 the i n t e g r a l c o u l d be parameterized by t B x cU = 3.71(9) - 15.15 (6.2) where the angle 9 i s measured i n degrees w i t h r e s p e c t to the d i r e c t i o n of B. At a f i x e d angle the v a r i a t i o n of the l i n e i n t e g r a l over the range of momenta r e l a t i v e to t h i s experiment was l e s s than 1%. Having determined the v a l u e s o f the l i n e i n t e g r a l the magnetic d e f l e c t i o n s were c a l c u l a t e d u s i n g the e x p r e s s i o n found i n S e c t i o n 3.4. FIGURE 6.2 LINE INTEGRAL OF THE B FIELD 36 38 4 0 42 44 46 48 50 LAB DEGREES 6.2 Background Sub t r a c t i o n s The opening angle and the coplanar angle, d e f i n e d as the angle between the r e c o i l nucleon momentum and the normal to the s c a t t e r i n g plane where the s c a t t e r -i n g plane i s t h a t plane which c o n t a i n s the momentum v e c t o r of both i n c i d e n t and s c a t t e r e d p a r t i c l e , were c o n s t r u c t e d f o r each event t h a t t r i g g e r e d the monitor. Momentum c o n s e r v a t i o n r e q u i r e s t h a t f o r proton proton e l a s t i c s c a t t e r i n g w i t h the t a r g e t p a r t i c l e a t r e s t the r e c o i l nucleon's momentum must l i e i n the s c a t t e r i n g p l a n e . Thus the copla n a r angle must be 90°. T h i s r e s t r i c t i o n does not apply t o a proton s c a t t e r i n g from a pro t o n i n a carbon nucleus whose c o n s t i t u e n t nucleons possess fermi motion. For the carbon events the ferm i momentum of the nucleons was c o n s i s t e n t w i t h a gaussian d i s t r i b u -t i o n c e n tered a t approximately 170 meV/C i n each of the three momentum c o o r d i n a t e s . As a r e s u l t the p r o b a b i l i t y of the s t r u c k nucleon having zero component of ferm i momentum i n the normal d i r e c t i o n was s m a l l . Histograms o f the coplanar and opening angle were made f o r each run i n beam s p i n "up", "down" and " o f f " c o n f i g u r a t i o n s . The copla n a r and opening angle histograms c o n t a i n e d e s s e n t i a l l y the same i n f o r m a t i o n , however the coplanar histograms were found t o be more s e n s i t i v e to geometry and thus p r o v i d e d b e t t e r r e s o l u t i o n . 52 For t h a t reason c o p l a n a r i t y histograms were used f o r a l l c a l c u l a t i o n s . A sample coplanar histogram i s shown i n F i g u r e 6.3. The p a r a m e t e r i z a t i o n i s a gaussian e l a s t i c peak and a q u a d r a t i c background. The f i v e parameter f i t used was N = P l + P3(8-P2) 2 + P4 EXP(-.5((0-P2)/P5) 2) (6.3) where N i s the number of counts. Though the s c a l e i n F i g u r e 6.3 i s i n degrees the data were a c t u a l l y binned i n h a l f degree s t e p s . Two c o n s i s t e n t f e a t u r e s o f these d i s t r i b u t i o n s were the p o i n t s above the gaussian f i t a t approximately ± 6° from the c e n t r e and the p o i n t s above the peak near the c e n t r e . These f e a t u r e s were p r e s e n t i n a l l s p i n c o n f i g u r a t i o n s . The p o i n t s l y i n g o f f the gaussian t a i l s were i n f a c t an i n d i c a t o r of good r e s o l u t i o n i n the chambers. These events have been assumed t o be hydrogen s i g n a l t h a t has been broadened by m u l t i p l e s c a t t e r i n g and energy l o s s e s . The chamber r e s o l u t i o n was adequate to d i s t i n g u i s h these events from the background. The p o i n t s l y i n g above the peak were a l s o assumed to be hydrogen s i g n a l s i n c e , as e x p l a i n e d e a r l i e r , the fermi momentum i n carbon would not favour events to be peaked i n t h i s r e g i o n . In the f i n a l a n a l y s i s the q u a d r a t i c background was i n t e g r a t e d a n a l y t i c a l l y u s i n g the parameter FIGURE 6 - 3 COPLANAR HISTOGRAM 54 v a l u e s PL, P2 and P3 as determined by the f i t t i n g procedure. The:,number o f protons e l a s t i c a l l y s c a t t e r e d i n t o the monitor was determined by n u m e r i c a l l y summing a l l the events l y i n g above the f i t t e d background. In t h i s way the m u l t i p l e s c a t t e r i n g events and the events l y i n g above the g a u s s i a n peak were i n c l u d e d i n the e l a s t i c hydrogen count. The r a t i o of hydrogen to background s i g n a l i n the monitor was t a b u l a t e d f o r each run i n the c o n f i g u r a -t i o n s of sp i n s p a r a l l e l and s p i n s a n t i p a r a l l e l . The run averaged v a l u e s of these r a t i o s a t each energy are tabu-l a t e d i n Table 6.1. The hydrogen to background r a t i o s r e f l e c t the f a c t t h a t the hydrogen i n the t a r g e t i s p o l a r i z e d but the background i s not. The terms p a r a l l e l and a n t i p a r a l l e l are o n l y nominal s i n c e n e i t h e r the i n -c i d e n t protons nor the t a r g e t protons were p o l a r i z e d e x a c t l y along the l o n g i t u d i n a l . The s p i n p r e c e s s i o n i n the bending magnet was d i f f e r e n t from 9 0° and the t a r g e t was r o t a t e d through 12°. The p o l a r i z a t i o n of the i n c i -dent protons i n the l o n g i t u d i n a l d i r e c t i o n depended f i r s t l y on the value of the beam p o l a r i z a t i o n b e f o r e p r e c e s s i o n , and secondly on the amount of p r e c e s s i o n i n the bending magnet which was energy dependent. Hence because of the v a r i a t i o n i n the l o n g i t u d i n a l component of beam p o l a r i z a t i o n the r a t i o of hydrogen to background s i g n a l v a r i e d from run to run and had a d e f i n i t e .'energy 55 Table 6,1 Run Averaged Hydrogen To Background R a t i o s Energy Spins P a r a l l e l .' " Spins.'.Anti P a r a l l e l 516.5 2.373: ± .082 1.960 ± .114 497.1 i- 2.All ± .098 1.966 ± .130 455.8 2.487 ± .140 2.020 ± .108 419.5 2.594 ± .255 2.026 ± .064 325.4 3.003 ± .276 1.851 ± .154 202.7 4.942 ± .828 2.303 ± .258 Table P r e c e s s i o n Angle i n Energy 519.9 500.1 459.4 423.3 329.6 208.2 6.2 the Bending Magnet (Degrees) 97.52 96.24 93.47 91.06 84.79 76.68 56 dependence. The r a t i o s were used t o check the c o n s i s t e n c y of the data and the run averaged val u e s are i n c l u d e d here to demonstrate i n a syste m a t i c way t h a t the monitor was s e n s i t i v e t o the p o l a r i z a t i o n of hydrogen i n the t a r g e t . 6.3 The F i t t i n g Equation Using the d e n s i t y matrix formalism d e s c r i b e d i n S e c t i o n 2.1 the number of protons e l a s t i c a l l y s c a t t e r e d i n t o the monitor f o r beam p o l a r i z a t i o n and t a r g e t p o l a r i z a t i o n P T i s I~ +=I o|l+[C L Lsin0p)cos(12°) - C s s cos (^ ) s i n (12°)^ - C S L cos (i|)+12°) ] P* P+} (6.4) T h i s equation was f i t t e d by a non l i n e a r c h i - s q u a r e d m i n i m i z a t i o n procedure t o determine the t a r g e t p o l a r i -z a t i o n . I ^ * i s the number of protons e l a s t i c a l l y + s c a t t e r e d i n t o the monitor f o r beam p o l a r i z a t i o n P^ and t a r g e t p o l a r i z a t i o n P+ . The s u p e r s c r i p t s + and -i n d i c a t e p o s i t i v e and negative s p i n p r o j e c t i o n s along the l o n g i t u d i n a l d i r e c t i o n . I i s p r o p o r t i o n a l to the u n p o l a r i z e d r a t e i n the monitor as d e s c r i b e d i n the fo l l o w i n g . C T T , C c and C C T are the s p i n c o r r e l a t i o n parameters whose r e l a t i o n t o the amplitudes of the . 7 s c a t t e r i n g m a t r i x i s d e s c r i b e d by H o s h i z a k i . The sub-s c r i p t s S and L r e f e r t o the conventions of F i g u r e 2.1. The c o n t r i b u t i o n s of the C O T and C c c components a r i s e 57 from the 12° r o t a t i o n of the t a r g e t and the p r e c e s s i o n i n the bending magnet 4B;3. Values o f i f c a l c u l a t e d f o r a 35° bend a t the p r e c i s e beam energies are t a b u l a t e d i n Table 6.2. The val u e s of I'-* i n p u t i n t o eq. 6.4 were the e l a s t i c hydrogen count o b t a i n e d from the cop l a n a r r e c o n s t r u c t i o n s , a f t e r c o r r e c t i o n f o r the d e t e c t i o n e f f i c i e n c y of the monitor by the method o u t l i n e d i n Chapter IV. The c o r r e c t i o n f o r monitor e f f i c i e n c y was c a l c u l a t e d on a run to run b a s i s and a p p l i e d to the hydrogen count., The monitor e f f i c i e n c y ranged from 95° to 99° throughout the experiment. The u n p o l a r i z e d count I i s the number t h a t would have been e l a s t i c a l l y s c a t t e r e d i n t o the monitor f o r an u n p o l a r i z e d beam and u n p o l a r i z e d t a r g e t . IQ de-pends on the geometry of the monitor and on the i n c i d e n t r a t e . I t was w r i t t e n as I Q = (KK) (N'$) (6.5) where KK i s a co n s t a n t and N<j> i s the number i n the i n c i d e n t beam as d e f i n e d by the three beam d e f i n i n g c o u n t e r s . The c o n s t a n t KK r e p r e s e n t s the d i f f e r e n t i a l c r o s s e c t i o n f o r u n p o l a r i z e d s c a t t e r i n g i n t e g r a t e d over the s o l i d angle acceptance o f the monitor. I t was l e f t as a f r e e parameter o f the f i t . 58 The s p i n parameter v a l u e s a t the p r e c i s e experimental e n e r g i e s and angles were ob t a i n e d from David Bugg's phase s h i f t a n a l y s i s t h a t makes use of the most r e c e n t nucleon-nucleon data from Argonne and 12 Geneva. Accuarate v a l u e s o f the beam p o l a r i z a t i o n were c a l c u l a t e d from the beam l i n e p o l a r i m e t e r u s i n g the method d e s c r i b e d i n S e c t i o n 3.1. Values o f the p o l a r i z a t i o n parameter a t 26° were a l s o o b t a i n e d from Bugg's phase s h i f t a n a l y s i s and are giv e n i n Table 6.3. T y p i c a l v a l u e s of the beam p o l a r i z a t i o n were 62°. To o b t a i n the t a r g e t p o l a r i z a t i o n , e q u a t i o n 6.4 was f i t t e d w i t h a l l data a v a i l a b l e a t each energy i n the f o u r s p i n c o n f i g u r a t i o n s . To a l l o w f o r a v a r i a t i o n i n t a r g e t p o l a r i z a t i o n from run to run, i t was w r i t t e n as ± + -5 P T = x 10 x NMR enhanced s i g n a l (6.6) where C i s a con s t a n t . Although C + and C~ are the same q u a n t i t y the t a r g e t s p i n up data was f i t s e p a r a t e l y from the t a r g e t s p i n down data as a systematic check and l a t e r recombined. W r i t t e n i n t h i s way the c o n s t a n t taken t o -gether w i t h the s c a l i n g f a c t o r i s j u s t P c x i n 5 •_ thermal (6.7) |NMR thermal s i g n a l C was a f r e e parameter o f the f i t and t h e r e f o r e was determined from the s c a t t e r i n g d a t a . T h i s measurement of 59 Table 6.3 Values of the P o l a r i z a t i o n Parameter a t 26° Lab Energy P (26°) 520,3 0.3992 ± .0077 501.0 0.3970 ± .0067 459.8 0.3670 ± .0064 423.7 0.3535 ± .0064 330.1 0.3314 ±.'.0036 209.0 0.2670 ± .0078 60 t a r g e t p o l a r i z a t i o n thus was completely independent of P t h e r m a l ^ r i <^ the NMR thermal s i g n a l . These were the two q u a n t i t i e s t h a t c o n t r i b u t e d the g r e a t e s t u n c e r t a i n t y to the NMR measurement. The n u c l e a r s c a t t e r i n g measurement r e q u i r e d o n l y t h a t the NMR enhanced s i g n a l be p r o p o r t i o n a l to the n u c l e a r p o l a r i z a t i o n . I t was f r e e of the n o r m a l i z a t i o n u n c e r t a i n t i e s a s s o c i a t e d w i t h the NMR method of measurement. Hence the n u c l e a r s c a t t e r i n g data was used to d e t e r -mine the constant C which i s "the constant-., of - p r o p o r t i o n a l i t y r e l a t i n g the t a r g e t p o l a r i z a t i o n to the NMR s i g n a l when the t a r g e t p o l a r i z a t i o n i s enhanced by the microwave r a d i a t i o n . 6.4 E r r o r A n a l y s i s The monitor accepted events s c a t t e r e d i n t o a f i n i t e angular range. The s p i n c o r r e l a t i o n parameters are s e n s i t i v e to angle. The forward arm of the monitor subtended an angular range of 19.7° a t the t a r g e t . The s p i n parameters are not l i n e a r over t h i s angular range. For t h a t reason cu t s were made i n the o f f l i n e a n a l y s i s t o l i m i t the angular acceptance t o ± 3° of a va l u e chosen to o p t i m i z e d e t e c t i o n of the r e c o i l p a r t i c l e i n e l a s t i c two body events. T h i s optimum angle v a r i e d s l i g h t l y w i t h energy due to r e l a t i v i s -t i c e f f e c t s . For i n p u t i n t o eq. 6.4, the average v a l u e o f the s c a t t e r i n g angle f o r a l l events accepted was determined a t 61 each energy. T h i s was done by i n s e r t i n g counters i n t o the computing r o u t i n e to average s e p a r a t e l y the s c a t t e r i n g and r e c o i l angle. In the c e n t r e of mass r e f e r e n c e frame the opening angle must be 180°. Hence measurement of the r e c o i l angle g i v e s a second measure of the s c a t t e r i n g angle. The two v a l u e s thus o b t a i n e d were averaged to form the q u a n t i t y 0. The d i s p e r s i o n of the two v a l u e s about 8 i s t h e i r d i f f e r e n c e over /2 and the u n c e r t a i n t y i n 0 i s o n e - h a l f of the d i f f e r e n c e i n the measured v a l u e s . The composite q u a n t i t y A, d e f i n e d by A = C L L sin ( i f))cos(12°) - C s g cos ) s i n (12°) - C S L cos(^+12°) appearing i n eq. 6.4 was e v a l u a t e d a t the angle 0 f o r each experimental energy and t r e a t e d as a c o n s t a n t i n the f i t t i n g program. Denoted h e r e a f t e r by A, t h i s q u a n t i t y and the u n c e r t a i n t y i n i t was computed by r e r u n n i n g the phase s h i f t 12 program, each time u s i n g a d i f f e r e n t s e t of the e x i s t i n g world data as i n p u t . To ensure t h a t the e r r o r on A was not underestimated, the e r r o r was e v a l u a t e d from the maximum change i n A p l u s the s t a t i s t i c a l v a r i a t i o n s . There was a l s o a s m a l l c o n t r i b u t i o n to the u n c e r t a i n t y i n A due to the u n c e r t a i n t y i n 0. T h i s e r r o r , denoted S A ( 6 ) , i s j u s t the d i s p e r s i o n of A about A(0) f o r the two measure-ments of 0 and was added i n quadrature w i t h the e r r o r s from the phase s h i f t a n a l y s i s to o b t a i n an o v e r a l l u n c e r t a i n t y i n A g i v e n by SA. 62 The f i t t i n g procedure determined a v a l u e o f C + and C along w i t h t h e i r s t a t i s t i c a l e r r o r s a t each energy-except 497 meV. A t t h i s energy o n l y one run of data was a v a i l a b l e with t a r g e t s p i n "up" and hence o n l y C~ was + — determined. The weighted mean o f C and C was formed a t each energy u s i n g the e x p r e s s i o n (C / 6 C | ) j ~ i- i The r e s u l t was s i x valu e s o f C w i t h i t s e r r o r 6C, where SC i n c l u d e s s t a t i s t i c a l e r r o r s on C^ and the t o t a l e r r o r on A. F i n a l l y the weighted mean o f these s i x valu e s was c a l c u l a t e d to o b t a i n the o v e r a l l n o r m a l i z a t i o n c o n s t a n t C and i t s e r r o r . 63 Chapter V I I RESULTS AND CONCLUSION 7.1 R e s u l t s The r e s u l t s o f the n u c l e a r s c a t t e r i n g measurement are presented i n t h i s s e c t i o n . The f i n a l v a l u e s of A, 0 and 5A(0) are giv e n i n Table 7.1. A i s the q u a n t i t y t h a t was d e f i n e d i n S e c t i o n 6.4 and obt a i n e d from p h a s e - s h i f t a n a l y s i s . 0 i s the average s c a t t e r i n g angle f o r a l l events accepted i n the monitor as d e s c r i b e d i n 6.4 and 6A(0) i s the v a r i a t i o n i n A due to the u n c e r t a i n t y i n 0. T h i s e r r o r i s much s m a l l e r than the e r r o r s on A a r i s i n g from the phase-s h i f t a n a l y s i s . The values of C + and C obt a i n e d by f i t t i n g the' e l a s t i c s c a t t e r i n g data are t a b u l a t e d i n Table 7.2 wit h t h e i r s t a t i s t i c a l e r r o r s . A l s o t a b u l a t e d are C,--'8G(A) and SC, where C i s the weighted mean of C + and C , 6C(A) i s the r e l a t i v e e r r o r i n C due to the u n c e r t a i n t y i n A and <5C i s the t o t a l e r r o r on C. The t o t a l e r r o r on C was obt a i n e d by adding 6C(A) and the s t a t i s t i c a l e r r o r on C i n quadrature. 64 Table 7,1 F i n a l Values of A, (T Energy A 516.5 0.2857 ± .0361 497.1 0.2822 ± .0366 455.8 0.2636 ± .0330 419.5 0.3255 ± .0343 325.6 0.5303 ± .0325 202.7 0.85.0 ± .0400 and 6A(9) F(c.m.) 5A(8~) 74.43 ±.30 .0022 72.42 ±.41 .0110 71.74 ±.45 .1600 71.92 ±.58 .0100 72.07 ±.56 .0098 70.56 ±.56 .0068 The f i n a l v a lue o f C was o b t a i n e d by t a k i n g the weighted mean of the s i x d e t e r m i n a t i o n s of C ana i t s e r r o r 6C appearing i n Table 7.2. The r e s u l t i s C = 0.9920 ± .0336 For comparison, the va l u e o f C c a l c u l a t e d from the NMR measurement u s i n g equation 6.7 was 1.035 but wit h an u n c e r t a i n t y of the order of 6%. Values of the NMR enhanced s i g n a l p o l a r i z a t i o n i n t e g r a l s f o r each run are gi v e n i n Table 7.3. The e r r o r quoted i s the standard d e v i a t i o n of a l l measurements made du r i n g the run. As mentioned p r e v i o u s l y the p o l a r i z a t i o n i n t e g r a l was w r i t t e n onto tape f o l l o w i n g each event. The small d e v i a t i o n i n d i c a t e s the s t a b i l i t y o f the t a r g e t p o l a r i z a t i o n and the s t a b i l i t y of the NMR enhanced s i g n a l . Table 7.2 F i n a l Values of C Energy C + C C 6C (A) SC 516.5 0. 9118 + .0579 0. 7970 ± .0645 0 .8606 + .0431 0.1092 0.1174 497.1 0. 9715 ± .0306 0 .9715 + .0306 0.1271 0.1308 455.8 1. 0283 + .0624 1. 0736 ± .0845 1 .0443 + .0502 0.1322 0.1414 419.5 1. 1212 + .0580 1. 0870 ± .0497 1 .1015 + .0377 0.1177 0.1236 325.4 0. 8611 + .0319 1. 0925 ± .0267 0 .9972 + .0205 0.0622 0.0655 202.7 0. 9443 + .0236 1. 0414 ± .0243 0 .9914 + .0169 0.0466 0.0496 66 Table 7,3 NMR P o l a r i z a t i o n I n t e g r a l s and Absolute Target P o l a r i z a t i o n NMR P o l a r i z a t i o n ..Absolute Target Run # ' ' I n t e g r a l : . . P o l a r i z a t i o n 500 MeV Tar g e t P o l a r i z a t i o n Negative 1 64737.1 + 26.7 .6422 + .0217 2 64825.4 + 28 .0 .6431 + . 0218 3 64829.5 + 21.8 .6431 + .0218 4 64757.6 + 22.2 -6424 + • 0218 5 64685.5 + 17.8 .6417 + • 0218 6 64975.8 + 20.8 .6446 +. .0218 7 65036.2 + 31.8 .6452 + .0219 8 65237.8 + 32.9 .6472 + .0219 9 65021.8 + 30.9 .6450 + .0219 10 64937.1 + 37.8 .6442 + .0218 11 65786.8 + 50.8 .6526 + .0221 12 65696.5 + 56.9 .6517 + .0220 13 65717.4 + 44.2 .6519 + .0221 14 65846.9 + 46.0 .6532 + .0221 15 65678.3 + 60.7 .6515 + .0221 16 65733.7 + 57.7 . 6521 + .0221 17 65706.2 + 34.6 .6518 + .0221 18 65738.4 + 32.3 .6521 + .0221 19 65657.9 + 45.8 .6513 + .0221 20 65672.8 + 54.7 .6515 + .0221 21 65741.8 + 40.0 .6522 + .0221 22 65693.1 + 19.1 .6517 + .0221 500 MeV Target P o l a r i z a t i o n P o s i t i v e 23 63530.9 + 16.8 .6302 + .0214 210 MeV Target P o l a r i z a t i o n P o s i t i v e 24 64504.1 + 31.8 .6399 + .0217 25 64504.8 + 21.5 .6399 + .0217 26 64624.8 + 24 .8 .6411 + .0217 210 MeV T a r g e t P o l a r i z a t i o n Negative 27 64940.2 + 26.4 .6448 + .0218 28 65001.3 + 31.5 .6448 + .0218 T a b l e 7.3 - Continued 67 330 MeV T a r g e t P o l a r i z a t i o n Negative 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 64849.2 64881.2 64804.9 64873.1 6 4 7 9 6 . 3 64873.7 ± 58 64772.6 ± 33 64821.4 64665.3 33.0 58.2 55.3 45.5 58 ± 57 ± 39 ,6433 ± ,6436 ± ,6429 ± ,6435 ± ,6428 ± -6435 ± 6425 ± ,6430 ± 6415 ± .0218 .0218 .0218 .0218 • 0218 .0218 .0218 .0218 .0217 33 0 MeV Tar g e t P o l a r i z a t i o n P o s i t i v e 62233.7 ± 46.0 63702.5 ± 46.6 63830.1 ± 46.1 63970.6 ± 31.6 .6174 ± .0209 .6321 ± .0214 .6332 ± .0215 .6346 ± .0215 520 MeV Tar g e t P o l a r i z a t i o n P o s i t i v e 63583.6 6 3 4 5 7 . 4 63401.8 63365 .8 ± 29.6 ± 32.9 ± 29.0 ± 37.1 .6307 ± .0214 •6295 ± .0213 .6289 ± .0213 .6286 + .0213 46 47 48 49 50 51 520 MeV Tar g e t P o l a r i z a t i o n Negative 59312.2 ± 36.8 60173.9 ± 24.0 60265.0 ± 32.2 .5885 ± .0199 .5969 ± .0202 •5978 ± .0203 460 MeV Tar g e t P o l a r i z a t i o n Negative 59942.4 ± 28.1 59781.9 + 41.2 59587.2 ± 25.9 .5946 ± .0202 .5930 ± .0201 .5911 ± .0200 460 MeV Target P o l a r i z a t i o n P o s i t i v e 52 53 54 55 56 57 60961.2 ± 76 62355.2 ± 27 62376.7 ± 37 61450.1 ± 30 61419.1 ± 37 61385.2 ± 25, .6047 ± .6186 ± .6188 ± .6096 ± .6093 ± .6089 ± .0205 .0210 .0210 .0206 .0206 .0206 68 Table 7.3 V Continued 425 MeV Tar g e t P o l a r i z a t i o n P o s i t i v e 58 61374.7 ± 34.9 .6088 ± .0206 59 61306.5 ± 50.0 .6082 ± .0206 60 61167.8 ± 32.4 .6068 ± .0206 425 MeV T a r g e t P o l a r i z a t i o n Negative 61 54965.0 ± 57.1 .5452 ± .0184 62 56643.1 ± 38.4 -5619 ± .0190 63 56928.2 ± 3 3 . 5 .5647 ± .0194 69 The f i n a l t a r g e t p o l a r i z a t i o n s c a l c u l a t e d u s i n g the v a l u e of C determined from the n u c l e a r s c a t t e r i n g measurement are a l s o t a b u l a t e d i n 7.3. 7.2 C o n c l u s i o n The a b s o l u t e p o l a r i z a t i o n of a d y n a m i c a l l y p o l a r i z e d p r o t o n t a r g e t has been measured to an accuracy o f ± 2% by o b s e r v i n g the asymmetry i n a proton beam s c a t t e r e d from the t a r g e t . The a n a l y s i s made use of e x i s t i n g i n f o r m a t i o n on the s p i n dependence of - the nucleon nucleon i n t e r a c t i o n t o account f o r the asymmetry t h a t was observed. The measurement i s i n e x c e l l e n t agreement w i t h an independent NMR d e t e r m i n a t i o n of the t a r g e t p o l a r i z a t i o n , however the u n c e r t a i n t y has been s u b s t a n t i a l l y reduced. The l a r g e s t c o n t r i b u t i o n to the e r r o r i n t h i s measurement came from the phase s h i f t a n a l y s i s used t o determine the asymmetry A. Improvement i n the world nucleon nucleon data would l e a d to more ac c u r a t e phase s h i f t analyses and f u r t h e r reduce the u n c e r t a i n t y i n t h i s measurement. 70 REFERENCES 1. I.P. Auer e t a l . , Phys. L e t t . 67B, 113 (1977); Phys. Rev. L e t t . 41, 341 (1978). 2. E.K. B i e g e r t e t a l . , Phys. Rev. L e t t . 44, 558 (1975). 3. E.F. Parker e t a l . , Phys. Rev. L e t t . 31, 783 (1973). 4. C.L. H o l l a s , Phys. Rev. L e t t . 44_, 1186 (1980). 5. B. C o y l e r , R u t h e r f o r d L a b o r a t o r y Report, RHEL/R 138 (1966) . 6. L . W o l f e n s t e i n , Ann. Rev. Nuc. S c i . 6_, 43 (1956). 7. N. H o s h i z a k i , Suppl. Progr. Theor. Phys. 4_2, 107 (1968) . 8. C D . J e f f r i e s , "Dynamic Nuclear O r i e n t a t i o n " , John Wiley & Sons, Inc., 1963. 9. A. Abragam, "The P r i n c i p l e s of Nuclear Magnetism", Oxford U n i v e r s i t y P r e s s , 1961. 10. C. Amsler e t a l . , J . Phys. G. 4_, 1047 (1978). 11. D. G i f f o r d , "Proceedings o f the Second Workshop on P o l a r i z e d T a r g e t s " , eds. G. Court, S. Cox, D. Cragg and T. N i i n i k o s k i , R u t h e r f o r d High Energy Lab P u b l i c a t i o n , RL-80-080, October 1980. 12. D.V. Bugg, P r i v a t e Communication. 71 APPENDIX A D e r i v a t i o n o f Equ a t i o n 6,1 2 - P^ + P + m + E P Th e r e f o r e = E 2 + m I but = P 2 + m 2 2 2 P 2 + m p = 2 , 2 . m p + m p 2 2 P - P -2 I 2m2 + 2E P r e p l a c e *1-2 I " 2 P 1 P I 1 + E 2 p I p I I p 1 2 2 2 2 p I I p J. ^ 2 _. 2 -1- 2 E T ^ ^ 2 E T E 1 ~ 2 l t l E 1 + PTL + nip + I p I I p 1 Tm - 2E TE, - 2m E. I p I I p 1 ~ P I ~ P l + P 1 + P l •'•2P.1P-ICQ-sC9,}.^ 2mp2 + 2E Im p - 2E IE 1-2m^E 1 - P n P T c o s 6 - m 2 + E Tm - E Tm - E T E , - m E. 1 J- p I p I p I I p l E. (ET+m ) = m 2 + E Tm + P_P_ cos (6) l i p p I p I I 72 2 mp .+. E m + P n P_ cos (0) p = I. p -1 I I ( E T + m ) 1 P 2 now square both s i d e s and s u b s t i t u t e f o r E^ 2 2 P : + ra .= 1 .p irip + E ^ t P 2P 2 cos 2 ( 9 ) + 2m2EImp + ^ P . ^ cos (6) + 2E Im pP 1P I cos ( 9 ) ( E i + v2 2 2 2 2 (P. + M ) ( E T + m + 2E Tm ) = 1 p I p I p' 4 2 2 2 2 2 "3 9 p m. + Ejin + P ^ cos ( 9 ) + 2m Ej + 2rn P ^ cos (6) + 2E^ m P ^ cos (9 ) m u l t i p l y i n g out the L.H.S. and c a n c e l l i n g terms g i v e s P X ( E 2 + m2 + 2EImp-P2 cos 2 (9 ) ) + (~2Vjn? cos ( 9 ) - 2E.[mp P]; 'cos (8)) = 0 

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