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Analysing powers of hydrogen and helium-4 for protons at intermediate energies Dubois, Richard 1978

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ANALYSING POWERS OF HYDROGEN AND HELIUM-4 FOR PROTONS AT INTERMEDIATE ENERGIES by RICHARD DUBOIS B.Sc, McGill University, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1978 © : Richard Dubois, 1978 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date A p r i l 26, 1978. ABSTRACT Analysing Powers df Hydrogen and Helium-4 for Protons at Intermediate Energies An absolute c a l i b r a t i o n of the analysing powers of hydrogen and helium-4 has been performed at 24° and 15° lab, respectively, for protons between 200 and 520 MeV with a t y p i c a l uncertainty of ±2%. In a double scattering experiment, the TRIUMF polarized proton beam was scattered, f i r s t at 15° on a l i q u i d helium target, then a second time at 24° on a polyethylene target. The method combines the advan-tages of the very high analysing power of helium and the small difference i n incident energies on both scatters. The analysing power of helium-4 was measured to be 0.964 ± .012, 0.792 ± .011 and 0.476 ± .018 at energies of 222, 325 and 518 MeV, respectively. The analysing power of hydrogen was found to be 0.294 ± 0.003, 0.354 ± 0.005 and 0.412 + 0.013 at proton energies of 205, 308 and 499 MeV, respectively. This represents a great improvement over previous p-p data which was not known to better than ±6%, and so w i l l permit a more accurate determination of nucleon-nucleon phase s h i f t s . - i i -TABLE OF CONTENTS Page T i t l e Page Abstract ^ Table of Contents ^ L i s t of Tables ^ v L i s t of Figures v Acknowledgements v - j ^ Dedication v i i i Chapter I Introduction ]_ Chapter I I Formalism 5 2.1 Calculation of Asymmetries g 2.2 P r i n c i p l e of the Experiment H Chapter I I I Experimental Equipment 13 3.1 Polarized Proton Beam 13 3.2 e i Polarimeter 15 3.3 Liquid Helium-4 Target 18 3.4 £2 Polarimeter 18 3.5 E3 Polarimeter 22 3.6 Nal Telescopes 24 3.7 Data Collection System 25 Chapter IV Data and Discussion 30 4.1 Event Selection 31 4.1.1 Random Coincidences 32 4.1.2 Pileup i n Nal Crystals 34 4.1.3 Background Contributions 35 - i i i -TABLE OF CONTENTS (continued) P a g e 4.2 Instrumental Asymmetry 36 4.3 R e j e c t i o n of Pions 37 4.4 C o r r e c t i o n f o r R a i s i n g zi Nal C r y s t a l s 37 4.5 E f f e c t of Beam S t e e r i n g 38 4.6 A n a l y s i s of the Data 38 4.6.1 Asymmetry S e n s i t i v i t y to Cuts 38 4.6.2 The Nal Energy Spectra 39 4.6.3 Corr e c t i o n s to ej 48 4.6.4 A n a l y s i s D e t a i l s at Each Energy 48 4.6.5 M o d i f i c a t i o n f o r Energy Loss f o r Empty Target C o r r e c t i o n 50 4.6.6 Background C o r r e c t i o n s 50 4.6.7 Raw and Corrected Asymmetries 50 4.6.8 C o r r e c t i o n to d3 59 4.7 Results 59 4.8 Comparison of the Results w i t h E x i s t i n g Data 64 Chapter V Conclusion 72 References 74 Appendix A Density M a t r i x Formalism 75 Appendix B D e r i v a t i o n of P(24) 80 Appendix C Determination of Proton Energies at the Target Centers 86 Appendix D R o t a t i o n of ej counters f o r I n s e n s i t i v i t y to Beam Movement 87 - i v -LIST OF TABLES Page 3.1 Logical Event Definitions 28 4.1 Typical Changes i n Asymmetries with a Small Change i n Cuts 38 4.2 I n e l a s t i c Thresholds i n the Nal Crystals 39 4.3 Kinematic Energy Differences i n the £3 Crystals 46 4.4 Increase i n Cuts for Empty Target Corrections 50 4.5 Correction Run Asymmetries 51 4.6 Rates for Background Subtractions 52 4.7 Measured and Corrected Asymmetries 53 4.8 Renormalization of PTT from Stetz, et a l . 61 He 4.9 Values of P u , n, P(24) used to Correct d 3 62 He 4.10 Results 69 C l Beam Energies at the Target Centers 86 D.l Change i n Solid Angle i n q as a Function of Beam Shift 89 - v -LIST OF FIGURES Page 2.1 Double Scattering Configuration 7 2.2 Left-Right Asymmetry i n P-P Scattering 9 3.1 Proton Experimental Area at TRIUMF !4 3.2 Upstream Polarimeter (ej) 16 3.3 Liquid Helium-4 Target 1 9 3.4 Liquid Helium Target Polarimeter ( £ 2 ) 21 3:5 Scattered Beam Polarimeter ( £ 3 ) 23 3.6 Nal Telescope Inefficiency 26 3.7 Electronics Configuration 27 4.1 Typical Time Spectrum at 520 MeV 3 3 4.2 Typical e 2 Energy Spectra - 520 MeV 40 4.3 Typical £3 Energy Spectra - 520 MeV 41 4.4 Typical e 2 Energy Spectra - 225 MeV 4 2 4.5 Typical e 3 Energy Spectra - 225 MeV 43 4.6 Typical £ 2 Energy Spectra -'. 327 MeV 44 4.7 Typical e 3 Energy S p e c t r a - 327 MeV 45 4.8 Kinematic Broadening due to the £3 Target 47 4.9 P(24), P(17) from PSA 6 0 4.10 P-P Analysing Power 6 5 4.11 P-He4 Analysing Power 6 6 4.12 Spin Up Beam Polarization 6 7 4.13 Spin Down Beam Pol a r i z a t i o n 68 4.14 Comparison of P(24). Results 7 0 4.15 Comparison of P u ( 1 5 ) Results 71 - v i -LIST OF FIGURES (continued) Page A . l Orthogonal Vector Sets (n,s\,S f), (n,P i 5P f) 78 D.l Effect.of Beam Shift on 88 - v i i -ACKNOWLEDGEMENTS I would l i k e to thank my colleagues i n this experiment: C. Amsler, L.G. Greeniaus, D.A. Hutcheon, CA. M i l l e r , G.A. Moss, B.T. Murdoch and G. Roy. In addition, I wish to thank the members of the BASQUE group at TRIUMF for t h e i r support and advice. My special thanks go to Dr. C. Amsler for his invaluable assistance i n the performance and analysis of this experiment, as wel l as guidance i n the wri t i n g of this thesis. I would l i k e to express my appreciation to Dr. M.K. Craddock for his supervision, advice and encouragement during the time leading to th i s degree. The author was supported by UBC and NRC during this time. - v i i i -ThJjs thej>-u> iA dtxLLaatzd to my Zov^ng wu^e. VEBORAH I. Introduction This thesis reports the results of a precise measurement of the polarization analysing powers of hydrogen and helium-4 for protons at intermediate energies. The experiment was performed by a collabora-tion of workers from several u n i v e r s i t i e s . * The experiment made use of the variable energy polarized proton beam at TRIUMF i n performing a double scattering experiment with primary beam energies of 225, 327 and 520 MeV. Experiments using spin polarized beams generally require knowledge of the magnitude of the pol a r i z a t i o n and obtain i t by monitoring with a polarimeter. The incident beam pol a r i z a t i o n P^ i s obtained from P, = E ./A 1.1 b measured where e , i s the measured l e f t - r i g h t asymmetry i n scattering from measured ° J J b the polarimeter target i n the plane perpendicular to the direc t i o n of P^, and A i s the analysing power of the target. This equation w i l l be derived i n the next chapter. Hence a precise knowledge of A w i l l y i e l d *R. Dubois, University of B r i t i s h Columbia; C. Amsler, Queen Mary College (London); L.G. Greeniaus, D.A. Hutcheon, CA. M i l l e r , G.A. Moss, G. Roy, University of Alberta; B.T. Murdoch, University of Manitoba. - 2 -the value of P^ through such a measurement and, indeed, the purpose of this experiment was to normalize the results of two previous experi-ments : measurements of the proton-proton (p-p) and neutron-proton (n-p) Wolfenstein parameters' 1'^* 1, and measurements of the analysing power of helium over a wide angular range 1. Wolfenstein 1 has shown that e l a s t i c p-p or n-p scattering can be described by a set of 6 parameters (see Chapter 2). These parameters can then be related to the phase s h i f t s due to the nucleon-nucleon (N-N) interaction. P r i o r to these experiments, world data had been insuf-f i c i e n t to define a good x2 minimum i n the phase s h i f t solution at 518 MeV for p-p scattering. The data were generally poor for the n-p system, and, i n f a c t , were i n s u f f i c i e n t to distinguish between two different solutions at 325 MeV. I t was f e l t that t h i s s i t u a t i o n could be cured by a d e f i n i t i v e set of measurements i n both the p-p and n-p systems to the l e v e l of t y p i c a l l y 0.03 on a l l the parameters, giving unique x2 solutions with we l l defined minima. Measurements of the p-p analysing power to better than 2% would ensure that the normalization error was n e g l i g i b l e . Measurements of the analysing power of helium may prove to be a more-sensitive tool than the d i f f e r e n t i a l cross section i n testing reaction mechanisms i n e l a s t i c scattering from helium. Also helium-4 i s the target used i n many polarimeters because of i t s high analysing power over a wide range of energies. For these reasons i t i s also desirable to make an accurate normalization of the p-He4 p o l a r i z a t i o n analysing power. - 3 -Previous measurements of the p-p analysing power made use of two methods: scattering from a polarized target 1 and double scattering G A techniques 2' 1. The analysing power may be obtained from a polarized target experiment by scattering an unpolarized beam from the target. Si m i l a r l y to eq. 1.1 where P i s the polarization of the target. The determination of P^ from an NMR signal t y p i c a l l y gives r i s e to an uncertainty of 5% i n the polar i z a t i o n . The double scattering technique involves scattering an unpolarized beam (energy E^) from an unpolarized target, at angle COi, .arid then rescattering the resulting polarized beam (energy E2) from a second unpolarized target, at angle 62 - The asymmetry measured at the second target i s where P(Ei,0i) i s the polarization of the scattered beam, A(E2,02) i s the analysing power of the second target. I f two different targets are used, then obtaining A requires a measurement of e and knowledge of P. This was the method used by Cheng et a l . , i n which a carbon target was used as the f i r s t scatterer, and measured 1.2 e = P(E1,e1)A(E2se2) 1.3 - 4 -for which a normalization error of ±3% was quoted. However, the cross section for scattering from carbon varies rapidly with angle (50% per degree) and so the method i s sensitive to experimental geometry. As w e l l , there i s an i n e l a s t i c contribution from the excited states of carbon which begin at 4 MeV. While the present experiment was being mounted, an experiment using a v a r i a t i o n of the double scattering technique was performed. In this v a r i a t i o n , targets of the same material are used i n both scatters. Assuming time reversal invariance, e i s given by e = A ( E 1 , e 1)A ( E 2,6 2) 1.4 and for Ei=E 2, 6i=9 2 we have A(E,0) = /e . 1.5 However, i n the p-p case, the condition E}=E2 cannot be achieved,:even approximately for the angles of intere s t . The data was analysed by an energy dependent parametriz.ation of the po l a r i z a t i o n , with the quality of the f i t deteriorating at the extreme energies, 200 and 500 MeV. The experiment described here also made use of the double scatter-ing method, (Fig. 2.1), but used a polarized beam. The incident beam polarization was monitored by an upstream polarimeter, with the beam then scattering from a l i q u i d helium-4 target. The secondary beam - 5 -scattered from hydrogen i n the second target. Helium has a f l a t t e r cross section than carbon and has no low-lying excited states. The accuracy obtained did not depend on the energy, so i t was f e l t that a useful check of the endpoint results of the p-p double scattering technique could be achieved. - 6 -I I . Formalism To perform this measurement a spin zero nucleus, He , was used to produce the secondary beam (Fig. 2.1). The reasons for t h i s w i l l Ho become clear. I t i s simplest to use the density matrix formalism . to describe the properties of the beam of p a r t i c l e s , and the necessary results of the formalism are derived i n Appendix A. In the experiment, the incident proton beam scatters from hydrogen at the f i r s t target, helium-4 at the second target, and then scattered protons are rescattered from hydrogen at the t h i r d target. 2.1 Calculation of Asymmetries From eq. A.9 the d i f f e r e n t i a l cross section i n scattering a polarized beam from an unpolarized target i s where I i s the unpolarized d i f f e r e n t i a l cross section, P. i s the o — i incident p o l a r i z a t i o n vector, and P_ i s the pol a r i z a t i o n resulting from scattering unpolarized protons. With <j> as the angle between I\ and _P, the asymmetry i n scattering at an angle 8, i s defined as 1 = 1 (1 + P.-P) b — I 2.1 I(9,<j>=6) - I(6,<fr=ir) I(e,<j>=o) + 1(6, $=71) ' 2.2 - 7 -Incident beam direction Figure 2.1: Double Scattering Configuration. The incident proton beam spin p o l a r i z a t i o n i s monitored with the upstream polarimeter measuring asymmetry e\. The beam scatters from the l i q u i d helium target at 15° and then i s rescattered from the CH2 target at 24° to measure the asymmetry £ 3 ! - 8 -As shown i n Fig. 2.2, t h i s represents a l e f t - r i g h t asymmetry i n the plane of scatter, since scattering to the right with I\ i s equivalent to scattering to the l e f t with -P^. To s a t i s f y p a r i t y conservation, P = Pft 2.3 with fl being the normal to the scattering plane, giving fromneq. 2.1 e = P P(9). 2.4 P(9) i s also c a l l e d the analysing power of the target. The asymmetries at the f i r s t and second targets are, then ei = P.P(6i) 2.5 To obtain the asymmetry i n scattering from the t h i r d target, the incident p o l a r i z a t i o n of the protons incident on i t must be calculated. With P.=P.fi, then from eq. A.15, the polarization after scattering — x x i s P„ + DP. P = _Hf 1 2.6 - f 1 + P„ P. He x - 9 -Scattering to the left Scattering to the right Figure 2.2: Left-Right Asymmetry i n P-P Scattering. Scattering to the l e f t with <oi> i s equivalent to scattering to the right with - <ai>. - 10 -where D i s the depolarization parameter. For spin-O-spin-% scattering, the interaction matrix, M, takes on the simple form M = a + ba_'t\ where a i s a constant 2 :x 2 matrix, b i s a scalar and cr i s the vector formed by the set of Pauli spin matrices. Then from the d e f i n i t i o n of D I D = TrMcr M+a 2.7 o n n we have D = 1 so that the asymmetry i n scattering from the t h i r d target i s PHe ( 9 2 > + P i E3 - 1 + p - p ( Q 2 ) P(9 3) 2.8 l He z Together with eq. 2.5,.one gets for 61=63, and equal incident proton energies i n a l l scatters PO3) =j — ( d + £ 2 ) ^ 3 " £1) 2.9 ^ 2 With three independent measurements of e^, £ 2 and £ 3 , the three unknowns P(S 3), PHe^ 9 2) a n d P^ can be determined. This.is only possible since D=lj requiring that the second target have zero spin. A - 11 -2.2 Pr i n c i p l e of the Experiment One goal of the experiment was to normalize the results of previous t r i p l e scattering experiments, which required that P be measured at 24°(lab). A polarimeter suitable for measuring ei was already i n exis-tence, with Qi = 17°. However, the form of P(8,E) i s very well known from phase s h i f t analysis 3, and so P(17°,Ei) can be related to P(24°,E2) by P(24°,E2) = nP(17°,E1) where E]_ and E 2 are the energies of the protons incident on the and £3 targets respectively. However, to minimize error, the energies should be as close as possible to each other. The angle 0 2 was chosen to be as small as possible to ensure that the rate of scattered protons was high, and the energy loss oh the scattered protons low. Physical constraints prevented 9 2 from being made smaller than 15°. The three unknowns P(24°) , PT7 and P. are then given by He 1 P(24°) = J - r - ((1 + E 2)£ 3 - nei) •J £2 P = nei/P(24°) 2.10 PHe = £ 2 / P i - 12 -Assuming that the errors i n e\ and e 2 can be neglected i n comparison with that of £ 3 , then the error i n P ( 2 4 ) i s given by n e i ( l + £2) AP(24) - A E 3 2.11 2P(24)e 2 showing that, as £ 2 increases, AP(24) decreases. This i s one reason why helium with a high analysing power i s used rather than other heavy spin zero n u c l e i , such as carbon, with much lower analysing powers. The author was responsible for the design of the e 2 polarimeter, a s s i s t i n g i n the design and testing of the on-line computer data acquisition program, p a r t i c i p a t i n g i n the alignment, cabling and testing of the e 2 and £3 polarimeters, and f i n a l l y , analysing the data r e s u l t i n g from the two weeks of data taking. - 13 -I I I . Experimental Equipment This chapter w i l l describe the beamline, the three polarimeters used i n the measurement, the l i q u i d helium (LHe) target, and the electronics. 3.1 Polarized Proton Beam The TRIUMF sector-focused cyclotron produces polarized or un-polarized protons with energy variable from 180-520 MeV and an energy spread of 2 MeV. The protons are extracted by a carbon or aluminium (stripper) f o i l whose r a d i a l position determines the energy of the extracted beam. The experiment was performed i n the proton h a l l using beamline 4B which i s shown i n Fig. 3.1, with the LHe target at the 4BT1 position. The entire experiment was i n t e r n a l to the shielding ';cave' of the beamline. In addition to the beamline elements shown, there were two g a s - f i l l e d multiwire proportional chambers, M4 and M6, which were used to determine the beam position upstream and downstream of 4BT1. The radio frequency (rf) cavity of the cyclotron operates at 23 MHz giving a beam pulse every 43 ns, of 5 ns duration. A signal i s supplied to the experimenters from the r f cavity to allow them to monitor r f timing s h i f t s i n the a r r i v a l of the bursts. As predicted by J.R. Richardson^ 1, beam i n the cyclotron tank which i s not stripped and extracted w i l l be accelerated to the outer edge, and, i f i t s l i p s Figure 3.1: into BL4B at Proton Experimental Area at TRIUMF. The proton beam enters BL4 at 4VQ1 and i s bent 4BVB2. The l i q u i d helium target i s located at the PT1 (or 4BT1) position. - 15 -i n phase with the r f , can be decelerated back to the stripper f o i l and Oo extracted. I t has been found i n recent studies ^ that this decelerated beam has lower polarization than that of the accelerated beam. Thus the r f timing becomes important i n distinguishing between the two. Any s h i f t s of the r f timing during a run would require that the run be rejected. Ro The polarized protons are produced by a Lamb s h i f t source z consist-ing of a Duoplasmatron producing an unpolarized proton beam, a cesium c e l l to produce excited hydrogen atoms i n the 2S and 2P states, two solenoids to mix selected energy levels of the atoms, an argon c e l l to produce the polarized H~ ions, and a Wien f i l t e r to control the spin dire c t i o n . The source can produce 960 na of polarized ions of which up to 150 na has been extracted from the cyclotron of t y p i c a l l y 75-80% pol a r i z a t i o n . Beamline 4B was limited to 15 na of beam on the 4BT1 target by radiation monitor levels i n the 4B cave. The beamline tune provided a double waist at the upstream polarimeter and 4BT1 targets with spot sizes of t y p i c a l l y 1.x 1: cm2 and 0.5 x 0.5 cm2, respectively. 3.2 ei Polarimeter The purpose of the ej_ polarimeter was to measure the asymmetry generated by e l a s t i c p-p scattering of the polarized primary beam from a CH2 target. A double arm polarimeter design was used (Fig. 3.2). - 16 -Key: 1st letter denotes right or left of beam 2nd letter denotes forward or recoil arms Figure 3.2: Upstream Polarimeter (e^). The incident proton beam scatters from a CH2 target. The forward scattered protons are detected at 17° while the conjugate r e c o i l protons are detected at 73°. - 17 -Scattered protons were detected right and l e f t to eliminate instrumental asymmetries. A coincidence was required between the forward scattered proton at 17° and the r e c o i l proton at 73°, the angle between the two arms being chosen to ensure that "the scatter had been e l a s t i c from hydrogen i n the CH2 target. The polarimeter was made insensitive to beam movement by rotating the defining counters, LF2 and KF2, downstream of the CH2 target (Fig. D.l). I f the beam spot moved to one side on the target (closer to LF2, say), then, looking from the spot, the area of counter LF2 would appear smaller. Since the s o l i d angle i s given by fi = A/r 2 the effect of smaller area would be closely cancelled by the spot moving closer to the counter, leaving the s o l i d angle unchanged. This effect i s explained i n Appendix D. The target and r e c o i l counters were cont tained i n an evacuated chamber; the forward counters were outside the chamber. The s c i n t i l l a t o r s were a l l 2.54 x 2.54 x .635 cm3. The upstream window of the chamber was 0.0019 cm thick T i , with the vacuum extending downstream to the LHe target. The target holder was remotely controlled so that one of a CH2 , carbon or phosphor-coated p l a s t i c target could be inserted into the beam. The CH2 f o i l was 0.0038 cm thick, for the best compromise between low enough rate for the photo-m u l t i p l i e r tubes and high enough for adequate s t a t i s t i c s i n the asymmet-ry measurement. The carbon f o i l was used to perform the carbon subtrac-- 18 -t i o n , while the phosphor-coated target was for beam alignment. A camera was placed to view the target and s c i n t i l l a t i n g phosphor, and hence ali g n the beam on cross hairs drawn on the target. 3.3 Liquid Kek Target G l The target (Fig. 3.3) presented a c e l l of.'•'.liquid He\ 8 cm thick, to the proton beam. The c e l l was fed by a 15 Z reservoir, which was insulated from the surroundings by two thermal shields, kept cool by. the cold b o i l o f f gas,^passing through copper pipes soldered to the walls of the shields. The inner and outer shields were 20°K and 80°K respec-t i v e l y . The target was remotely movable i n the v e r t i c a l direction with three positions: furthest down placed the LHe c e l l i n the beam; the middle position placed a dummy c e l l , i d e n t i c a l to the LHe c e l l , i n the beam; and the highest position, a p l a s t i c f o i l coated with phosphor for beam alignment. A camera viewed the target for t h i s purpose. The c e l l windows were 0.025 cm thick stainless s t e e l soldered to the body of the c e l l . A vacuum surrounding the c e l l was maintained by an o i l d i f f u s i o n pump and was t y p i c a l l y 1 0 - 6 t o r r . The target was interlocked into the beamline vacuum system. 3.4 e 2 Polarimeter This polarimeter was designed to measure the asymmetry i n scatter-ing of the incident proton beam from the l i q u i d helium target. A - 19 -10 cm Schematic of cryostat interior, showing (a) fill inlet, (b; interior pressure gauge, (c) boiloff vent line, (d) mechanical coupling (Cajon VCR) (Cajon Co. Inc., Cleveland, Ohio 44139, U.S.A.), (e) port to pumping system, (0 outer radiation shield, (g) inner radiation shield, (h) liquid 4He vessel, (i) cell boiloff standpipe (j) fill tube receptable, (k) target cell, (I) dummy cell, (m) vertical position stop ring, (n) sprocket key, (o) mounting flange. Figure 3.3: Liquid Helium-4 Target. - 20 -double arm design was not used, as the more massive r e c o i l helium nucleus had i n s u f f i c i e n t energy to escape the target. I t was necessary to measure the energy of the scattered protons d i r e c t l y . The t o t a l energy of the proton was measured by stopping i t i n a Nal c r y s t a l . The i n e l a s t i c contribution from the helium nucleus begins 20 MeV below the e l a s t i c peak, so i t was necessary to resolve only t h i s energy difference. The polarimeter had two separate arms (Fig. 3.4) so that the instrumental asymmetry could be measured. Precautions were taken to keep the singles rates i n the Nal crystals low, as too high a rate results i n degradation (and loss) of some of the pulses. The t e l e -scopes were therefore placed as far as physically possible from the target. The defining counters were made very small and a s t e e l collimator shadowed the remainder of the faces of the cr y s t a l s . Lead bricks were stacked around the sides of the telescope to reduce back-ground from the beam pipe. The scattered beam was collimated to avoid excessive background rates i n the £3 polarimeter, so a simi l a r dummy collimator was placed on the 15° l i n e to the right of the beam pipe to reduce instrumental asymmetries i n £ 2. In order to avoid the scattered beam passing through the £3 polarimeter target and losing energy, the £ 2 counters were raised by 10 cm ensuring that protons entering them from the LHe target did not pass through the E 3 target. - 21 -Incident beam direction Figure 3.4: Liquid Helium Target Polarimeter ( £ 2 ) • The incident beam scatters from the l i q u i d helium target and forward scattered protons are detected at 15° by Nal telescopes. - 22 -The polarimeter arms consisted of two s c i n t i l l a t i o n counters followed by a Nal c r y s t a l . The counter sizes were 0.5 cm diameter by 0.635 cm and 0.8 cm diameter by 0.635 cm back and front respectively. The small s o l i d angle acceptance of these s c i n t i l l a t o r s made the polarimeter a sensitive monitor to beam movement. 3.5 £3 Polarimeter The measurement of t h i s asymmetry was complicated by the p o s s i b i l -i t y of protons scattered i n e l a s t i c a l l y from the helium target being scattered e l a s t i c a l l y from the CH2 target; these protons had to be separated out. The target was made small enough so that the r e c o i l protons would be able to escape. This permitted the use of a double arm polarimeter (Fig. 3.5). In addition, Nal crystals were used to measure the energy of the scattered protons. The forward arms had an acceptance of ±3°, allowing a large spread i n proton energies accepted i n the Nal, due to the rapid v a r i a t i o n of energy with scattering angle. Wedged absorbers were used to correct for this v a r i a t i o n . Thus, the energy spread was due almost e n t i r e l y to energy loss i n the £3 target, and was 16 MeV at 32.7 MeV, which would therefore be approximately the expected width of the e l a s t i c peak. A resolution of 10 MeV FWHM was required of the Nal crystals to separate the e l a s t i c and i n e l a s t i c peaks. At 32:7 MeV and 520 MeV, the target was a 1 x 2 x 6 cm3 block of CH2, while at 225 MeV the dimensions were 0.5-x 2 x 6 cm3, the thinner - 23 -( Incident (scattered) beam direction Figure 3.5: Scattered Beam Polarimeter ( £ 3 ) . The scattered beam from the l i q u i d helium target i s rescattered from a CH2 target. Forward scattered protons are detected at 24° by Nal telescopes, while r e c o i l protons are detected at 62° by single s c i n t i l l a t o r s . - 24 -target being used to reduce energy loss i n the target (which would be greater at the low energy i n the target used than at the other two energies). A carbon target, of the same size and equivalent density of carbon atoms, at each energy, was used to perform the carbon subtraction. The target was constructed of carbon s l i c e s , separated to give the same density of carbon as i n the CH2 target. The forward s c i n t i l l a t o r s were 5.8 x 5.8 x 0.63 cm3, while the r e c o i l ones were 9.6 x 17.8 x 0.63 cm3. A l l the £3 s c i n t i l l a t o r s , except Si, were hung from a r i g i d l y supported superstructure. The dimensions of the scattered beam defining counters, and S 2, were 2.54 x 5.08 x 0.635 cm3. As previously mentioned, a 20 cm lead c o l l i -mator was used to reduce the singles rates i n the £3 counters due to background from the LHe target. In addition, the polarimeter was shielded from the target upstream and from the beam pipe. Further-more, lead bricks were placed around the Nal cry s t a l s . To eliminate electrons scattered out of the LHe target and flooding S\ and S 2, a magnet was placed before the lead collimator to sweep them out of the way. 3.6 Nal Telescopes The Nal crystals were a l l the same s i z e , v i z . 12.5 cm<j> x 7.5 cm. Copper plates were used to degrade the proton energy to 110 MeV as the crystals were thick enough to stop 150 MeV protons. - 25 -The resolution of the telescopes had been studied i n a preliminary run at 320 MeV with a monochromatic proton beam produced by scattering from a hydrogen target. The resolution was found to be 10 MeV FWHM. The actual pulse height c a l i b r a t i o n was obtained d i r e c t l y i n the experiment by inse r t i n g 0.476 cm copper plates i n the 15° l i n e s on both sides, immediately upstream of the collimators, and observing the s h i f t s of the e l a s t i c peaks i n a l l four Nal cry s t a l s . I t was not necessary to measure the e f f i c i e n c i e s of the Nal telescopes. The losses were mainly due to scattering i n the copper, and the eff i c i e n c y at 500 MeV was 30% 2 (Fig. 3.6). Differences i n e f f i c i e n c i e s r e s u l t i n instrumental asymmetries which could be measured and corrected f o r , assuming the differences remain constant i n time. 3.7 Data Collection System The data c o l l e c t i o n system consisted of three basic subsystems; a Honeywell 316 minicomputer and magnetic tape uni t ; a CAMAC system 1; and fast NIM electronics. The fast electronics l o g i c (Fig. 3.7) determined the occurrence of any one of four different types of events, which are described i n Table 3.1. None of the Nal signals were included i n defining these events. To equalize the rates of e 2-type and £3~type events, the e 2 events were prescaled by a factor of 2 8 before being passed to the event trigger. 1 0 0 . 0 I N E F F 'T IV.) 80 .0 6 0 . 0 40 .0 2 0 . 0 0 . 0 100 .0 2 0 0 . 0 3 0 0 . 0 4 0 0 . 0 E [ M E V ) 5 0 0 .0 Figure 3.6: Nal Telescope Inefficiency. The in e f f i c i e n c y of Nal telescopes similar to the ones used i n t h i s experiment was measured by Stetz as a function of the proton energy incident on the telescope. - 27 -NolR3 Na! L3. NaIR2 NaIL2 M . H _ .FL, B R L 5 O.R. output register f rom computer F.O. fon out ELECTRONICS CONFIGURATION Figure 3.7: Electronics Configuration. The four event def i n i t i o n s are shown giving r i s e to the master gate, which causes a l l ADC's and TDC's to be read. - 28 -Upon i d e n t i f i c a t i o n of an event, a master gate was generated which inhib i t e d the computer from accepting further triggers and provided the gate for the analogue to d i g i t a l converters (ADC's) and stops for the time to d i g i t a l converters (TDC's). The i n h i b i t was removed after the event was processed. The data was stored event by event and consisted of a pattern unit word (DCR word) describing the event type scalers TDC's ADC's re a l time clock. Table 3.1: Logical Event Definitions Event Type Def i n i t i o n e3-R Sj_-S2-LF3-RR3 e 3-L S1«S2*RF3-LR3 e2-R FR2•BR2 e 2-L FL2«BL2 where • denotes a l o g i c a l 'and'. At the end of each run the f i n a l values of a l l scalers were stored on tape. The data was recorded on 2400 f t , 800 bpi magnetic tapes, each of which held about 100,000 events. - 29 -In order that corrections could be made o f f - l i n e for random coincidences i n the e 2 a n c^ e3 polarimeters, the pulse width of one s c i n t i l l a t o r i n each event type was made greater than 86 ns (two beam bursts). Only scalers were recorded for the polarimeter,: so one signal i n each arm was delayed by 43 ns, and the resulting coincidences were scaled. Pileup i n the Nal crystals were determined using gates 600 ns wide. I f a second p a r t i c l e f i r e d the Nal during t h i s time, an appropri-ate b i t i n the DCR was set. The computer software allowed the user to examine histograms and scalers at any time during the run. On-line cuts to the data could be applied, but only affected the hardcopy outputs. No cuts were applied to the events before storage on tape. - 30 -IV. Data and Discussion In this chapter the data i s presented and the method of analysis and results are presented. To reduce the effects of instrumental asymmetries and take into account the fact that P/*" i s not equal to P^ (see Appendix B) , the + asymmetries ( i = 1,2,3) resulting from runs with incident p o l a r i -zation up (P/1") and down (P^ -) were combined to y i e l d d. = e. - e. 1 1 I S. = e. + + e.~. i i I I t i s shown i n Appendix B that the expression for the analysing power of hydrogen, P(24) , given i n eq. 2.10 must be replaced by eq. B.12, i. e . n d i 2 J * n d 3 ' d i Si*' 2 " d l " 2" U J d l d7 + ~2_ T~ with the analysing power of helium and beam pol a r i z a t i o n being simply related to P(24) . A l l the variables, but n, on the right hand side of eq. (1) are measured quantities derived from t h i s experiment. The object of the analysis was to calculate these variables from the Nal pulse height spectra. Histograms of the Nal pulse height spectra were used to determine e l a s t i c events i n the e 2 a n d £3 measurements. Fig. 4.2 shows t y p i c a l - 31 -spectra from the Nal crystals for the incident beam of 520 MeV. The e l a s t i c peaks are c l e a r l y discernible, while the f a i l s are mainly due to reactions i n the copper absorber and the c r y s t a l . 4.1 Event Selection To remove i n e l a s t i c events from the pulse height spectra, i t was necessary to make 'cuts' on the e l a s t i c peak, i . e . only events within applied l i m i t s , whether i n time or i n energy, were accepted. This required knowledge of the energy c a l i b r a t i o n of the ADC's, and where the i n e l a s t i c contribution began. To calibrate the ADC's, 0.476 cm copper plates were placed at the entrance to the collimators on both 15° l i n e s , and the resulting s h i f t s i n a l l four e l a s t i c peaks were observed. The.expected s h i f t was calculated from range-energy tables, giving the energy s h i f t per channel of the ADC. To place the cuts, the expected energy difference i n the crystals between the e l a s t i c peak and i n e l a s t i c beginning 20 MeV below was calculated. The cuts were made at approximately half the difference to minimize the number of i n e l a s t i c s . With this choice the cut position could be varied s l i g h t l y so long as the r e l a t i v e l e f t - r i g h t cuts were correct. The asymmetries were defined to be e = (L - R)/:(L + R) 4.1 - 32 -where L,R are the numbers of events remaining i n the l e f t and right e l a s t i c peaks respectively. Corrections were made to these asymmetries to account for randoms, pileup, empty target events and carbon events from the CH2 target. Each type of correction w i l l be discussed. (i) random coincidences In the E2 and £3 polarlmeters the pulse width of one of the s c i n t i l l a t o r s i n each event coincidence was made: more than two beam bursts (86 ns) long. The res u l t i n g time spectrum (at 520 MeV) i s shown i n Fig. 4.1, where the small peak indicated i s 43 ns below the larger one. The large peak consists mainly of coincidences between protons from the same burst; the smaller peak consists of coincidences between protons from one burst i n one counter, with protons from the next burst in the other counter. By counting the number of events i n thi s '".random''epeak corresponding to the e l a s t i c 'prompt' energy peak, the number of random coincidences i n the e l a s t i c peak was estimated. In the ej_ polarimeter, Fig. 3.2, one arm was delayed by 43 ns with respect to the other, and the resulting coincidences were scaled. This measured, e.g. for constant current T LF' LR where T = 43 ns and T i s the duration of the measurement. L and L r R are the singles t o t a l s , during this time T, i n the forward and r e c o i l 8 0 0 . 0 6 4 0 .0 4 8 0 . 0 3 2 0 .0 1 5 0 . 0 0 . 0 COUNTS 10 NS 0 . 0 2 0 . 0 4 0 . 0 T 6 0 . 0 8 0 . 0 Figure 4.1: Typical Time Spectrum at 520 MeV. The major peak shows the majority of the events, which arose from coincidences between the forward and r e c o i l arms of the e 3 polarimeter of protons due to one beam burst. The small peak, 43 ns below the major peak, arises from coincidences between a proton from one burst i n one arm with a proton from the next burst i n the other arm. - 34 -arms, treating each arm of the polarimeter as a single counter. The true number of randoms i s given by L = ^  CL, - L) (I, - L) A T F R with L being the number of coincidences between L_ and L-. Then,cas r K i n this case, for L p » L L R » L, LA " T LF' LR which i s just the quantity scaled. For both the Nal's and ei p o l a r i -meter, the number of randoms i s subtracted d i r e c t l y from the number of time events, i . e . q - L.) - (R - RA> £ (L - L A) + (R - RA) ( i i ) pileup i n the Nal crystals Those events with the pileup b i t set i n the DCR were removed ('gated') from the histograms of true events. I t was found that the pulse height spectrum of these events was not noticeably different from that of the true events, indicating that the pileups consisted of - 35 -e l a s t i c s and p a r t i c l e s just above the discriminator threshold. There were v i r t u a l l y no events showing pileup of two e l a s t i c events. For this reason, the pileups were added back to the true event spectrum and sub-jected to the same cuts. No correction for pileup was found to be necessary. ( i i i ) background contributions (a) carbon i n the CH2 targets Some of the ' e l a s t i c ' events detected i n the and £3 polarimeters were due to 1 2C(p,2p)X reactions. To measure th i s contribution, the CH2 targets were replaced by carbon targets. Care was taken to account for d i f f e r i n g carbon densities i n the two target types. (b) contribution from the LHe target walls I t was possible for events i n the £2 and £3 polarimeters to res u l t from a scatter off the walls of the LHe target vessel. This contribu-tion was measured by replacing the f u l l LHe c e l l by an i d e n t i c a l empty one. To correct for these backgrounds, the following formula was used^ 1 .n £ , - y R.E. measured ^ x 1 e^ = 1=1 / o true — 4.3 n l - t R . i = l 1 - 36 -where the index i runs over the corrections being made. i s the r a t i o of the background event rate to the normal target event rate. To correct £ 3 , there are two correction terms: one for the carbon background, and one for the empty target subtraction. E 2 i s corrected for empty target background and for carbon background. 4.2 Instrumental Asymmetries P(24) was calculated by grouping consecutive runs of opposite spins and then averaging over a l l pairs of runs. Where such pairing was not possible consecutive runs of the same spin were averaged together and then grouped with a following run of opposite spin. This method assumes that the instrumental asymmetries remain constant between the two runs. I t also eliminates the need to assume that the beam p o l a r i -zation stays constant for times longer than one run. To calculate the correction to A3 due to the instrumental asymmetry i n £3 spin off runs near the runs being corrected were averaged OFF together to give e 3 , then from eq. B.3, 6 * e 3 0 F F - v p(24) 4.4 He with the correction on d3 being d 3 * = d 3 - C 4.5 - 37 -where C = -<5P2(0+- - a g) - S 2P(a + - a ) • [1 - P 2(a+ 2 + a V + a"2)] 4.6 with the quantities defined as i n Appendix B. To calculate C, the values of P(24) and PTT were taken from the l i t e r a t u r e . He 4.3 Rejection of I T ' S Above 290 MeV, T T ' S are produced i n i n e l a s t i c reactions, and must b rejected from the spectrum of good events. For the Nal crystals used here, the maximum energy deposited by a TT i s 75 MeV (in which the TT i s just stopped), which i s far below the e l a s t i c peak i n the c r y s t a l , and so these TT events were rejected when the cuts were applied. 4.4 Correction for Raising the e 2 Nal's To avoid energy loss by the scattered beam protons i n the £3 CH2 target, the e 2 Nal telescopes were raised by 10 cm. For counters out o the scattering plane, the apparent po l a r i z a t i o n of the incident beam i s reduced by cos <j> where <j> i s the angle from the plane to the counter position. In this case, for counters placed 300 cm from the target, and raised by 10-' cm', cos cj> = .9994, a completely neg l i g i b l e effect. - 38 -4.5 Effect of Beam Steering Using an upstream steering magnet the beam was steered to observe the effect on the asymmetries. I t was found that a 1 mm beam s h i f t at the target resulted i n a 0.5% change i n E 2 . £ I and £3 showed smaller effects. Thus i t was necessary to monitor the beam position frequently. 4.6 Analysis 4.6.1 Asymmetry S e n s i t i v i t y to Cuts A l l runs were tested for i n t e r n a l consistency at each energy: singles rates, peak to t a i l r a t i o s , and event rates were a l l compared. Runs were analysed i n groups whose Nal e l a s t i c peaks f e l l i n the same positions. Any small errors incurred i n applying the cuts would result i n instrumental asymmetries, for which corrections are made, making the error negligible i n comparison with the s t a t i s t i c a l error, as shown i n Appendix B. The s e n s i t i v i t y of the asymmetries to random variations of the peak by one histogram d i v i s i o n i s shown i n Table 4.1. Table 4.1: Typical Changes i n Asymmetries with a Small Change i n Cuts. E(MeV) Ad 2/d 2 (%) Ad 3/d 3 (%) 225 0.3 6.0 327 0.2 0.8 520 0.3 0.8 - 39 -Small s h i f t s at 327 and 520 MeV are c l e a r l y n e g l i g i b l e i n each run, however, care was taken at 225 MeV to monitor the peak position and move the cuts with i t . The error i s smaller than the s t a t i s t i c a l error i n a l l cases. 4.6.2 Nal Energy Spectra From range-energy curves, the points i n the Nal spectra where the i n e l a s t i c contribution began could be estimated. These are shown i n Table 4.2. Table 4.2: In e l a s t i c Thresholds i n the Nal Crystals. Primary Beam Energy below Energy below Energy (MeV) peak i n e 2 (MeV) peak i n £3 (MeV) 225 35 24 327 44 35 520- 60 44 Fig. 4.2-4.7 show t y p i c a l Nal spectra at 225, 327 and 520 MeV respectively. In analysing the ca l i b r a t i o n runs for the ADC's at "225 MeV, i t was found that the peaks shif t e d by the following amounts 46 channels i n NaIR3 56 channels i n NaIL3 67 channels i n NaIR2 104 channels i n NaIL2 - 40 -50.0 n 40.0 30.0 \ 20.0 10.0 0.0 COUNTS 35.0 65.0 95.0 125.0 NflIR 2 ENERGY SPECTRUM 100.0 -, 80.0 £0.0 40.0 20.0 0.0 COUNTS 40.0 70.0 100 .0 130.0 160.0 NflIL 2 ENERGY SPECTRUM Figure 4.2: Typical e 2 Energy Spectra - 520 MeV. The calibrated pulse height spectra of NaIR2 and NaIL2 are shown, gated on prompt and non-piled up events, i denotes inelastics and c denotes cuts. 60.0 COUNTS 48.0 35.0 24.0 12.0 0.0 0.0 t c 20 HEV I 1 M L 40.0 80.0 120.0 160.0 NRIR3 ENERGY SPECTRUM 100.0 COUNTS 60.0 60.0 40.0 20.0 0.0 o c 25 MEV 1 1 0.0 50.0 100.0 150.0 200.0 N flIL 3 ENERGY SPECTRUM Figure 4.3: Typical £3 Energy Spectra - 520 MeV. The calibrated pulse height spectra of NaIR3 and NaIL3 are shown, gated on prompt and non-piled up events. i denotes inelastics and c denotes cuts. - 42 -150.0 n 120.0 90.0 60.0 30.0 0.0 COUNTS P 0_ HEV rT-t-| 20.0 40.0 60.0 80.0 100.0 E Nfl IR2 ENERGY SPECTRUM 1000.0 800.0 eoo.o 400.0 200.0 0.0 COUNTS m m tu. 25.0 45.0 65.0 85.0 105.0 E NA1L2 ENERGY SPECTRUM Figure 4.4: Typical e 2 Energy Spectra - 225 MeV. The calibrated pulse height spectra of NaIR2 and NaIL2 are shown, gated on prompt and non-piled up events, i denotes i n e l a s t i c s and c denotes cuts., - 43 -180.0 144.0 108.0 J 72.0 35.0 0.0 COUNTS 0.0 30.0 60.0 2L 90.0 120.0 NRIR3 ENERGY SPECTRUM 300.0 , 240.0 i 180.0 120.0 60.0 0.0 COUNTS 0.0 30.0 60.0 90.0 120.0 N flIL 3 ENERGY SPECTRUM Figure 4.5: Typical £3 Energy Spectra - 225 MeV. The calibrated pulse height spectra of NaIR3 and NaIL3 are shown, gated on prompt and non-piled up events, i denotes i n e l a s t i c s and c denotes cuts. - 44 -150.0 COUNTS 120.0 4 90.0 60.0 30.0 0.0 30.0 60.0 90.0 120.0 N R I R 2 ENERGY SPECTRUM 150.0 120.0 90.0. J 60.0 30.0 0.0 COUNTS IS MEV j - _ ^ 30.0 60.0 90 .0 120.0 N R I L 2 E NE RGY 5<P E C-T RUM-Figure 4.6: Typical e 2 Energy Spectra - 327 MeV. The calibrated pulse height spectra of NaIR2 and NaIL2 are shown, gated on prompt and non-piled up events, i denotes i n e l a s t i c s and c denotes cuts. - 45 -50.0 40.0 30.0 20.0 10.0 0.0 COUNTS JI 1 30.0 60.0 30.0 13=1 120.0 N flIR 3 ENERGY SPECTRUM 80.0 64.0 48.0 J 32.0 16.0 0.0 COUNTS n piLJULY, I t-n-30.0 60 .0 90.0 120.0 N R I L 3 ENERGY SPECTRUM Figure 4.7: Typical e 3 Energy Spectra - 327 MeV. The calibrated pulse height spectra of NaIR3 and NaIL3 are shown, gated on prompt and non-piled up events. i denotes i n e l a s t i c s and c denotes cuts. - 46 -These s h i f t s correspond to changes of 20.9 MeV i n the £2 crystals and 15.7 MeV i n the £3 c r y s t a l s . For the other two energies i t was assumed that the gain on the ADC's remained constant i n c a l i b r a t i n g the h i s t o -grams . Fig. 4.2-4.7 show the expected threshold of the i n e l a s t i c s and also where the cuts were applied. From the very 'clean' £2 spectra, at 327 and 520 MeV, i t appears that i n e l a s t i c s were not present to a s i g n i f i c a n t l e v e l . As w e l l , there was no 'shoulder' indicating the onset of i n e l a s t i c s i n the £3 spectra. The width of the peak, i n the £3 c r y s t a l s , at 520 MeV, can be explained by kinematics. Fig. 4.8 shows an expanded view of the c 3 target and one set of copper absorbers. The width of the peak w i l l mainly be determined by protons emerging from points (1) and (2) as shown. These two positions show the greatest difference i n energy loss i n the target. Since the wedges accounted for v a r i a t i o n of energy with scattering angle, the energy difference can be calculated for protons emerging from (1) and (2) at 24°. These differences are shown i n Table 4.3. Table 4.3 Kinematic Energy Differences with £3 Crystals. Primary Beam Energy (MeV) AE (MeV) 225 13 327 24 520 37 - 47 -Figure 4.8: Kinematic Broadening due to the £3 Target. The greatest difference i n energy loss i n the £3 target i s due to protons entering the Nal's from points (1) and (2). At 327 MeV, the difference was 24 MeV. - 48 -I t should be noted, however, that the number of protons from positions (1) and (2) should be much smaller than from the center of the target. Hence at 520 MeV a broad energy peak was expected. 4.6.3 Corrections to ej In correcting for carbon contribution i n e i 5 the different densities of carbon i n the two targets, carbon and CH2, were corrected for using the formula PCH 2 12 R = R X X -zrr-true meas 14 PC where p n T T = 5.3g/cm2 and p._ = 19.4 g/cm2 are the area densities of the CH2 C CH2 and carbon targets, respectively. The r a t i o 12/14 accounts for the extra two protons of H 2 i n CH2 • R denotes rate. Furthermore, the asymmetries used were corrected for instrumental asymmetry. 4.6.4 Analysis Details at Each Energy At 225 MeV, i t was found that the carbon correction i n £3 was en t i r e l y n e g l i g i b l e . In addition, the £i_ polarimeter f a i l e d i n t e r -mittently at this energy, so that a l l rates were normalized using an Mi ion chamber 1 . E J information was not used for these runs and i t was assumed that beam conditions remained constant so that the E\ data from previous runs could be used. - 49 -The runs at 327 MeV were analyzed i n two groups, since, during the experiment, one s c i n t i l l a t o r , FR3, moved. Runs just before this time showed clear inconsistencies and were rejected. When the carbon correction for the ej polarimeter was taken, the asymmetry was tested for s e n s i t i v i t y to beam movement. Data was taken with the beam steered to the right and l e f t of the center of the carbon target, but none taken with the normal run settings. Therefore an interpolation was performed between the two data sets to obtain the asymmetries at the correct settings. I t i s believed that this i n t e r -polation introduced errors much smaller than the s t a t i s t i c a l errors. At 520 MeV, the gain on the NaIL2 c r y s t a l was raised, making i t impossible to extrapolate the c a l i b r a t i o n for that c r y s t a l from 327 MeV. For runs at 327 MeV, i t was found that the e l a s t i c peaks i n NaIR2 and NaIL2 were of equal width. Assuming th i s to be also true at 5 MeV, the ca l i b r a t i o n of NaIL2 was related to that of NaIR2 through the r a t i o .of e l a s t i c peak widths. Since no s h i f t i n the peaks with time was observed, excellent s t a t i s t i c s for the peak widths were obtained by summing, channel by channel, the counts i n each c r y s t a l for a l l runs. I t was found that the r a t i o of NaIL2 width to NaIR2 width was 11/13, so that the NaIL2 ca l i b r a t i o n was obtained by dividing that of NaIR2 by this r a t i o . Furthermore, the e\ polarimeter was unreliable during the empty target correction runs and so the rates for those runs were normalized to the ion chamber. - 50 -4.6.5 Correction forHEnergy Loss for Empty Liquid Helium Target Correction In performing the empty LHe target correction, the cuts were modified to account for the higher energy of the protons emerging from the target. Had the target been f u l l , these protons would have l o s t energy i n the l i q u i d helium. The amounts that the cuts were raised are shown i n Table 4.4. Table 4.4 Increase i n Cuts for Empty Target Corrections Primary Beam Increase i n E 2 Increase i n £3 Energy (MeV) cuts (MeV) cuts (MeV) 225 4 4 327 6 4.5 520 5 4 4.6.6 Background Corrections Table 4.5 shows the asymmetries measured i n the correction runs, while t y p i c a l rates are shown i n Table 4.6. 4.6.7 Raw and Corrected Asymmetries Table 4.7 shows the asymmetries measured, as wel l as the corrected asymmetries. In subsequent experiments using unpolarized beam, i t has been found that the instrumental asymmetry remains constant to a l e v e l below the s t a t i s t i c a l error. This implies that the va r i a t i o n i n ET_ Table 4.5 Correction Run Asymmetries Primary Beam Spin Energy (MeV) 225 225 32.7 32.7-52G 520 up down up down up down ei "(carbon) 0.049O:':±. .0020 -0.0490 ± .0020 0.0602 ± .0035 -0.0602 ± .0035 0.1462 ± .0168 -0.1462 ± .0168 £ 2 (empty target) £3 (empty target) £3 (carbon) 0.187 ± 0.067 -0.295 ± .046 .5085 ± .0168 -.6637 ± .0120 .3604 ± .0274 -.2402 ± .0187 0.289 ± .066 -.264 ± .070 0.289 ± .031 0.041 ± .051 .339 ± .038 -.224 ± .038 0 0 .207 ± .072 -.156 ± .095 .104 ± .034 -.179 ± .039 ..^Instrumental Asymmetry in E ^ 225 .0034 ± .0005 327 (runs 1001-1009) .0014 ± .0002 327 (runs 1024-1044) .0090 ± .0004 520 .0102 ± .0002 Table 4.6 Rates for Background Subtractions. Primary Beam Energy (MeV) 225 32.7 520 ei (carbon) (%) 6.5 9.5 14.0 E 2 (empty target) (%) 10.5 5.0 5.0 £3 (empty target) (%) 2.0 6.0 6.0 £3 (carbon) (%) 0 3.5 14.0 TABLE 0.7 MEASURED AND CORRECTED ASYMMETRIES AT 225 MEV Run it E!m delc E2m de2m e2c d e 2 C 63m <*e3m e 3 C d£3c - 0 . 2 5 6 7 0 . 0 0 0 6 - 0 . 2 7 6 9 - 0 . 0 0 0 9 - 0 , 7 7 2 7 ' 0 . 0 1 3 3 • - 0 . 7 8 5 3 0 . 0 1 0 2 0 . 1 0 5 6 0 . 0 6 8 9 0 . 1 9 8 3 0 . 0 7 8 3 i os L 0 . 2 5 6 1 0 . 0 0 OS 0 . 2 6 3 7 0 . 0 0 0 7 0 , 7 3 5 0 0 . 0 1 0 0 0 . 7 0 8 2 0 . 0 1 0 8 0 . 2 7 2 0 0 ' .0179 0 . 2 7 1 9 0 . 0 1 8 7 10?.? - 0 . 2 5 7 1 0 . 0 0 0 3 - 0 . 2 7 5 7 0 , 0 0 0 7 - 0 , 7 7 0 2 0 , 0 0 6 8 - 0 . 7 8 5 0 0 . 0 0 7 1 0 . 0 3 7 5 0 ' .0353 0 . 0 7 3 9 0 , 0 0 0 5 1053 0 .2=530 0 . 0 0 0 6 ' 0 . 2 6 3 2 0 . 0 0 0 8 0 . 7 0 0 9 0 . 0 1 2 6 0 . 7 5 6 6 0 , 0 1 2 9 0 , 2 0 2 1 0 . 0 2 2 1 0 . 2 0 0 7 0 . 0 2 2 8 los-i - 0 . 10 30 0 . 0 0 0 3 - 0 . 1 5 3 0 0 , 0 0 0 6 - 0 , 7 5 1 6 0 . 0 0 6 0 - 0 . 7 6 0 2 0 . 0 0 6 7 0 . 1 2 1 3 0 . 0 3 1 3 0 . 1 7 0 7 0 . 0 3 7 0 I 05u 0 . 2 5 3 3 0 . 0 0 0 5 0 . 2 6 0 0 0 . 0 0 0 7 0 , 7 2 0 9 0 , 0 1 0 9 0 . 7 3 2 6 0 . 0 1 1 2 0 . 2 6 0 6 0 . 0 1 8 7 0 . 2 6 3 9 0 . 0 1 9 0 loss - 0 . 2 0 9 7 0 . 0 0 0 3 - 0 . 2 2 0 5 0 . 0 0 0 6 - 0 , 6 0 0 0 0 . 0 0 6 7 - 0 . 6 5 1 9 0 . 0 0 7 0 0 . 1 9 3 0 0 . 0 230 0 . 2 2 8 9 0 , 0 2 5 8 IDS' 3 - 0 . 2 5 0 6 0 . 0 0 0 0 - 0 . 2 7 0 6 0 . 0 0 0 7 - 0 , 7 7 0 3 0 . 0 0 9 0 - 0 . 7 8 0 9 0 . 0 0 9 7 0 , 0 7 8 0 0 . 0 0 7 7 0 . 1 1 6 6 0 . 0 5 3 7 I do3 y . 2 0 9 0 0 . 0 0 0 5 0 . 2 5 9 5 0 . 0 0 0 8 0 , 7 0 0 3 0 . 0 1 2 2 0 . 7 1 7 0 0 . 0 1 2 6 0 . 2 3 8 7 0 . 0 2 0 9 0 . 2 3 6 9 0 . 0 2 1 7 1 U-J-4 - 0 . 2 5 5 3 0 . 0 0 0 3 - 0 . 2 7 3 0 0 . 0 0 0 6 - 0 . - 7 5 1 8 0 . 0 0 7 1 - 0 . 7 6 2 9 0 . 0 0 7 3 0 , 1 6 5 6 0 , 0 3 0 5 0 . 2 2 0 0 0 , 0 3 9 9 UH>7 0 . 2 0 9 1 0 . 0 0 1 9 0 . 2 5 9 9 0 . 0 0 2 1 0 . 7 3 5 2 0 . 0 1 1 2 0 , 7 3 5 9 0 . 0 1 1 2 0 , 0 0 0 6 0 , 0 7 5 1 • 0 . 0 0 5 8 0 . 0 7 7 7 lOufl - 0 . 2 5 13 0 . 0 0 2 0 - 0 . 2 6 9 3 0 . 0 0 2 2 - 0 . 7 0 8 3 0 . 0 1 2 0 - 0 , 7 0 9 0 0 . 0 1 2 0 0 . 3 8 0 6 0 . 2 5 6 0 0 . 5 1 3 7 0 . 3 0 7 3 1070 0 , 1 3 2 3 0 . 0 0 0 5 0 . 3 395 0 , 0 0 0 8 0 , 7 3 9 0 0 , 0 1 0 3 0 . 7 5 1 7 0 . 0 1 1 2 0 . 2 3 0 8 0 . 0 1 9 1 0 . 2 2 3 8 0 . 0 1 9 8 1071 - 0 . 2 5 0 3 0 . 0 0 00 - 0 . 2 7 2 9 0 . 0 0 0 7 - 0 , 7 0 9 3 0 , 0 0 8 5 - 0 . 7 6 1 0 0 . 0 0 8 9 0 . 0 8 8 2 0 . 0 0 0 6 0 . 1 3 0 6 0 . 0 0 6 9 107-4 0 . 2 5 1 7 0 . 0 0 0 6 0 . 2 6 2 0 0 , 0 0 0 8 0 , 7 5 1 0 0 . 0 1 3 2 0 . 7 6 0 1 0 . 0 1 3 6 0 , 2 7 5 8 o'.osoo 0 . 2 7 5 3 0 . 0 2 5 0 1075 - 0 . 2 5 3 3 0 . 0 0 0 3 - 0 . 2 7 1 8 0 , 0 0 0 6 - 0 . 7 2 9 0 0 , 0 0 7 7 - 0 . 7 0 0 3 0 . 0 0 8 0 0 . 1 9 0 3 . 0 . 0 3 5 1 0 . 2 0 8 3 0 . 0 0 0 6 *This t\ data not used. TAF5LF *!,7 CONT D Run i- delc G2m de2n 107S 0 .2520 0 ,000'4 0 .2627 0 .0007 0.7/178 0,0119 1079 -0 .2537 0 .0005 - 0 .2716 0 .0007 -0,7/156 0.0112 1 08 0 -0 .25*15 0 .0003 - 0 .2725 0 .0006 -0.7325 0.0088 1083 0 . 19GS 0 '.0007 0 .200 ' ! 0,0009 0.7706 0.01/18 1087 0 .2507 0 . 0007 0 .2612 0.0009 0.7130 0,0155 1055 0 .0031 0 .0006 - 0 . 0 165 0,0206 1 (H,0 0 .003 3 0 .0017 -0.0/1*17 0.0287 l o o t o .0027 0 . 0012 -0.0915 0.016/1 10u2 0 ,0o3 ' l 0 .0017 -0.0239 0.0233 10w5 0 , 0035 0 .0007 -0,0088 0.0228 1 (loo 0 . 0 0 '18 0 .0021 0,0562 0.0171 !) .0007 0 .0031 0,0938 0.0263 i 0 M n . 0 062 0 .0005 0,0379 0.0170 lOSo 0 .0077 0 .0006 -0.0560 0.0203 *This c\ data r.ot used. £2c de2c G3m 0.7711 0.0127 0.255/1 0.757a 0.0116 0.1698 0.7506 0.0093 0.1501 0,78/13 0.0152 0.2323 0.72/16 0,0160 0.27/11 0,2235 0,2609 0,2600 0.2222 0.1966 0.16/16 0.3125 0.2950 0.2/109 de3n e3c <Je3c 0'.0212 0.253/J 0.0229 o'.0512 0.2213 0.0578 0',0*I06 C.2'U5 0.0519 0.0281 0.2303 0.0291 0.0 265 0.2736 0.027*1 0'.0330 0'. 1006 0,0966 0.1625 0.0369 0.1110 0.1679 0.0263 0.0313 TARLH: *I,7 MEASURED AND CORRECTED Run 9 eic clelc S2m dE2m 1002 -0.2893 0.0003 - o . 3155 0.0006 -0,5819 0.0112 1003 0,28 1'4 0.0003 0. 3030 0,0005 0.5859 0.0102 10 0-4 -0.295a 0.0003 -0. 3218 0.0006 -0,5976 0.0108 lo or. -0,2965 0 .0 0 0.3 -0. 3230 0,0005 -0.599/1 0.0087 1 OOo -0.2943 0.0003 -0. 320U 0,0006 -0.57*10 0.0105 10 03 0.2896 • 0.0 003 . 0. 3120 0,0005 0.6073 0.0100 10 2*1 0.2939 0.0003 0. 313/1 0,0007 0,5679 0,0110 1025 -0.29 12 0.0010 -0. 3250 0,0012 -0,5705 0.0307 1 02u 0.2978 0,0001 0. 3130 0.0007 0.5730 0.012/1 1023 -0.2969 0.0 0 03 -0, 3323 0.0007 -0,6098 0,0087 1030 0,2956 0.000-4 o. 3108 0,0007 0,5720 0.0139 1033 -0.2898 0.0002 -0. 3245 0,0006 -0,6093 0>073 10 3'4 0.2921 0 .000/4 0. 3072 0,0007 0.5602 0,0125 100 1 0,00 15 0.0003 0.0121 0.011/1 1007 0,0 0 12 0 .0005 0.0027 0,0113 1009 0,0013 0.0009 0,0238 0,0136 102° 0.0076 0.0007 -0,01/19 0.0280 10 32 0,0093 0.0 0 05 -0,01 i a 0.0209 1035 0.0090 0,0010 -0.026/1 .0,01/16 AT 327 MEV e2c dc2c e3m -0.5770 0,01 19 0.0319 0,5857 0.0107 0.3212 -0,5936 0,0115 0.0221 -0.5956 0.0093 0,003/1 -0.5686 0.0112 0,0215 0.6081 0.010/1 0.3173 0.5671 0.0115 0.331/1 -0.5733 0,032/1 0,0258 0.5723 0.0130 0,32/15 -0.6067 0.0093 0.0222 0.571/1 0.0105 .0.2588 -0,6062 0,0078 -0,0012 0,5590 0.0131 0.3/103 0.266 0 0.2280 0,2/181 0,2970 0,2869 0./1211 de3m e3c d£3c 0.0230 0.0390 0.0263 0.0103 0.3271 0,0117 0.0227 0.0282 0.0260 0.018/1 0.0071 0.0213 0.0216 0.027*1 0,02*18 0,010/1 0.323*1 0,0119 0.0107 0.3375 0.0119 0,0607 0.0311 0.0668 0.0122 0.3302 0.0135 0.0186 0.0279 0.021*1 0.01*16 0.2591 0.0162 0,015*4 0.0018 0,0178 0.0120 0.3'47<4 0.0133 0.0118 0.02*19 0.0603 0.0286 0.0216 0', 0 6 0 1 TABLE 0.7 MEASURED AND CORRECTED Run it delm E2m de2m 200 1 0 .3069 0 .0003 0. 3687 0,0008 0,3023 0.0213 2 0 02 0 ,3513 0.0003 0. 3751 0.0008 0,320 7 0.0200 2003 -0,3305 0 .0008 -0. 3778 0.0011 -0,3602 0.0050 2000 -0 .3315 0 .0 003 -0. 3700 0.0008 -0,3576 0.0167 2005 0 ,3397 0 .0003 0. 36 0 6 0,0008 0,3200 0.0196 2006 -0 ,3329 0 .0003 -0. 3759 0.0007 -0.3929 0.0159 2003 0 ,3331 0 .0 003 0. 3585 0.0008 0.3693 0.0198 20 09 -0 ,33 I 1 0 . 0003 -0. 3737 0,0007 -0,3088 0.0158 20 1 I U .3369 0 .0000 0. 3573 0,0008 0.3321 0,0233 2012 -0 .3312 0 .0003 -0. 3739 0.0007 -0.3678 0,0162 20 13 -0 .3292 0 , 0003 -0. 3716 0,0008 -0,3271 0.0163 2015 0 .3393 0 . 0005 0. 3607 0.0009 0.3090 0.0295 20 lo -0 , 330 0 0 .0003 -o, 3726 0.0008 -0,3291 0.0210 20 17 0 ,3393 0 .0005 0. 3603 0,0009 0,3091 0,0330 2013 -0 .3235 0 .0000 -0. 3712 0,0008 -0.3663 0.0222 20 20 -0 .3263 0 . 0000 -0. 3685 0,0008 -0,3601 0,0221 AT 520 MEV tzc 0.3056 0.3271 -0.3616 -0.3590 0,3220 -0,3961 0,3739 -0,3098 0,3350 -0.3698 -0.3270 0.3105 -0,3291 0,3528 -0,3681 -0.3659 de2c 0.0226 0,0211 0,0070 0.0175 0,0206 0,0167 0.0209 0.0166 0.0207 0,0170 0.0192 0.0310 0.0220 0,0309 0.0233 0,0232 £3m 0,1091 0,3262 -0.2029 -0,1981 0,3006 -0,2006 0.3019 -0,1576 0.3637 -0.1699 -0.1017 0,2823 -0.1620 0,30 1 1 -0.1970 -0,1858 de3 m 0,0201 0,0158 0,0580 0.0205 0,0157 0.0192 0,0169 0.'0192 0.0180 0.0198 0.0213 0.0233 0.0253 0.0261 0,0268 0.0270 e 3 C 0.1019 0.3602 -0.2552 -0.1988 0,3830 -0.2020 0.3370 -0.1083 0.0115 -0.1635 -0.1291 0.3101 -0.1502 0,3826 -0.1975 -0.1830 0.0283 0,0207 0.0730 0.0266 0,0208 0.0250 0.0226 0.0251 0,0230 0.0260 0,0280 0.0305 0,0325 0,0332 0.030 1 0,0308 TABLE '1.7 CONT D Run if "!02.l 20 22 2023 202*1 "'025 !02o !027 02° 030 0 3 1 032 033 035 o3u 037 0 39 0,3375 -0.3236 .0,3395 -0,3290 0,3^ 39 -0.3302 -0,3237 0,3'! 79 -0,34 10 -0 , 3.377 -0,3386 0.3526 -0.3330 -0.3392 0.3536 -0.3'417 d e l m 0.0003 0.0003 0.0003 0-.0003 0.0005 0.000" 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0'.0003 0.0003 0.0 0 03 0.0003 0.358/1 -0.3715 0.36!0 -0.3720 0,366/1 -0,3737 -0.3720 0.371 1 -0.3366 -0.3327 -0.3339 0.3770 -0.3833 -O.33/10 0,3772 -0.3865 delc 0.0008 0.0008 0.0008 0,0008 0.0009 0.0008 0.0008 0,0008 0,0008 0.0008 0,0008 0.0008 0,0 0 08 0.0008 0.0008 0. 0008 e2m 0.3130 -0,3736 0.3062 -0,3621 0,3390 -0,383/1 -0,385/1 0,3569 -0,3622 -0.3656 -0,3728 0,3/182 -0,3595 -o./iono 0,3110 -0,39/17 dE2m 0,0212 0,0165 0,0211 0.0172 0'.0289 0.02/11 0.0195 0,0208 0,0165 0.0169 0.0 168 0,0209 0.020*1 0.0162 0.0227 0,0211 e2c de2c 0'.31*I8 0,022*1 0 . 3 3 1 0 -0.3758 0,017*1 -0,1782 0,3076 0,0223 0.3361 -0,3633 0,0181 -0,1586 0,3*120 0.030*1 0,3186 -0.3912 0.0252 -0,1627 -0.3882 0.0205 -0,1828 0.3610 0.0220 0,3192 -0,3637 0,0173 -0,2221 -0.367*1 0.0177 -O.J 679 -0,37*19 0.0176 -0,1857 0.3517 0,0220 0,3265 -0,3610 0,021*1 -0,1862 -0./I117 0.0169 -0,1963 0,3126 0,0239 0.3217 -0.3980 0.0221 -0,2316 d£3m E 3C d£3c 0.0166 0.3733 0.0215 0,0201 -0,1711 0.0261 0.0162 0.3757 0,0211 0.0210 -0, 1196 0.0272 0,0238 0.3560 0.0305 0,0306 -0. 1515 0,0391 0,0211 -0.1797 0.0313 0,0171 0„358a 0,0230 0,0207 -0,2288 0,0269 0.0209 -0.1615 0.0268 0,0210 * -0.1831 0.0273 0.0169 0,3618 0.0221 0.0251 -0.1811 0.0319 0.0206 -0,1966 0.0267 0,0188 0,3635 0.0252 0.0259 -0,2115 0,0332 TABLF 0 . 7 CONT D Run # 21) '10 2007 2010 "•013 '01° M23 '034 0 38 'O'l 1 0.3507 0,0 08 1 0.0036 0.0 105 0.0106 0.0123 0.0120 0,0 1 O'l 0.0095 delm 0.0005 0.0008 0.0005 0.0006 0.0005 0.0007 0.0006 0.0007 0.0005 0 . 3 7 8 5 0 , 0 0 0 9 G2B 0 . 3 2 5 6 0 , 0 5 7 1 0 , 0 5 5 9 - 0 . 0 0 5 5 - 0 , 0 0 9 9 - 0 . 1 0 6 2 0 , 0 0 2 6 - 0 , 0 0 6 1 - 0 , 0 2 6 1 d e 2 m 0 . 0 3 0 6 0 , 0 5 0 7 0 . 0 3 0 1 0 . 0 3 7 0 0 . 0 3 5 2 0 . 0 0 0 5 0 . 0 3 6 3 0 .0O21 0 . 0 3 2 3 E 2 C 0 . 3 2 8 2 d e 2 c 0 . 0 3 2 0 E3m 0 , 2 8 9 9 0 . 1 192 0 , 2 0 0 6 0 , 1 2 8 0 0 . 1 5 3 5 0 . 1 3 2 1 0 . 2 1 0 6 0 , 2 3 2 0 0 . 1 0 2 1 d e 3 m 0 , 0 2 5 7 0 , 0 5 0 5 0 ' .0317 0 . 0 3 0 3 0 . 0 3 1 0 0 . 0 O 1 1 0 . 0 3 3 9 0 . 0 3 9 3 0 , 0 2 9 9 E 3C 0 . 3 2 0 3 d e 3 c 0 . 0 3 0 0 - 59 -shown i n Table 4.7 i s due to changes i n the beam polarization rather than systematic errors i n the polarimeter. The £ 2 and £3 asymmetries show variations consistent with their s t a t i s t i c a l errors. 4.6.8 Corrections to d3 As explained i n Appendix B, £3 was corrected for instrumental asymmetry. The values of P(24) and P^e have been taken from the l i t e r a -ture. Phase s h i f t analysis (PSA) has given the curves of P(24) and P(17) as shown i n Fig. 4.9. These have a common normalization, and so are used to produce n. In taking P from ref. Si, the values were normal-ized to P(24) as shown i n Table 4.8. To decide on which values of r\ to use i t was necessary to calculate the energies of the protons at the centers of the E J , £ 2 and £3 targets. This calculation appears i n Appendix C, with the energies l i s t e d i n Table C l . Interpolation of the r e s u l t s , together with the values used for P(24) and the t y p i c a l corrections C, give Table 4.9. 4.7 Results ± The f i n a l values of P(24), P R e and P.^  were calculated from eq. B.12, B.14 and B:.15i v i z - 60 -0.6 0.48 0.36 0.24 0.12 PC243 . P U 7 ) 0.0 "T 1 1 1 1 T 200.0 300.0 400.0 500.0 E ( M E V ) Figure 4.9: P(24) , P(17) from PSA. The data points are taken from the phase shift analysis of Bugg. The curves shown are a spline least squares f i t to the data points. Table 4.8 Renormalization of PTT from Stetz, et a l . P.(17°) - PHe P(17°) from Al and E (MeV) from S 2 from Si t) from PSA 200 .901 ± .030 .305 .305 ± .015 350 .730 ±..030 .435 .440 ± .007 500 .500 ± .030 .505 .499 ± .015 PHe renorm. .901 ± .050 .744 ± .030 .506 ± .050 Table 4.9 Values Primary Beam Energy (±1 MeV) 225 327 520 of P R e, n, P(24) used Energy at e 2 t g t . center P (±1 MeV) 222 .885 ± 325 .772 ± 518 .480 ± to correct d3> Energy at £3 tgt. center (±1 MeV) . .030 205 .030 308 .030 499 P(24) n .285 ± .007 .856 ± .010 .354 ± .004 .354 ± .004 .417 ± .009 .817 ± .010 C .0020 ± .0007 .0040 ± .0004 .0032 ± .0011 - 63 -± _ ^ ( S i * ± P i " 2P(24) and P H e ( 1 5 ) = d2P(24)/ndi with d i , d 2 and d 3* being differences of the respective spin up and down asymmetries, and S i * the sum. The errors were calculated from eq. B..13 and B.16. As an example of how the various error "terms contribute to the t o t a l error i n P(24) a t y p i c a l run i s shown below: P(24) = 0.413 ± .040 2V^- dn = 0.010 2V^- = 1.00 9n an 2P |5- ddi = 0.001 2P|J- = 1.10 3di * ddl 2P|5~ d d 2 = 0-006 2P |5- = 0.242 3d 2 od2 2 P § 3 ^ ' d d 3 * = 0 ' 0 3 1 2 P t f e = °'927 The above indicates that each p a r t i a l derivative contributes about the same amount, and that i t i s the error measured i n each of n, d i , d 2, d 3 and S i * which determine the r e l a t i v e importance of each. Clearly i t i s - 64 -d3 which contributes most to the error in'P(24). The results of P(24), PTT (15°), P. + and P.~ are shown i n Fig. 4.10-. He ' I x • 4.13, and are summarized i n Table 4.10. No evidence was found for systematic errors. From on-line informa-t i o n , giving asymmetries as a function of ".time, runs were compared for consistency of q and z^, with the result that no effects were detectable within the accuracy of the plots. An alternative method of analysing the data has been started, which performs a x 2 minimization on the data and gives an estimate of correlated errors, and possibly systematic errors as w e l l . U n t i l this method i s completed, the results presented here must be regarded as preliminary. 4.8 Comparison of the Results with Previous Data Previous data for P(24) used for comparison at these intermediate energies come mainly from Cheng et a l . . The p-p double scattering experiment, which was a competing experiment with the one described here, i s also used. Measurements of PTT have come from Chamberlain et a l . and Stetz et He a l . These data are plotted together with those presented here i n Figs. 4.14 and 4.15. The results for P(24) are i n good s t a t i s t i c a l agreement over the entire range 200-500 MeV. The results for P H e agree, within s t a t i s t i c s , with those of Stetz et a l . at 323 and 515 MeV, but are 3 standard deviations different at 223 MeV. They are, however, i n disagreement with the results of Chamberlain et a l . , at 312 MeV. 0.5 P ( 2 4 ) 0.44 0.38 0.32 0.26 Where error bars are missing, the error i s smaller than the point. 0.2 200 .0 300 .0 400.0 500.0 E ( M E V ) Figure 4.10: P-P Analysing Power. The curve-shown i s a spline least squares f i t to the data points. Figure 4.11: P-He4 Analysing Power. The curve shown i s a spline least squares f i t to the data points. 0.9 0 .84 0 .78 0.72 0.66 0.6 R Figure 4.12: points. 200 .0 1 Where error bars are missing, the error i s smaller than the point. 300 .0 400.0 500.0 E ( M E V Spin Up Beam P o l a r i z a t i o n . The curve shown i s a spline least squares f i t to the data 0.9 0.84 0.78 0.72 0.66 0.6 R 200 .0 300 .0 Where error bars are missing, the error is smaller than the point. 400.0 500.0 E ( M E V ) Figure 4.13: Spin Down Beam Polarization, points. The curve shown is a spline least squares f i t to the data Table 4.10 Results Primary Beam Energy (±1 MeV) Energy at e 2 t g t . center (±1 MeV) rHe Energy at £3 t g t . center (±1 MeV) P(24) 225 0.763 ± 0.002 -0.796 ± 0.003 222 327 0.727 ± 0.003 -0.763 ± 0.003 325 520 0.715 ± 0.010 -0.735 ± 0.010 518 0.964 ± 0.012 205 0.792 ± 0.012 308 0.476 ± 0.018 499 0.294 ± 0.003 0.354 ± 0.005 0.412 ± 0.013 0.5 0.44 0.38 0.32 0.26 0.2 P (24) 7HI5 EXPT flMSLER ET Bl CHENG ET RL Where error bars are missing, the error i s smaller than the point. 200.0 300.0 400.0 500.0 E (MEV) Figure 4.14: Comparison of P(24) Results. The results of this experiment are compared with those of Amsler and Cheng. The curve shown i s a spline least squares f i t to the data points. 1 .0 0.88 0.76 0.54 0.52 0.4 HE ° THIS EXPT A STETZ ET RL f C H P M B E R L R I N 200.0 300.0 400.0 500.0 E ( M E V ) Figure 4.15: Comparison of P^e Results. The results of this experiment are compared with those of Stetz and Chamberlain. The curve shown i s a spline least squares f i t to the data points. i - 72 -V. Conclusion The analysing power of hydrogen for protons, at 24°, has been measured at the energies of 205, 308 and 459 MeV, to an accuracy ±0.003, 0.005 and 0.013, respectively. These results are i n agreement with those previously e x i s t i n g . The error l e v e l i s comparable to the p-p double scattering result at 308 MeV, higher by a factor 2 at 499 MeV, and lower by a factor 2 at 205 MeV. The error i s larger at 499 MeV since, due to limited experimental running time, less data was taken than desired. As desired, an improvement has been made i n the error l e v e l at 205 MeV, over the p-p r e s u l t . This experiment does provide P(24) to better than 2% at 205 MeV and 308 MeV, but f a i l e d to do so at 459 MeV. The analysing power of helium for protons, at 15°, has been mea-sured at 223, 325 and 518 MeV, to an accuracy of ±0.012, 0.012 and 0.018,respectively. These results are i n agreement with Stetz et a l . above 300 MeV, but show disagreement below, and also with Chamberlain et a l . at 312 MeV. The spin p o l a r i z a t i o n of the proton beam at TRIUMF has been measured at 225, 327 and 520 MeV. The s t a t i s t i c a l errors quoted do not account for variations i n the polarization i t s e l f which can be larger than the s t a t i s t i c a l errors. The polarization magnitudes are indeed different between spin up and spin down, the difference being t y p i c a l l y 0.02-0.03. The magnitudes decrease with increasing energy as expected, since the polarized protons are exposed to more depolarizing effects i n the cyclotron as their energy (and r a d i a l distance from the - 73 -cyclotron center) increase. If the experiment were to be repeated, i t would be desirable to take a larger r a t i o of data with spin o f f . This would provide an alternative method of analysing the data which would be free of the necessity of using the parameter n, though i t requires the use of a X 2-minimization to determine the res u l t s . With s u f f i c i e n t shielding from the beam pipe, ej and &2 targets, higher beam currents would decrease the time required to perform the measurement. In sum, this measurement i s a novel va r i a t i o n of the double scatter-ing technique which produces accurate r e s u l t s , and can be done i n such a way as not to require any input from existing data, or assumptions on the form of the data. - 74 -References Ai C. Amsler et a l . , J . Phys. G. (to be published). Bx F. Betz et a l . , Phys. Rev. 148, 1289 (1966). B 2 D.V. Bugg (private communication). B 3 D.V. Bugg, "Proton-Proton E l a s t i c Scattering from 150 to 515 MeV", Rutherford Lab. Report, RL-77-146/B, Dec. 1977. B^ J. B y s t r i c k i et a l . , " E l a s t i c Nucleon-Nucleon Scattering Data 270-3000 MeV", Saclay Report, CEA-N-1547(E). Ci "CAMAC Tut o r i a l Issue", IEEE Trans. Nucl. S c i . , 18, 1 (1971). C 2 D. Cheng et a l . , Phys. Rev. 163, 1470 (1967). C 3 .0. Chamberlain et a l . , Phys. Rev. 102, 1659 (1956). Ck M.K. Craddock et a l . , Nature, 270, 671 (1978). Gi C.A. Goulding et a l . , Nucl. Instr. and Meth. 148, 11 (1978). H l N. Hoshizaki, Prog, of Theor. Phys., 42, 1 (1968) (Suppl.). L i G.A. Ludgate, (Thesis), Rutherford Lab. Report, HEP/T/62, Oct. 1976. Mi R.H. McCamis, (Thesis), University of Alberta (1978). 01 C.J. Oram, (Thesis), Rutherford Lab. Report, HEP/T/65, May 1977. 0 2 C.J. Oram, (private communication). Rl J.R. Richardson, Nucl. Inst, and Meth., 24, 493 (1963). R 2 G. Roy, "Polarization Measurements at the Ion Source", TRI-DNA-75-7 (1975). 51 A.W. Stetz et a l . , Nucl. Phys., A290, 285 (1977). 5 2 A.W. Stetz, "Optimizing the Resolution and Calibrating the Ef f i c i e n c y of Nal Counter Telescopes", TRI-DNA-75-4 (1975). Wi L. Wolfenstein, Phys. Rev., 85, 947 (1952). - 75 -Appendix A H 2 This appendix describes the density matrix formalism"^ used to determine the expectation values of polarization and d i f f e r e n t i a l cross section for beams of p a r t i c l e s with spin % il. Expectation values are given by < A > - T r ( W A ) A . l TrW where W i s the density matrix and A an arbitrary operator. The scatter-ing process i s characterized by the matrix M such that the f i n a l density matrix i s related to the i n i t i a l matrix by WV = MW.M+. A.2 f 1 The d i f f e r e n t i a l cross-section i s given by the r a t i o TrW-I = —-±- A.3 TrW. I and W. i s normalized so that l TrW± =1. A.4 For an incident beam of spin p a r t i c l e s - 76 -W. = h (1 + P. -cr) . A.5 i ^ — i — Here 1 i s the unit matrix, cr i s the vector formed by the set of Pauli spin matrices, and P_ = <— >± ^ s t^ i e incident p o l a r i z a t i o n , defined as the expectation value of a_ for the incident beam. For an unpolarized beam, the d i f f e r e n t i a l cross-section i s I = hTrmt A. 6 o and for a polarized beam I = !$TrMM+ + %P.*TrMaM+. A. 7 Ho Under time reversal invariance TrMaM+ = TrMM+a A.8 and so 1 = 1 (1 + P.-P) A.9 o — 1 — where P = Tr(MM +o)/l . — — o Note that, since the polarization after the interaction i s given by P E <o>f = Tvimjt^/Trimjt) A. 10 the vector P_ i n fact represents the pol a r i z a t i o n which would result from the scattering of unpolarized protons. Choosing the co-ordinate axes (Fig. A.l) to describe the scatter as p = (Z ± + I f > / l l i + P f IJ K = (i\ - l_f)/\7± - P f I; A. 11 fi = P. x P,/|P. x P J . —x — f '—x — f 1 Substituting for i n eq. A.5 gives - • TrMM+a + I P Tr (MakM+£) P = k = ! L A.12 f TrMM+ + P1«Tr(MaM+) Using the d e f i n i t i o n s 1 1 2 of the Wolfenstein parameters I D = TrMa M+a o n n I R = TrMac M +a c o S. Sj. x f I A = TrMa„ M +a c A-!3 O P. Sj. X f - 78 -Figure A . l : Orthogonal Vector Sets (n,S. 5S f), (h,P i,P f). The two sets of orthogonal vectors used to describe the scattering of a beam of p a r t i c l e s are shown. - 79 -I R' = TrMa0 M+a„ o S. Pj. 1 f I A* = TrMa„ M+a„ o P. P_ 1 f where S. = n x P. / . A. 14 S f = n x P as shown i n Fig. A . l . Using eq. A . l and defining P ^ = TrMM+£ one gets P. S. P. S ^ (P„ + DP. n)n + (AP. 1 + RP. 1 ) S ^ + (A'P. 1 + R'P. )P. _ . He l i i f i i f • . -, r p _ : ^ A. 15 - f 1 + P. • Jv. — l —He - 80 -Appendix B This appendix describes the derivation of an equation for P(83> and the approximation used to correct d 3 for instrumental asymmetry under the assumption that |p/'"|^|Pi |, with the +,- superscripts denoting spins up and down. To account for differences i n e f f i c i e n c i e s and misalignments i n the polarimeter used to measure asymmetries, instrumental asymmetries are introduced. Assuming that the detection apparatus measures G r o f ( 8 + d 0 1 ' * = d < h ) " C 2 f § ( e + d 9 2 ' * = * + d ^ 2 ) e - — - j -5 B . l 1 1 1 6 3 8 + d 8'l' * = d < f > l ) + C 2 d § ( 8 + d e 2 ' * = U + M 2 ) where Gi and C 2 characterize the e f f i c i e n c i e s of the polarimeter arms and d0i, d02, d<J>i_ and d<|>2 characterize misalignments. Write C l | f i ( 0 + d0i,. * = d*i) = Ci^(6,<|> = 0) E C L L c 2 ^ ( e + de2, * = TT + d<j)2) = cR-|y(e,<j> = TT) = C RR then C L L - CR R 'meas CT L + CDR which can be rewritten as C_R L - R CT + H L + R £ m e a S 1 + . C L - CR ; L - R CT + CL L + R - 81 -From eq. 2.4 and with « C I - - C R one gets 6 + P.P(G) 1 B.3 'meas 1 + 6P iP(6) where P^ i s the polarization of the beam incident on the polarimeter and P(6) i s the analysing power of the polarimeter target. Note that with P. = 0 l e =6 meas which indeed shows that 6 i s an asymmetry inherent to the detection device. Including instrumental asymmetry, 6 3 , i n e 3 gives , "from .eq. 2.8 6-3 + a ±P ( 6 3 ) e 3 j • B.4 1 + a P ( 6 3 ) 6 3 ± ± ± where CT = (PTT + P. ) / ( l + P„ P. ), which i s the polarization of the He 1 He 1 ' R beam scattered from the helium target. Also P^e = ^ e ^ ® 2 ) • Taking d 3 = e 3 + - e 3 gives 63 + o +P 6 3 + o~P d s = 1 + Sqa+P'l + 6qcTp B ' 5 - 82 -where P = P(9 3) Assuming that 63 i s small, then CI3 can be approximated by d 3 = (a+ - a )P + C B.6 where C = - 6 3 P 2 ( a + 2 - a - 2) - 6 3 2 p ( a + - a ) [1 - P 2 ( a + 2 + a+o-- + a - 2 ) ] B.7 To correct for instrumental asymmetry, C i s subtracted from the measuredd3, with the values of P(9 3) and P(02) obtained from the l i t e r a t u r e . With d 3* = d 3 - C, d o * = P u + P. He 1 1 + P u P-H He x P„ + P. He 1 1 + P u P." He x with P. > 0 and P. < 0. 1 x d 3* = (P. - P. )P 1 - P 2 t He i i ' 1 + P (P. + P. ) + P 2P. P. He x 1 He 1 1 B.8 Assuming that the instrumental asymmetry i s small, then from equations si m i l a r to B.6 and B.7, taking only the f i r s t order term i n 63 gives - 83 -d 2 = e 2 + - e 2 = ( P . + - P . ) P R E - 6 2 P 2 ( P . + 2 - P ±-2) with 6 2 j the instrumental asymmetry i n e 2. Since p/*" - - P ^ , then the term multiplying 6 2 i s small, so that i t can be neglected. Then d 2 = ( P . + - P . " ) P H E B.9 S i m i l a r l y , di = ei+ - E I ~ = ( P ± + - } ? ± ~ ) ? / T ] B.10 s l = £l + + e i ~ = ( P - + + P . ~ ) P / T V+ 26i and i s the instrumental asymmetry i n e^. Assuming that <Si has been measured with spin o f f , subtracting 261 from Si gives S i * = Si - 2<5i = ( P + + P ~)P/n B . l l Substituting these quantities into eq. B.8, one gets ndi 2 nd 3* 'd i S i * ' 2 d.l' 2 Ld2 J d 2 2 [2 • B.12 - 84 -The error i n P(24) i s obtained from the p a r t i a l derivatives 2 A N = — 'dl' 2 d 3 * "d 2 S j * " 2 " d i " 2 ' 2n . — + _ L d 2j dl d 2 2 2 dn 2d, n d 2 2 nd 3* + di-dl S i * dz" + T ~ nd 3* 2 'dl S j * " | - d! dl d 2 d 2 2 - ddi 2 P f 2 d d 2 = 3P 2 P 3 d ^ d d 3 * = 2_ d 2 rid] 1 2 T)do* + 2di d2< di S i * + d 2 2 .dd5 n dl * "dl Sj*" 2-. "di 2" d 2 + 2 — 2 dd 3* B.13 2 P f ^ S l * -nd 3 * dl S i * d^ + 2~ ds x* With the f i n a l error on P(24) obtained by adding these errors i n quadrature. Furthermore, the remaining unknowns are solved to give ( S i * ± di) P. = n _ / P (24) B.14 P H e(15°) = d 2P(24)/nd 1 B.15 S i m i l a r l y , the errors for P. and PTT (15°) are obtained from l He - 85 -9P ±~ (Si* ± di) 9n 2P(24) 1,™ n ••: 9P(24) P(24) 3n 9P. (Si* ± di) 9P(24) 9di ~ 2P(24) " 2P(24) 2 9Si* B.16 9P." 1 (Si* ± d x) 3di " 2P(24) n 2P(24)2 9d x 3P(24) The contribution from the remaining variables i s obtained by using 9P. 3X JJ /Q * + J N 3P(24) 9P(24)2 ( S1 * d l ) 9X B.16 where X i s either d 2 or d 3*. 9P H e _ d 2P(24) + d 2 m ^ 9n n2,di ndi 9n 3P R e _ d 2P(24) + d 2 3 p ( 2 4 ) 3di ndi 2 n d i dd1 B.17 9P„ T,/O/N . d 2 He = P(24) + ^ _ 9P(24) 9d 2 ndi ndi 9d 2 and the contribution from S i * and d 3* are obtained from 3 PHe = ^2_ 3 P(24) 9X ndi 9X B.17 as above. - 86 -Appendix C Determination of the Proton Energies at the Target Centers. The incident proton beam l o s t energy due to material i n the beam. At the ei polarimeter this consisted of a 0.0019 cm Ti window and the 0.0038 cm CH2 target. When traversing the LHe target, the protons passed through 0.0076 cm Kapton, 0.0254 i n stainless s t e e l LHe c e l l windows and 0.0127 cm Kapton on e x i t . Before scattering i n the £3 target the protons passed through the two 0.635 cm p l a s t i c s c i n t i l l a t o r s and 1.5 m of a i r . Table C l shows the beam energies at the centers of the three targets. Table C l Beam Energies at the Target Centers. Primary Beam Energy (± 1 MeV) Energy at £1 Target Center (± 1 MeV) Energy at E 2 Energy at £3 Target Center Target Center (± 1 MeV) (± 1 MeV) 225 225 222 205 327 327 325 308 520 520 518 499 - 87 -Appendix D Rotation of ej Counters for I n s e n s i t i v i t y to Beam Movement. This appendix shows the geometry behind rotating the defining counters i n ej_ to make the polarimeter insensitive to movement of the beam. The geometry i s shown i n Fig. D.l. For the beam centered on the target, the s o l i d angle seen i s n = A/r 2 where A i s the effec t i v e area of the counter and i s A = h£.s"in60° with h and £ the height and width of the counter- respectively. I f the beam s h i f t s by x mm, then the new distance, r', to the center of the target i s r ' 2 = r 2 + x 2 - 2r x cos 73° and the angle between r and r' i s Figure D.l: Effect of Beam Shift on e j . A s h i f t of x on the ej target results i n the shorter distance (r') to the counter center as w e l l as a smaller e f f e c t i v e counter area, with the s o l i d angle remaining constant. - 89 -The new effective area A' i s then A' = hH sin(60-6) so that the solid angle becomes hi sin(60-8) Qi = A>/r.2 r 2 + x 2 - 2r x cos 73° fi and fi' can be compared numerically: with r = 1270 mm, h = £ = 25.4 mm. Table D.l results, fi" i s evaluated using 90° instead of 60°. Table D.l Change in Solid Angle in ei as a Function of Beam Shift. fi' (st.) ft"; (st.) x (mm) O(deg) xl0~h x l O - 4 1 .04 3.4642 4.002 2 .09 3.4643 4.004 5 .22 3.4645 4.009 10 .43 3.4646 4.018 15 .65 3.4647 4.027 The solid angle for 60° i s 3.4641 x 10 4 st. and for 90° i s 4.00 x 10~k st. The choice of 60°, completely eliminates any effects from a reasonable shift in beam position, leaving any changes due to changes i n the differen-t i a l cross section with the angle 0. 

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