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The pion double charge exchange reaction on ¹⁸O at 50 MeV Hessey, Nigel P. 1985

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THE PION DOUBLE CHARGE EXCHANGE REACTION ON 1 8 0 AT 50 MeV. by NIGEL P. HESSEY B.Sc. University of Nottingham, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1985 © Nigel P. Hessey In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date fl OcXcA/y*^ I ^ S f DE-6(3/81) i i ABSTRACT This thesis discusses the pion double charge exchange (DCX) reaction 1 8 0 ( i r + , u ~ ) 1 8 N e at 50 MeV. Transitions to the ground state of 1 8Ne, which i s the double-isobaric-analogue state (DIAS) of 1 8 0 , have been i s o l a t e d . The d i f f e r e n t i a l cross sections for DIAS t r a n s i t i o n s have been measured at 6 s c a t t e r i n g angles from 18.2° to 122.6°. The experiment was performed at TRIUMF i n December 1984 using the QQD low energy pion spectrometer [26]. The d i f f e r e n t i a l cross section angular d i s t r i b u t i o n i s forward peaked, f a l l i n g from 4.7±0.5 yb/sr at 0° (by extrapolation) to 0.61±0.11 yb/sr at 122.6°. The t o t a l (angle-integrated) cross section i s 16.2±1.2 \ib. DCX measurements are expected to give information on nuclear structure that i s hard to obtain by other reactions. This information includes short range c o r r e l a t i o n s and neutron-proton density d i f f e r e n c e s . However, before such information can be extracted the mechanism for DCX must be understood. The aim of t h i s experiment was to provide more data to test the various theories of the DCX mechanisms. The implications of the r e s u l t s for several theories of DCX are discussed. The forward peaking of DCX angular d i s t r i b u t i o n s at 50 MeV was unexpected. 50 MeV s i n g l e charge exchange (SCX) angular d i s t r i b u t i o n s are forward dipped e.g. [14], a r e s u l t of the c a n c e l l a t i o n of the 0° s and p wave sca t t e r i n g amplitudes for the r e a c t i o n p(ir+,n°)n. E a r l y DCX c a l c u l a t i o n s were based on the simple sequential mechanism. This assumes DCX proceeds v i a 2 successive SCX reactions, with the i s o b a r i c analogue as the intermediate state. These c a l c u l a t i o n s predicted forward dipping and small cross sections f o r DCX [13,15]. The data shows t h i s mechanism i s an o v e r - s i m p l i f i c a t i o n . The standard model for ir-nucleus scattering i s the o p t i c a l p o t e n t i a l . Johnson and S i c i l i a n o are developimg a p o t e n t i a l with which to calculate i i i e l a s t i c , SCX and DCX cross sections [48,38,22]. They include second, order terms, important i n DCX because the reaction must involve scattering by at le a s t two nucleons. By using a general form for the o p t i c a l p o t e n t i a l they include contributions from excited intermediate st a t e s . M i l l e r has suggested the forward peaking i s due to the presence of six-quark c l u s t e r s i n the nucleus [16]. His model reproduces the data for 50 MeV DCX on 1 8 0 and lhC at forward angles. Karapiperis and Kobayashi have used the A-hole model to calculate e l a s t i c , SCX and DCX cross sections [19]. They obtain f a i r agreement with data f o r a range of n u c l e i and energies. Jennings et a l . [22] are developing a model i n which short range c o r r e l a t i o n s produce the forward peaking. This work i s at an early stage. More DCX measurements are needed to choose between the various models. Measurements at 50 MeV are p a r t i c u l a r l y valuable because the simple sequential mechanism i s small, allowing other mechanisms to be observed. Further data such as e x c i t a t i o n functions below 80 MeV and angular d i s t r i b u t i o n s f o r other n u c l e i are needed. i v TABLE OF CONTENTS Abstract i i Table of Contents i v L i s t of Tables v i L i s t of Figures • v i i Acknowledgements i x 1 INTRODUCTION 1.1 Pions 1 1.2 Charge Exchange Reaction 1 1.3 Target Nuclei 4 1.4 Energy 4 1.5 Models and Previous Experiments 4 1.6 Aim of This Experiment 9 2 THE EXPERIMENT 2.1 Beamline 11 2.2 Spectrometer 13 2.3 Targets 15 2.4 Runs 18 2.5 Data A q u i s i t i o n 18 2.6 On-line Analysis 20 3 ANALYSIS 3.1 Wire Chamber Cali b r a t i o n s 3.1.1 Signal Production In Wire Chambers 21 22 V 3.1.2 Sumcuts 25 3.1.3 Converting TDC Output to Position Coordinates 25 3.2 Calculating Pion Momenta 28 3.3 Derivation of Magnet Transfer Coefficients 29 3.4 Cuts 32 3.5 Peak Size Measurements 37 3.6 Beam Monitoring 39 3.7 Target Thickness 41 3.8 II-Decay Correction 41 3.9 Solid Angle 42 3.10 Cross Section Calculations 1 43 3.11 Error Analysis 43 4 RESULTS 47 5 THEORY 5.1 Introduction 55 5.2 Optical Potential Model Calculations 57 5.3 The Six-Quark Cluster Mechanism 60 5.4 Effect of Short Range Correlations 66 5.5 The A-hole Model 70 6 CONCLUSION 75 Bibliography 7 6 v i LIST OF TABLES Table 2.1 Target De t a i l s 17 Table 3.1 Wire Chamber Cali b r a t i o n s used and L a b e l l i n g System 24 Table 3.2 Transfer C o e f f i c i e n t s used to Calculate 6 5 30 Table 3.3 Transfer C o e f f i c i e n t s used to Calculate 6^  31 Table 3.4 Values of Cuts used During Analysis 33 Table 3.5 Target Traceback C o e f f i c i e n t s used 34 Table 4.1 Experimental D i f f e r e n t i a l Cross Sections f o r the Reaction 1 8 0 ( T T + , H + ) 1 8 0 (g.s.) at 48.3 MeV 48 Table 4.2 Experimental D i f f e r e n t i a l Cross Sections f o r the Reaction 1 80(TT +,TT") 1 8Ne (DIAS) at 48.3 MeV 50 v i i LIST OF FIGURES F i g . 1.1 The s h e l l model structure of the double i s o b a r i c analogues 1 8 0 and 1 8Ne 2 F i g . 1.2 E x c i t a t i o n function for SCX 5 F i g . 1.3 Dependence of DCX on A 6 F i g . 1.4 Angular d i s t r i b u t i o n f o r 1 80(TT +,TT~) 1 8Ne (g.s.) at 164 MeV 6 F i g . 1.5 DCX to DIAS e x c i t a t i o n functions for 1 8 0 and 2 6Mg 8 F i g . 1.6 SCX to the IAS angular d i s t r i b u t i o n f o r 1 1 +N, at 48 MeV 8 F i g . 2.1 The QQD low energy pion spectrometer showing p o s i t i o n of target 12 F i g . 2.2 Typ i c a l TCAP spectrum used to estimate the contamination i n the incident f l u x 13 F i g . 2.3 Target D e t a i l s (a) Target holder plan view. (b) Target holder front e l e v a t i o n . (c) Schematic target and holder cross section. 16 F i g . 2.4 Configuration of E l e c t r o n i c s 19 F i g . 3.1 Arrangement of wires i n QQD wire chambers 23 F i g . 3.2 The e f f e c t s of cuts made during the analysis on DCX 6 histograms 36 F i g . 3.3 T y p i c a l 6 histogram f o r an e l a s t i c run 38 F i g . 3.4 6 Histogram for 1 8 0 e l a s t i c s c a t t e r i n g at 30°, with OPDATA f i t 38 F i g . 4.1 E l a s t i c cross sections used f o r normalization 49 F i g . 4.2 DCX cross sections to the DIAS f or 1 8 0 51 F i g . 4.3 DCX cross sections f o r 1 8 0 compared to those f o r ll*C from [10] 53 F i g . 4.4 6 Histogram for DCX at 122° 53 v i i i F i g . 5.1 A meson exchange current contribution 55 F i g . 5.2 The simple sequential mechanism for DCX......... 59 F i g . 5.3 Formation of six-quark c l u s t e r s 61 F i g . 5.4 Simplest six-quark c l u s t e r contribution to DCX 62 F i g . 5.5 Comparison of t h e o r e t i c a l predictions with measured 50 MeV 1 8 0 DCX data 65 F i g . 5.6 The mechanisms considered by Jennings and de Takacsy 67 F i g . 5.7 1 8 F energy l e v e l diagram 70 F i g . 5.8 The basic DCX mechanisms i n the A-hole model 71 i x ACKNOWLEDGEMENTS I would l i k e to take this opportunity to thank a l l those who made this thesis possible, and made my time at TRIUMF so enjoyable. In p a r t i c u l a r I would l i k e to thank Dick Johnson, my research supervisor, f o r his guidance; Ami Altraan f o r so much help with the a n a l y s i s ; U l i Wienands f o r cross-checking the analysis; Dave G i l l for his commitment to the QQD spectrometer; and to the rest of the PISCAT group who did so many s h i f t s f o r data taking. F i n a l l y , a very big thankyou to my wife J u l i e f o r typing and e d i t i n g the th e s i s . 1 1 INTRODUCTION This thesis describes an experiment to measure the pion double charge exchange (DCX) cross-section of 1 8 0 , and discusses the r e s u l t s . The aim of the experiment was to obtain data that w i l l help our understanding of the DCX mechanism. 1.1 Pions Pions are the l i g h t e s t mesons, with masses around 140 MeV. They are sp i n l e s s , and they have three possible charge states, +1, -1 and 0. They are la r g e l y responsible for binding nucleons together i n n u c l e i . The charged pions decay by the weak i n t e r a c t i o n with a mean-life of 2.60xl0 - 8 s, the ir° decays by the electromagnetic i n t e r a c t i o n with a mean-life of 0.9xl0'" 1 6 s. Pions are produced at TRIUMF when protons with k i n e t i c energy 500 MeV react with a production target. This allows beams of i r + and TT~ to be produced, whilst the ir° decay too rapidly f o r beam production - at least at TRIUMF energies. For an introduction to pion-nucleus physics, see e.g. [1]. There are many reasons f o r using pions as nuclear probes. They i n t e r a c t with the nucleus v i a the strong i n t e r a c t i o n , whereas electrons i n t e r a c t v i a the electromagnetic i n t e r a c t i o n . So pions Interact with neutrons as well as protons whereas electrons i n t e r a c t almost only with protons. Furthermore the TT+ reacts d i f f e r e n t l y with a given type of nucleon - i . e . neutron or proton -to a TT~, due to isospin-space e f f e c t s ; so pions can probe differences between proton and neutron states [2,3,4]. 1.2 Charge Exchange Reactions Double charge-exchange (DCX) i s one of many reactions pions have with n u c l e i . Experiments usually use i r + beams; the i r + gives up two units of 2 charge to the nucleus and emerges as a u~. In the nucleus two neutrons are converted to two protons. The reaction 1 80(TT+,TT~) 1 8Ne i s an example of pion double charge-exchange. Pion s i n g l e charge-exchange (SCX) i s the related reaction i n which the TT+ changes to a ir°, and one neutron changes to a proton i n the nucleus. SCX experiments look f o r the two gamma rays from the ir° decay to s i g n a l an event. The most studied DCX reactions are those i n which the product nucleus i s the double isobaric-analogue state (DIAS). For an i s o s p i n T=l, T =-1 nuclear state, the DIAS i s the nuclear state with T=l, Tz=+1. For example, the DIAS of 1 8 0 i s the ground state of 1 8Ne ( F i g . 1.1). DIAS t r a n s i t i o n s can d 5 / 2 -•—• •—• P l / 2 O '• -e—o — P 3 / 2 — * ~ ° • -Energy — 0 - 0 0 — 0— s 1 / 2 — 0 — 0 — protons neutrons protons neutrons 18 0 1 8 N e Fig. 1.1 The s h e l l model structure of the double i s o b a r i c analogues 1 8 0 and 1 8Ne. be separated from non-analogue t r a n s i t i o n s by requiring the emerging pion to have the c o r r e c t momentum. DIAS t r a n s i t i o n s are s t u d i e d because: t h e o r e t i c a l c a l c u l a t i o n s are simplfied; there are fewer possible reaction mechanisms; and the cross-sections f o r DIAS t r a n s i t i o n s are usually larger 3 than for any other t r a n s i t i o n s . In SCX, t r a n s i t i o n s to the i s o b a r i c analogue state (IAS) are studied, e.g. 1 5N(TT +,TT° ) 1 5 0 ( g . s ) . DCX was f i r s t discussed i n the 1960's [5]. P h y s i c i s t s saw i t s p o t e n t i a l both to produce proton r i c h n u c l e i , and to give nuclear information such as nucleon-nucleon c o r r e l a t i o n s and neutron-proton density d i f f e r e n c e s . There are very few reactions which give t h i s information. In 1976 M i l l e r and Spencer showed that DCX should indeed be s e n s i t i v e to c o r r e l a t i o n s and density differences [6]. It i s s e n s i t i v e to c o r r e l a t i o n s because DCX must involve two nucleons - one nucleon cannot remove two units of charge and remain a nucleon. It i s s e n s i t i v e to density differences at energies near the delta (3/2,3/2) resonance: at this energy, the pion i n t e r a c t s so strongly that i t does not penetrate the nucleus, and DCX must occur on the surface of the nucleus. If the excess neutron density P n -Pp i s large at the nucleus' surface then DCX w i l l be enhanced. DCX d i f f e r e n t i a l cross sections are small - t y p i c a l l y of the order 1 ub/sr. This meant DCX experiments had to wait u n t i l the 'meson f a c t o r i e s ' TRIUMF, LAMPF and SIN had been commissioned. These laboratories produce high f l u x beams of pions with a narrow momentum b i t e . The high f l u x allows a complete DCX cross-section a n g u l a r - d i s t r i b u t i o n to be measured i n a reasonable time: about 3 weeks i n the case of t h i s experiment. The narrow momentum b i t e allows the DIAS t r a n s i t i o n s to be resolved from non-analogue ones. The f i r s t successful use of DCX has been i n producing proton-rich nuclei [7]. This i s useful i n t e s t i n g mass formulae. Other uses have been prevented by gaps i n our knowledge of the reaction mechanism; the main aim of DCX cross-section measurements so f a r , i s to understand the reaction mechanism. 4 1.3 Target Nuclei One c r i t e r i o n f or choosing targets i s ease of t h e o r e t i c a l c a l c u l a t i o n s . Nuclei with a f u l l core plus e i t h e r two valence neutrons or two holes s a t i s f y t h i s . In most models the core nucleons are P a u l i blocked from taking part, so DCX can involve only the valence neutrons, s i m p l i f y i n g c a l c u l a t i o n s . Many nu c l e i have been studied f o r DCX, i n c l u d i n g the two hole nucleus l k C [8J, and 1 8 0 and 2 6Mg [9] which have 2 valence neutrons. 1.4 Energy The energy at which DCX i s studied ranges from pion k i n e t i c energies of a few tens of MeV to a few hundreds of MeV. Many measurements have been made at 165 MeV where DCX i s dominated by the (3/2,3/2) resonance. However, 50 MeV may prove to be the most i n t e r e s t i n g energy. Here, c a n c e l l a t i o n of the 0° form factors f o r the reaction p(n~,Tr 0)n ( f g = -0.312 + 0.013i, f p = 0.302 + 0.0311), [10] gives a very small cross-section ( F i g . 1.2). This e f f e c t remains i n heavier n u c l e i ( F i g . 1.2), making the contribution of the 'simple sequential mechanism' small. This mechanism consists of two SCX reactions, with the intermediate state dominated by the IAS ( F i g . 5.2). Since the DCX cross section remains large at 0° at 50 MeV f o r 1 1 +C and 1 8 0 , other mechanisms must be present; the near absence of the simple sequential mechanism makes these easier to study. 1.5 Models And Previous Experiments Burman et a l . at LAMPF made the f i r s t measurement of DCX cross sections [7]. The experiment had low r e s o l u t i o n ( s 4 MeV) so did not disinguish the DIAS t r a n s i t i o n , and measurements were only made at 0° and 164 MeV. Nonetheless, they showed that the cross section i s both small and - by using several target n u c l e i - s e n s i t i v e to nuclear structure. 5 0 KK) 200 300 400 Tff (MeV) Fig. 1.2 E x c i t a t i o n functions f o r SCX. ( i ) curve for p ( i r - , T 7 ° ) n ( i i ) data points f o r 1 4 C ( T T + , TV° ) 1 4N (g.s.) [ 5 ] Seth et a l . at LAMPF measured the f i r s t angular d i s t r i b u t i o n of DCX cross sections [ 1 1 ] . They used 1 8 0 at 1 6 4 MeV, over the angular range 1 3 ° to 4 5 ° . The most s t r i k i n g r e s u l t was that non-analogue t r a n s i t i o n s to the 1 . 8 9 MeV 2 + state were as common as analogue t r a n s i t i o n s . In e a r l i e r predictions, based on the simple sequential model, the DIAS t r a n s i t i o n was always found to dominate; t h i s i s because of the large overlap i n nuclear wavefunctions for target, IAS and DIAS. This suggested the presence of other important mechanisms. Since Seth's 1 8 0 measurements, several other targets have been used i n the energy range 8 0 - 3 0 0 MeV [ 1 2 ] . The r e s u l t s show that DCX cross sections decrease with increasing A; roughly o « A _ 1 0 / 3 for T = 1 n u c l e i ( F i g . 1 . 3 ) . The 1 6 4 MeV angular d i s t r i b u t i o n s are d i f f r a c t i v e with the f i r s t minimum 6 F i g . 1.3 Dependence of DCX on A. Line i s a = 2 x l O 4 A ~ 1 0 / 3 . C i r c l e s are normalized d i f f e r e n t i a l cross sections for DIAS t r a n s i t i o n s from i 8 0 , 2 6Mg and 2 0 9 B i at 5° at 292 MeV. Squares are for 1 8 0 and 2 6Mg at 180 MeV. 7 between 20° and 30° (e.g. F i g . 1.4). The e x c i t a t i o n functions ( i . e . v a r i a t i o n of cross section with energy) f o r several n u c l e i have s i m i l a r shapes with large v a r i a t i o n s i n t h i s energy range ( F i g . 1.5). To be considered successful, a complete theory should reproduce a l l these features. The f i r s t DCX cross-section a n g u l a r - d i s t r i b u t i o n at 50 MeV was measured i n 1983, when Navon et a l . at TRIUMF measured the d i s t r i b u t i o n for 1 4 C [13]. L e i t c h et a l . at LAMPF then repeated t h i s [8] but with better s t a t i s t i c a l errors and at more forward angles. The main feature of the d i s t r i b u t i o n i s the peak, at 0°, where the cross section i s greater than at 120° by a factor of about 7 ( F i g . 4.3). In 1982, Doron et a l . measured the 1 5N SCX cross section at 165 MeV [14]; They showed that the angular d i s t r i b u t i o n i s d i f f r a c t i v e . In 1984, Cooper et a l . measured i t at 48 MeV; they showed that the d i s t r i b u t i o n i s forward dipped - the 0° c r o s s s e c t i o n i s about one-tenth the 90° cross s e c t i o n ( F i g . 1.6). This i s s i m i l a r to the p(Tr~,Tr°)n reaction. Navon's r e s u l t s were very d i f f e r e n t to predictions such as S i c i l i a n o ' s [15]. S i c i l i a n o ' s c a l c u l a t i o n s were based on two sequential SCX; he predicted a 0° cross-section of about 0.2 yb/sr. Navon's data suggested the actual value could be as high as 12 yb/sr, although L e i t c h l a t e r showed i t to be nearer 4 yb/sr. M i l l e r introduced the six-quark bag model to explain the difference [16]. He argued that no conventional mechanism could give DCX forward peaking as high as 12 yb/sr whilst keeping SCX forward dipped and small. He went on to show that Navon's r e s u l t s could be reproduced with the new model. His model Introduces quark degrees of freedom into the nuclear wave-function. It postulates that some of the time, the 3-quark bags of the valence neutrons merge together to form a 6-quark bag. The incoming i r + can give up two units of charge to the 6-quark bag i n e f f e c t i v e l y one stage ( F i g . 5.4), greatly 8 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 40 60 80 100 120 140 160 180 200 220 240 260 280 300 T f f (MeV) Fig. 1.5 DCX to DIAS e x c i t a t i o n functions f o r 1 8 0 ( c i r c l e s ) and 2 6Mg (squares). b 10" I • I I I I I I I I I T 1' 1 I 1 . 0 20 40 60 80 100 120 140 160 180 6c* (degrees) Fig. 1.6 SCX to the IAS angular d i s t r i b u t i o n f o r 1 5N, at 48 MeV. 9 enhancing the DCX cross-section. Six-quark bags have been used to explain other e f f e c t s , including the EMC e f f e c t [17] and the apparent difference of the magnetic moments of free nucleons to those of nucleons i n n u c l e i [18]. Leitch's measurements of 1 1 +C DCX at 50 MeV showed a 0° cross section of 4 ub/sr; t h i s means conventional mechanisms may explain the data. When M i l l e r allowed for pion absorption into other channels, he reproduced Leitch's 0° cross section. At the same time Karapiperis & Kobayashi reproduced L e i t c h 1 s r e s u l t s using a more conventional model, the A-hole model [19,20,21]. P a u l i blocking prevents DCX from occurring on core nucleons. In the A-hole model, core nucleons are excited to a A p a r t i c l e , leaving a 'hole state' i n the core; the A - p a r t i c l e i s not P a u l i blocked, so this mechanism enhances DCX ( F i g . 5.8). Another model being developed by Jennings and de Takacsy shows that c o r r e l a t i o n s of the valence neutrons also produce forward peaking [22] for 1 8 0 DCX with l i t t l e e f f e c t on SCX. These ca l c u l a t i o n s are i n an early stage. The various models d i f f e r i n t h e i r s e n s i t i v i t i e s to the nuclear core contents. The A-hole mechanism involves a l l nucleons whilst the six-quark bag mechanism and Jennings model involve only the two valence nucleons. So the A-hole model i s much more s e n s i t i v e to core contents than the other two. These models are discussed i n Section 5. 1.6 Aim Of This Experiment The aim of t h i s experiment was to give more information on the DCX mechanisms. At 50 MeV the co n t r i b u t i o n of the simple sequential mechanism to forward angle DCX was small, so that other mechanisms stand out. Previously, 50 MeV DCX a n g u l a r - d i s t r i b u t i o n data only existed f or lt*C. Adding to this the d i s t r i b u t i o n f o r 1 8 0 provides information on the A-dependance of DCX. The r e s u l t s show that the 1 8 0 d i s t r i b u t i o n i s very s i m i l a r to that of 10 1 H C , i n both si z e and shape. This i s as predicted by the six-quark, bag model. I n i t i a l c a l c u l a t i o n s with the A-hole model gave d i f f e r e n t predictions f o r the two n u c l e i ; however more recent c a l c u l a t i o n s by Karapiperis & Kobayashi do reproduce the 1 8 0 r e s u l t s reasonably well [19]. This possibly a r i s e s because the A-hole model involves a large contribution to DCX from the 2 + excited state. Both 1>*C and 1 8 0 have a low-lying 2 + state. C l e a r l y further data w i l l be useful i n deciding the importance of the various mechanisms for DCX. The 50 MeV DCX cross-section angular-d i s t r i b u t i o n f o r 2 6Mg has recently been measured ( A p r i l 1985) at TRIUMF [23]. Also proposed are measurements of the e x c i t a t i o n function f o r 1 8 0 and measurements on 3l*S and 5 6 F e [24]. These measurements w i l l lead to a better understanding of the mechanisms for DCX. 11 2 THE EXPERIMENT 2.1 Beamllne The experiment was c a r r i e d out on the M13 low-energy pion channel at TRIUMF [25]. The channel was tuned to 50 MeV TT+, with a momentum bite of 0.5%. The primary proton current was t y p i c a l l y 130 uA, and the pion-production target used was 2mm graphite: t h i s gave a rate of around l . l x l O 6 i r + / s . For spectrometer angles less than 30°, the pion beam passes through the f i r s t wire chamber. At these angles keeping the wire chamber working properly required a reduction i n the rate to 0.35xl0 6 iv +/s; t h i s was acheived by reducing the momentum b i t e . The beam was monitored mainly by two s c i n t i l l a t o r s ( F i g . 2.1), BI just before the s c a t t e r i n g chamber and B2 just a f t e r . The number of pions which passed through the target was calculated from the number of coincidences B1.B2. B2 had to be removed at spectrometer angles 50° or l e s s , because the spectrometer then occupies the usual B2 space. For these angles, two 'muon telescopes' were used to monitor the beam, one fixed above the beam pipe and the other below i t . Each telescope consisted of two s c i n t i l l a t o r s at an angle of about 7° to the beam pipe. At t h i s angle the Jacobian for the CM to laboratory frame transformation i s nearly constant; t h i s reduces s e n s i t i v i t y to beam d i r e c t i o n changes [20]. The telescope counts P ] / U 2 a n d ^ 3 * ^ 1 + w e r e c a l i b r a t e d against Bl*B2 at large angles; at small angles the telescope counts were used with the c a l i b r a t i o n s to estimate what the Bl«B2 counts would have been. As w e l l as pions, the beam contains some muons and electrons. The pion f r a c t i o n - i . e . the f r a c t i o n of beam p a r t i c l e s that are pions - was 12 F i g . 2.1 The QQD Low Energy Pion Spectrometer Showing P o s i t i o n Of Target 13 determined from 'beam sample events' - i . e . coincidences between BI, B2 and a gate generator which was set to - Is. The beam sample event starts a t i m e - t o - d i g i t a l converter; the stop s i g n a l comes when the next pulse of protons passes a capacitive probe i n the main beamline. So pions can be distinguished from muons and electrons by time of f l i g h t along the M13 channel, c a l l e d TCAP. F i g . 2.2 shows a t y p i c a l spectrum. GO 500 TCAP (TDC channels) 600 F i g . 2.2 T y p i c a l TCAP Spectrum used to estimate the contamination i n the incident f l u x . 2.2 Spectrometer The QQD low-energy pion spectrometer ( F i g . 2.1) [26] was used to measure the momentum of the scattered pions. The spectrometer has two quadrupole magnets and one dipole magnet. The quadrupoles increase the s o l i d angle and the dipole separates pions of d i f f e r e n t momenta. This experiment used only the dipole and the second quadrupole QT2, which focusses i n the v e r t i c a l d i r e c t i o n . QT1 focusses i n the horizontal 14 d i r e c t i o n , increasing the s o l i d angle s l i g h t l y (~ 5%); but i t also reduces r e s o l u t i o n i n the target traceback (Section 3.4) since i t l i e s between wire chambers 1 and 3 - so i t was not used here. There were four s c i n t i l l a t o r s at the e x i t from the spectrometer, c a l l e d E l , E2, E3 and E4. In the data a c q u i s i t i o n system, a spectometer event was defined as a coincidence B1»E1 ,E2'E3; such a coincidence was most l i k e l y to be caused by a p a r t i c l e passing through the spectrometer. Each time one occured the data a c q u i s i t i o n system read d e t a i l s of the event and recorded them on magnetic tape. The QQD uses four multi-wire proportional chambers - c a l l e d wire chambers from now on - to measure the t r a c k s of p i o n s . These p o s i t i o n - s e n s i t i v e detectors are described i n d e t a i l i n [3]. Whenever a spectrometer event occurs, each wire chamber gives the x and y coordinates^ of where the pion passed through i t ; the z coordinate i s simply the z-coordinate of the wire chamber. Wire chambers 1 and 3 are at the front-end of the spectrometer. The coordinates from these were used i n the target traceback - i . e . , the determination of where on the target the pion o r i g i n a t e d . This allows pions o r i g i n a t i n g from other than the target to be cut - i . e . rejected from a n a l y s i s . Wire chamber 1, 3 and 5 coordinates were used to determine the momentum of the pion. A check was made using wire chamber 1, 3 and 4 coordinates: i f the two r e s u l t i n g momenta were very d i f f e r e n t , the event was cut. This i s one of the main ways of cuttin g events i n which a pion decayed i n the spectrometer. D e t a i l s of cuts made are given i n Section 3.4. The spectrometer r e s o l u t i o n was ~ 1 MeV, r e s u l t i n g mainly from % e use the standard right-handed orthogonal t r i p l e coordinate system for beams: the z - d i r e c t i o n i s along the beam; the y - d i r e c t i o n i s v e r t i c a l l y up; and the x - d i r e c t i o n i s to the l e f t of an upright observer facing along the beam. 15 straggling i n the target, wire chambers and the helium gas that f i l l e d the target chamber and spectrometer. This e a s i l y separated the DIAS t r a n s i t i o n s from the DCX t r a n s i t i o n s to the 1.89 MeV f i r s t excited state. The s o l i d angle subtended by the spectrometer at the target was 16 msr (Section 3.9). 2.3 Targets This experiment used 1 6 0 and 1 8 0 targets i n the form of gelled water. The 1 6 0 target was included for background measurements but was not needed because the background was very low. The 1 8 0 was used f o r DCX wire-chamber e f f i c i e n c i e s and e l a s t i c normalizations. Table 2.1 gives the target compositions. The water was gelled to prevent i t bowing the windows of the target holder: at room temperature, the gel l e d water was self-supporting. The holder was machined from perspex, to the dimensions shown i n F i g . 2.3. 50 pm kapton windows were then glued under tension to the perspex with epoxy r e s i n . The target water was heated to nearly b o i l i n g , and the agar agar g e l l i n g agent dissolved i n i t . The s o l u t i o n was allowed to cool s u f f i c i e n t l y to not damage the windows, but not enough to s t a r t g e l l i n g . It was then syringed into the holder v i a the f i l l i n g hole. The f i l l i n g hole was blocked with epoxy r e s i n . 12 pm aluminium f o i l was then glued to the kapton, to prevent evaporative loss of water. No glue was used over the target area. The target mass thicknesses were calculated by d i v i d i n g the mass of water contained, by the target area. The target area was calculated from the design dimensions. The targets were mounted on the remote c o n t r o l l e d target ladder described i n [27]. The target ladder was held i n the h e l i u m - f i l l e d s c a t t e r i n g chamber. The target angle - i . e . the angle between the . 2.3 Target D e t a i l s (a) Target holder plan view. (b) Target holder front e l e v a t i o n . (c) Schematic target and holder cross s e c t i o n . 17 Table 2.1 Target D e t a i l s . 1 8 0 Target 1 6 0 Target Target area ( c m 2 ) : 23.50 23.50 Composition (%): H 2 1 8 0 94.0 0.2 H 2 1 7 0 2.54 H 2 1 6 0 2.50 98.8 Agar Agar 1.0 1.0 Thicknesses Gelled Water (g/cm 2) 0.694 0.611 Kapton Window (pm) 50 50 Aluminium Window (pm) 12.5 12.5 To t a l Window (g/cm 2) 0.017 0.017 18 t a r g e t - n o r m a l and the beamline - was s e t to h a l f the s p e c t r o m e t e r a n g l e . T h i s m i n i m i z e s energy s t r a g g l i n g because a l l s c a t t e r e d p i o n s t r a v e l the same d i s t a n c e i n the t a r g e t , i r r e s p e c t i v e o f where they s c a t t e r e d . However, a t s p e c t r o m e t e r a n g l e s 40° and 5 0 ° , the t a r g e t a n g l e was 35° and 45° r e s p e c t i v e l y . T h i s was t o i n c r e a s e the t h i c k n e s s t o g i v e s h o r t e r r u n - t i m e s . 2.4 Runs DCX runs were made at nominal s p e c t r o m e t e r a n g l e s of 20°,30°,40°,50°,90° and 120° on the 1 8 0 t a r g e t . The d i p o l e was s e t so t h a t p a r t i c l e s of momentum 121 MeV/c passed a l o n g the c e n t r a l a x i s ; t h i s c o r r e s p o n d s t o 45 MeV p i o n s , the energy of the f i n a l s t a t e p i o n a f t e r a DIAS t r a n s i t i o n . The a n g l e was changed when the DIAS peak from o n - l i n e a n a l y s i s c o n t a i n e d about 80 e v e n t s , a f t e r background s u b t r a c t i o n by eye. Each a n g l e took about 3 days. E l a s t i c runs were made r o u g h l y e v e r y 12 hours d u r i n g each DCX r u n . Both the 1 6 0 and 1 8 0 t a r g e t s were used. The p o l a r i t i e s o f the s p e c t r o m e t e r magnets were changed; and the d i p o l e and QT2 were s e t to a c e n t r a l momentum of 128 MeV/c c o r r e s p o n d i n g to 50 MeV p i o n s . T h i s change of the c e n t r a l momentum between e l a s t i c and DCX runs kept the s o l i d a n g l e the same. The purpose of t h e s e runs was to measure w i r e chamber e f f i c i e n c i e s and f o r DCX c r o s s - s e c t i o n n o r m a l i z a t i o n . Some e l a s t i c runs were a l s o made a t the s t a r t o f the experiment, u s i n g CH 2, 1 6 0 and 1 8 0 t a r g e t s . The r e s u l t s of these were used t o o b t a i n w i r e chamber c a l i b r a t i o n s and magnet t r a n s f e r c o e f f i c i e n t s . 2.5 Data A q u i s i t i o n The d a t a a c q u i s i t i o n system was s i m i l a r to t h a t used i n [ 3 ] . A PDP 11/34 computer r e a d d a t a f o r each event and r e c o r d e d i t on magnetic tape u s i n g the s t a n d a r d TRIUMF d a t a a c q u i s i t i o n program DA. T h i s responds t o a 19 RF E4R JM7]—o -o -o E1R E1L E2R El -o E2 0 D E2L E3R+ E3R--E3L+ •{2 E3R l i n Z _ MT -» E3 DISCRIMINATOR AND MEAN TIMER GATE GENERATOR OR BIT 0 E3L-- r Q _ C C212 •{Si BI {2 SPECTROMETER EVENT BEAM SAMPLE EVENT LAM Or- TDC starts GG — C C212 BIT 1 ——1> Is W4 TI ION TI C TI CAP {>—o O CAMAC ADC O CAMAC TDC STOP ADC gates Inhibit scalers GG GG L — J lms manual start/stop 4 CAMAC output register (comp busy) D V VISUAL AND CAMAC SCALERS CAMAC BIT PATTERN UNIT MWPC SIGNALS WC 5Y F i g 2.4 Configuration of El e c t r o n i c s 20 'look-at-me' (LAM) s i g n a l by reading information stored i n various modules i n a CAMAC crate. The LAM was generated for each spectrometer event or beam sample event by a C212 b i t - p a t t e r n unit i n the CAMAC crate. A modified version of the Fermilab program MULTI was used for on-line a n a l y s i s . F i g . 2.4 shows the e l e c t r o n i c s configuration. It includes the following additions which were made to the system used i n [3]. The extra muon telescope, l^'U^ was connected to v i s u a l and CAMAC s c a l e r s . The extra back s c i n t i l l a t o r E4 was connected to both a t i m e - t o - d i g i t a l converter (TDC) and an analogue-to-digital converter (ADC). The l e f t and r i g h t signals from the Cerenkov detector were connected to ADC's. The three back s c i n t i l l a t o r s E l , E2 and E3 were connected to TDC's as well as the usual ADC connections. Various extra v i s u a l and CAMAC scalers were also used. 2.6 On-line Analysis The program MULTI was used for on-line a n a l y s i s . Early on i n the experiment, wire chamber c a l i b r a t i o n s and magnet transfer c o e f f i c i e n t s were calcul a t e d . These enabled the momenta of the pions passing through the spectrometer to be calculated and histogrammed. Various cuts were applied to reduce background. The cuts were s i m i l a r to those applied i n o f f - l i n e analysis described i n Section 3.4. The cut on t i m e - o f - f l i g h t from Bl to E l was p a r t i c u l a r l y important i n DCX runs: t h i s removed electrons very e f f e c t i v e l y to give a clear DIAS peak. This^ made i t easier to judge when s u f f i c i e n t DCX data had been taken f o r each angle. 21 3 ANALYSIS The data was stored on magnetic tapes, and analysed using the TRIUMF programs MOLLI [28] and OPDATA [29]. Several extra subprograms were written fo r MOLLI. MOLLI reads data from MULTI-written tapes for each event. It sorts these events into various categories according to user-written subprograms, r e j e c t i n g 'bad' events from the analysis - e.g. spectrometer events i n which only three wire chambers f i r e d . It cal c u l a t e s c e r t a i n parameters for the remaining good events, such as pion momentum; and i t histograms parameters as requested by the user. These histograms are used f o r example to count the number of events i n a p a r t i c u l a r peak. MOLLI also gives scaler values for each run. The histograms can be stored f o r l a t e r use such as p e a k - f i t t i n g using OPDATA. OPDATA finds least-squares f i t s of user-supplied functions to the histograms. T h i s was used here to separate background from the e l a s t i c - s c a t t e r i n g peaks. Peak f i t t i n g was not used for the DCX analysis because of the low background and small numbers of events contained i n the peaks. The f i r s t analysis was done early on i n the experiment to derive wire chamber c a l i b r a t i o n s and magnet transfer c o e f f i c i e n t s - allowing accurate on-line a n a l y s i s . DCX and e l a s t i c runs were analysed a f t e r the experiment. 3.1 Wire Chamber C a l i b r a t i o n s This section describes the method used to convert wire-chamber TDC outputs i n t o x,y-coordinates. This method has become standard because i t i s simple and accurate. It does not require any radioactiive sources; instead the taped data from any e l a s t i c s c a t t e r i n g run can be used, provided i t has 22 enough spectrometer events (> 50,000) to give clear histograms. Several runs can be analysed and t h e i r histograms summed to b u i l d up enough events i f wanted. It would be possible to determine pion momenta d i r e c t l y from TDC outputs, by deriving s u i t a b l e magnet transfer c o e f f i c i e n t s . However, these c o e f f i c i e n t s would then be highly dependent on the e l e c t r o n i c s : e.g. changing a TDC u n i t would require deriving a new set of c o e f f i c i e n t s . It i s easier to derive new wire chamber c a l i b r a t i o n s and keep the old transfer c o e f f i c i e n t s . This gives greater consistency i n the magnet transfer c o e f f i c i e n t s from experiment to experiment. 3.1.1 S i gna l Product ion In Wire Chambers F i g . 3.1 i l l u s t r a t e s the construction of the wire chambers. They consist of three p a r a l l e l planes of p a r a l l e l wires. The c e n t r a l plane has ho r i z o n t a l wires and i s the anode, which i s operated at a high p o t e n t i a l (4 to 5 kV). The two cathode planes are earthed. The cathode which measures the y-coordinate has i t s wires p a r a l l e l to the anode wires, whilst the x-cathode wires are perpendicular. The cathode wires are insulated at one end and connected to a delay l i n e at the other. The delay l i n e s are earthed at both ends v i a operational a m p l i f i e r s . The s i g n a l i s produced by e l e c t r o n m u l t i p l i c a t i o n forming an avalanche. This occurs over a short length (<0.1 mm) of one anode wire, i n the region clos e s t to the track of the p a r t i c l e being detected. The p o t e n t i a l of t h i s length of anode wire drops; capacitive coupling then Induces a po t e n t i a l drop on the nearby wires i n both cathodes. The s i g n a l i n each cathode wire then tr a v e l s to the delay l i n e , s p l i t s , and tr a v e l s to both operational a m p l i f i e r s . These produce the 'minus' and 'plus' pulses. The delay between wires (0.55 ys) i s much smaller than the s i g n a l dispersion (15 ys) produced 23 by the delay l i n e . So the signals from several wires merge together before reaching the a m p l i f i e r . F i g 3.1 Arrangement of wires i n QQD wire chambers (N.B. Table 2 . 1 i n [ 3 ] gives the wrong anode and cathode wire spacings). The x and y signals d i f f e r as a r e s u l t of the d i f f e r e n t orientations of wires i n the two cathodes. Whatever point along the anode the avalanche occurs at, i t w i l l be the same distance from y-cathode wires. But t h i s point may be r i g h t opposite an x-cathode wire, or anywhere between two of them. So the y-signals are always the same shape, whilst the x-signals can vary. Id e a l l y the peak of the y-sig n a l corresponds to the anode wire on which the avalanche occured; whilst the peak of the x-signal corresponds to how far along the anode the avalanche occurred. For y, t h i s gives a 'picket-fence' structure to a histogram of the time-difference between minus and plus pulses; whilst f o r x, t h i s histogram i s continuous. The picket-fence structure was used to c a l i b r a t e the wire chambers. 24 Table 3.1 Wire Chamber C a l i b r a t i o n s used and L a b e l l i n g System. Labe l Wire Chamber Slope Intercept Cathode (mm/TDC channel) (mm) 1 2 3 4 5 6 7 8 9 10 11 12 WC1X WC1Y WC3X WC3Y WC4X ri g h t WC4X middle WC4X l e f t WC4Y WC5X r i g h t WC5X middle WC5X l e f t WC5Y 0.09917 0.09917 0.1059 0.1059 0.09022 0.09022 0.09022 0.08889 0.09144 0.09144 0.09144 0.08260 6.00 5.10 14.30 -1.50 -172.80 0.50 203.50 -3.90 -197.50 0.00 203.00 -1.70 25 3.1.2 Sumcuts Histograms of the sum of the times of the two pulses are s i m i l a r f o r x and y. This sum i s a constant amount plus a v a r i a b l e amount. The constant part represents the time taken by a s i g n a l to t r a v e l the length of the delay l i n e and to the data a c q u i s i t i o n e l e c t r o n i c s . The v a r i a b l e amount i s the d r i f t time - the time between the p a r t i c l e passing through the wire chamber and the l i b e r a t e d electrons d r i f t i n g close enough to the anode to s t a r t an avalanche. The d r i f t time varies from zero to 25 ns. Thus i d e a l l y sum-histograms would have a square-wave form, with width 25 ns. The observed shape i s approximately t h i s but rounded and wider, with a small number of events well outside the 25 ns area. These events are due to m i s f i r e s of various forms: they can be conveniently removed from analysis by requesting that the sum l i e s between two values, c a l l e d sumcuts. 3.1.3 Convert ing TDC Output To P o s i t i o n Coordinates The coordinates are derived from a l i n e a r function of the d i f f e r e n c e i n TDC outputs f o r the minus and plus pulses: q ± - m±At± + c± where i runs from 1 to 12 and l a b e l s the cathodes (Table 3.1), At i s the di f f e r e n c e i n TDC outputs, q i s the p o s i t i o n coordinate ( X j or y j , j e {1,3,4,5} depending on the wire chamber), m and c are the slope and int e r c e p t , m and c were determined by analysing an e l a s t i c s c a t t e r i n g run. F i r s t , values f o r the sumcuts were determined by analysing the e l a s t i c run with the following cuts: o Only analyse spectrometer events - i . e . test C212 b i t pattern 26 • Ensure a l l four wire chambers f i r e d : check the minus and plus TDC outputs were within the range given i n Table 3.4 for both the x and y cathodes of each wire chamber. (For each of the back chamber x-cathodes, only one of the three sections need f i r e . ) o If the l e f t x-cathode of a back wire chamber f i r e d , check that the r i g h t did not f i r e too. The sums of the TDC outputs f o r each cathode were histogrammed, with 10 TDC channels per histogram bin. The edges of the main body of events for each cathode were then chosen as the sumcuts. The e l a s t i c run was then re-analysed with sumcuts added; th i s gives clearer picket fences. The following were histogrammed: «• At^ f o r a l l cathodes, with 10 TDC channels per bin • Picket fences for a l l y-cathodes, i . e . At^ with 2 TDC channels per bin • Double h i t s f o r back wire chamber x-cathodes: i . e . At^ with 10 TDC channels per bin, f o r each event i n which two adjacent x-cathodes f i r e d . These mark the edges of the cathodes. The slopes and intercepts were determined from these as follows, ( i ) Slopes: A l l y-cathode slopes were determined from the picket fence histograms. The peaks i n each picket fence were numbered i n order, and a graph plotted of TDC channel at centre of peak versus peak number. The slope of t h i s graph i s the number of TDC channels per 2mm i . e . 27 2 mi = | mm/TDC channel 1 = 2,4,8,12 The x- and y-cathodes In wire chamber 1 were b u i l t to the same design with the same methods. The signals were processed by the same TDC u n i t . So the slope f o r the x-cathode i s taken to be the same as that of the y-cathode; s i m i l a r l y f o r wire chamber 3, i . e . mi = m 1 + 1 i = 1,3 The slopes for the middle x-cathodes of wire chambers 4 and 5 were determined from the double-hits histograms. The two peaks represent the edges of the cathode. The cathode has 203 wires so i s 203 mm long. So i f the double peaks are at d^ and d^ TDC channels, then 203 m. = -. — mm/TDC channel i = 6,10 1 d L i " d R i Again the construction of the three sections of each back x-cathode i s si m i l a r , so the slopes are taken to be the same: V l = mi+l = m i 1 = 6 ' 1 0 ( i i ) Intercepts: The intercepts f o r a l l y-cathodes and the front-chamber x-cathodes were determined from the mean At.^ of a l l events i n the At.^ histogram. The mean was converted to millimetres using the slope: c i = m i ^ i 1 = i . 2 * 3 . 4 * 8 ' 1 2 For the back-chamber x-cathodes, the intercepts were chosen to put the double h i t s at ± 101.5 mm. If d L and d R are the TDC values of the 28 left-middle and right-middle double hits then c. = -101.5 - d„.m. mm i = 5,9 i Ri i ' L i Ri = 2 m £ 111111 i = 6,10 c± = +101.5 - dLim± mm i = 7,11 Table 3.1 gives the values used for the slopes and in t e r c e p t s . 3.2 Ca l cu l a t i n g Pion Momenta The pion momenta were calculated i n a subroutine i n the program MOLLI, using the magnet transfer c o e f f i c i e n t s and some of the wire-chamber coordinates. Rather than working d i r e c t l y i n terms of the momentum, the parameter 6 i s used, defined by: P-Po 6 = Po where p i s the pion momentum and p Q i s the c e n t r a l momentum of the spectrometer. The magnet transfer c o e f f i c i e n t s are explained b r i e f l y below; [3] and references therein contain more d e t a i l s . The pion track through the spectrometer i s determined by specifying a set of 5 independent c o o r d i n a t e s , e.g. r = ( r l , r 2 , r 3 , r ^ , r 5 ) = ( X j . y ^ X g . y g . a ) . Any other such set, e.g. r' = ( x 5 ,y g, 8 5 ,<|>5,6), can be derived from these. The r e l a t i o n between r and r' i s of the form V • M i + \ M i j r j + ^k MijkVk + ( 3 - 1 } 29 M i ' M i j ' M i j k **' a r e t h e m a 8 n e t transfer c o e f f i c i e n t s . In p a r t i c u l a r , x 5 i s highly dependent on 6. With r = (xx ,yl ,x 3 ,y 3,6), eqtn. 3.1 for i=5 can be written i n the form v i j k J l . m „ „ „ v x 5 = I c x x y i J x 3 y 3 6 ...(3.2) where Table 3.2 l i s t s the coeff i c i e n t s c used and the values of i to go with each. In thi s experiment, only lin e a r coefficients of 6 were used, i . e . m=0 or 1. So 3.2 can be written as x 5 = A + B6 => 6 = where A and B are independent of 6. MOLLI calculated A and B for an event and hence 6. A second value of 6 was calculated using x^ instead of x 5 with the coefficients given i n Table 3.3. This was used as a muon cut (Section 3.4). The two values of 6 are denoted 6^  and 6 5. 3 . 3 Derivation Of Magnet Transfer Coefficients. The co e f f i c i e n t s c of Tables 3.2 and 3.3 were derived using the method of Tacik and Barnett described i n [3] and further developed by M. Rosen [30]. B r i e f l y , the method started with either TRANSPORT [31] coefficients or coeff i c i e n t s from a previous experiment. These coeff i c i e n t s were used to analyse a CH 2 -target run made i n the current experiment. The 6 spectrum has three peaks: 1 2C ground-state, 1 2C f i r s t - e x c i t e d state and ground state. The co e f f i c i e n t s were then varied to minimize the widths of the three peaks. 30 Table 3.2 Transfer C o e f f i c i e n t s Used to Calculate 6 i j k 1 m c 0 0 0 0 0 -0.86950 E+02 0 0 1 0 0 0.12197 E+01 1 0 0 0 0 -0.83828 E+00 0 0 0 2 0 0.61252 E-02 0 0 2 0 0 -0.12644 E-01 0 1 0 1 0 -0.69003 E-02 1 0 1 0 0 0.80184 E-02 1 1 0 0 0 0.29219 E-02 2 0 0 0 0 -0.18679 E-02 0 0 0 0 1 -0.95040 E+01 0 0 0 1 1 -0.11576 E-02 0 0 1 0 1 0.58393 E-01 0 1 0 0 1 0.21457 E-02 1 0 0 0 1 -0.26014 E-01 0 0 2 0 1 -0.76500 E-04 2 0 0 0 I -0.14258 E-03 Notation: x 5 = £ c x x x 3 y 3 6 & f o r p o s i t i o n coordinates given i n mm and 6 given i n %. 31 T a b l e 3.3 Transfer C o e f f i c i e n t s Used to Calculate 6, 1 1 ' ~ ' ~ ~ • •" — - . i - - • i 1 k 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 2 0 0 2 0 0 1 0 1 0 1, 1 0 1 0 1 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 2 0 0 2 0 0 1 0 1 1 0 1 0 1 1 0 0 2 0 0 0 Notation: x^ = I c x 1 y 1 J x 3 given i n mm and 6 given i n %. m c 0 -0.56212 E+02 0 0.15673 E-01 0 0.16950 E+01 0 -0.85318 E+00 0 0.47875 E-02 0 -0.77813 E-02 0 -0.61697 E-02 0 0.54586 E-03 0 0.45635 E-02 0 0.80900 E-03 1 -0.65940 E+01 1 0.18382 E-02 1 0.36602 E-01 1 -0.15201 E-01 1 0.12953 E-03 1 0.68259 E-04 -0.22782 E-03 -0.70176 E-04 -0.14925 E-03 -0.36080 E-04 y 3 6^  f o r p o s i t i o n coordinates 32 3.4 Cuts Several cuts were used to remove 'bad' events from the 6-histograms. A 'good' event i s one where a pion from the target passes through the back s c i n t i l l a t o r s without decaying, f i r i n g a l l four wire chambers. Table 3.4 gives the values of the cuts used. The same cuts were used f o r both DCX runs and e l a s t i c runs, so that the normalization was v a l i d . ( i ) Wire Chamber Cut and E f f i c i e n c e s . Events i n which one or more wire chambers misfired (Section 3.1.3) were cut, since a l l x and y coordinates were needed i n the a n a l y s i s . Many of the events cut here would have been good events except that a wire chamber m i s f i r e d . These events are accounted for with 'wire chamber e f f i c i e n c i e s ' , defined as where F = number of times the wire chamber f i r e s when N p a r t i c l e s pass through i t . For e l a s t i c runs i t i s assumed a p a r t i c l e passed through a wire chamber whenever the other three chambers f i r e d i . e . f o r wire chamber 1 N x = WC 3«WC l |«WC 5 so that E n = wc1»wc3»wclt»wc5 i wc3«wctf«wc5 The t o t a l e f f i c i e n c y i s then E. . = E, •E ; )»E l i'E C; tot 1 3 •+ 5 DCX runs have a small rate of good events. This makes chance coincident f i r i n g s of 3 wire chambers too s i g n i f i c a n t f o r t h i s method to be r e l i a b l e . Instead, e f f i c i e n c e s from the e l a s t i c runs made during each DCX run were used. This i s v a l i d because the e f f i c i e n c y only changes slowly with time. 1 33 Table 3.4 V a l u e s of Cuts used D u r i n g A n a l y s i s . Wire Chamber Cuts Low High (TDC channels) ( i ) Minus and p l u s p u l s e s ( i i ) Sums of p u l s e s ALL WC's WC1X WC1Y WC3X WC3Y WC4XR WC4XM WC3XL WC4Y WC5XR WC5XM WC5XL WC5Y 100 1480 1580 1570 1710 1440 1630 1450 1430 1620 1560 1590 1630 1900 1880 2010 2030 2180 1950 2120 1950 1960 2120 2050 2040 2150 Pion s e l e c t i o n TCAP 80 240 E l e c t r o n Cuts E l 480 600 Target traceback E l a s t i c runs DCX runs A l l runs |x 0| < 2 5 c o s ( 9 Q Q D |x Q| < 2 5 mm |y Q | < 15 mm 6 t g t ) Muon cuts j Yi^ -y5 | < 28.0 mm I V - 6 5 | < 5.0 % 34 ( i i ) Pion S e l e c t i o n (TCAP cut) The p a r t i c l e - t y p e f i r i n g BI was determined by t i m e - o f - f l i g h t along the M13 channel. The BI pulse started a TDC; the stop was the next pulse from a capacitive-probe i n the proton beam l i n e . Events i n which the TDC value was outside the pion peak ( F i g . 2.2) were cut. ( i i i ) Target Traceback Cuts Wire chambers 1 and 3 coordinates can be used to determine ( x 0 , y 0 ) , i . e . where i n the target plane the p a r t i c l e o r i g i n a t e d . This 'target traceback' i s found using x 0 = a l x l + a 3 x 3 y 0 = + b 3 y 3 The two c o e f f i c i e n t s were found from e l a s t i c runs using 'picket fence' targets: these have 5 p a r a l l e l bars ~ 1/ 2cm wide spaced by ~ 112cm. a-i and a 3 were determined with a v e r t i c a l picket fence; they were chosen to make as many events as possible appear to come from the bars and not the gaps. b 1 and b 2 were determined using the h o r i z o n t a l picket-fence target. Table 3.5 gives the values of the coordinates used. Table 3.5 Target Traceback C o e f f i c i e n t s used. 1.42 0.259 1.28 0.478 a l a 3 b l b 3 35 Events which did not or i g i n a t e at the target were cut by l i m i t i n g x Q and y Q . This cut i s more important for e l a s t i c than DCX s c a t t e r i n g because the target holder materials have a high DCX Q-value, and so do not contribute to DCX. Consequently the change i n target p o s i t i o n with target angle was allowed f o r i n e l a s t i c analysis but not i n DCX a n a l y s i s . ( i v ) Muon Cuts. About a t h i r d of the pions passing through the spectrometer decay v i a IT •*• uv; t h i s i s allowed for by the decay c o r r e c t i o n f a c t o r (Section 3.8). Many of the muons do not reach the back s c i n t i l l a t o r s , but those that do need to be cut. Two methods were used. (a) Difference i n y^ and y 5 The quadrupole focusses i n the y - d i r e c t i o n . I t was set to give h o r i z o n t a l pion tracks, which maximizes the spectrometer s o l i d angle. Muons from decays with the u,v plane near v e r t i c a l , have non-horizontal tracks. So tracks with large | y 5 - y j j were cut. (b) Difference i n 6^  and 6 5 If the p,v plane i s near h o r i z o n t a l , then x 4 and x 5 d i f f e r from t h e i r values f o r no decay. This makes 6 5 d i f f e r from 5^; so events with large | 65—6^| were cut. (v) E l e c t r o n Cuts Cuts ( i ) to ( i v ) removed background events well enough to give clean histograms of 6 i n e l a s t i c runs. But the number of T T - events i n DCX runs was small, making the remaining e~ events highly s i g n i f i c a n t . In a n t i c i p a t i o n , a Cerenkov counter was placed a f t e r the back s c i n t i l l a t o r s to d i s t i n g u i s h electrons from pions. 128 MeV/c electrons have a v e l o c i t y very close to c and gave a s i g n a l ; 128 MeV/c pions have a v e l o c i t y of ~ 2 / 3 c and 36 Fig. 3.2 The e f f e c t s of cuts made during the analysis on DCX 5 histograms. In (a) a l l events which f i r e d a l l four wire chambers have been histogrammed. In (b) a l l cuts have been applied. 37 were not detected. However, the Cerenkov detector was smaller than the back s c i n t i l l a t o r s . This e f f e c t i v e l y reduces the spectrometer s o l i d angle i f used i n the a n a l y s i s . Instead, a t i m e - o f - f l i g h t method was used. The journey times for p a r t i c l e s to t r a v e l through the spectrometer, i . e . from Bl to E l , were measured. 128 MeV/c electrons took ~ 5 ns whilst 128 MeV/c pions took ~ 8 ns: The two can be r e l i a b l y distinguished with a TDC. This method was checked against the Cerenkov detector, and found to be as e f f e c t i v e , with no loss of spectrometer s o l i d angle. F i g s . 3.2(a) and (b) demonstrate the importance of these cuts for a p a r t i c u l a r DCX run. In (a), only the four wire chamber cut has been used. In (b), a l l the cuts have been used. 3.5 Peak Size Measurements Runs f o r each angle were analysed together using the above cuts. F i g s . 3.2(b) and 3.3 show t y p i c a l 6 histograms produced by MOLLI, for DCX and e l a s t i c runs. In both cases, the ground state peak i s the most prominent feature. To c a l c u l a t e cross sections the number of events i n t h i s peak, Np, must be determined, allowing f o r background events, N^. For DCX runs, the t o t a l number of events i n the peak, N t, was counted and found from N = N - N p t b was estimated by f i t t i n g a s t r a i g h t l i n e to the data outside, but i n the v i c i n i t y of, the ground state peak. i s then the area under t h i s l i n e and between the l i m i t s of the peak. T y p i c a l l y N p = 50, N D = 12. OPDATA was used to f i t the e l a s t i c - r u n ground-state peaks. The f i t t e d function was a Gaussian plus l i n e a r background: 38 -8 -6 - 4 - 2 0 2 <5 (%) F i g . 3 . 4 6 histogram f o r 1 8 0 e l a s t i c s c a t t e r i n g at 30° (data points), with OPDATA f i t ( s o l i d curve). 39 n(6) = m6 + c + N P exp a/2T m and c are the slope and intercept f o r the background, p i s the p o s i t i o n of the peak centre, a i s the standard deviation of 6 i n the peak, and N D i s was taken to be the area under the linear-background f i t between the peak edges. F i g . 3.4 shows a pl o t of an OPDATA f i t . T y p i c a l l y , N p = 5000, N p = 600, with x per degree of freedom = 2. The peaks are Poisson d i s t r i b u t i o n s so f i t t i n g with a Gaussian underestimates the area by x2 J N p was corrected f o r t h i s . 3.6 Beam Monitoring A very high proportion of a l l pions entering the target chamber pass st r a i g h t through to B2. Thus the s c a l e r monitoring B1«B2 gives the number of pions incident on the target. No dead-time c o r r e c t i o n was needed: the scal e r s were stopped whenever the data a c q u i s i t i o n system would not detect a spectrometer event - e.g. whilst already processing an event. If two pions pass through Bl and B2 within a short enough time, they give only one pulse. No c o r r e c t i o n was made f o r such 'double h i t s ' , because the rate i s small and f a i r l y constant: i t i s absorbed i n the normalization. The rate was corrected f o r contamination using the ' i r - f r a c t i o n ' , found from the TCAP histogram ( F i g . 2.2). The three peaks of t h i s histogram are due to i r + , u + and e + . The T r - f r a c t i o n i s found from the best estimate of the number of events without background. Again, N^ N (3.4) f rac N to t where N i s the number of events i n the i r-peak, and N i s the t o t a l number IT Z O Z 40 of events i n the histogram. T y p i c a l l y TT = 97% i n t h i s experiment. B2 could not be used at angles 50° or l e s s ; instead the muon telescopes were used. These were c a l i b r a t e d at 90° and 120° against Bl«B2, by c a l c u l a t i n g B1«B2(90°) B1«B2(120°) 1 ^ • ^ ( 9 0 ° ) + y i . p 2 ( 1 2 0 ° ) J with a s i m i l a r formula for R 3 I +. These were used to estmate what B J ' B J would have been from B1.B2(9) = j [ R 1 2 y i . p 2 ( 0 ) + R 3 1 ty 3'y l t(6)] At ^ t g t = 0°» almost a l l the beam f e l l within the target cut boundaries. As 8 t increases, more and more beam f a l l s outside the x Q l i m i t s . This was allowed f o r u s i n g the ' t a r g e t e f f i c i e n c y ' ^ t g t » calculated by two methods which gave the same r e s u l t s . Both methods allowed for the changes i n beam p r o f i l e from run to run. The f i r s t method used REVMOC [32]. This simulated the passage of i n d i v i d u a l pions along the M13 channel, and calculated t h e i r coordinates i n the target plane. Parameters were adjusted to give the observed beam spot s i z e . E i s the r a t i o of pions f a l l i n g w i t h i n the area defined by the target cuts, to the t o t a l number of pions reaching the target. The second method assumed the beam-flux p r o f i l e was Gaussian i n both the x and y d i r e c t i o n s . The widths were adjusted to the measured beam p r o f i l e shape. E was taken as the r a t i o of the i n t e g r a l over the target area to t g t the t o t a l i n t e g r a l of the beam f l u x . The f l u x f o r a run i s then * = Bl*B2 x it, x E_ _ ...(3.5) f r a c tgt 41 3.7 Target Thickness The target thickness depends on the target angle. If t(0) i s the thickness when the target plane i s perpendicular to the beam, then the thickness at a target angle 9 i s t(9) = t(0)/cos 0 ...(3.6a) The target angle was usually chosen to minimize straggle due to electromagnetic s c a t t e r i n g : i . e . 9 = I 6 tgt 2 QQD - t h i s gives a l l pions scattered i n t o the spectrometer the same path-length i n the t a r g e t , whatever depth they s c a t t e r e d a t . At 9QQ D = 40° and 50°, larger target angles were used (35° and 45° r e s p e c t i v e l y ) . This was to decrease running time, by increasing the thickness. The number of scatterers per unit area of target i s then calculated from t(9)N u(6) = A ...(3.6b) where i s Avogadro's number and A i s the r e l a t i v e molecular mass of the target. 3.8 II-Decay Correction About a t h i r d of pions s c a t t e r i n g into the spectrometer decay before the back s c i n t i l l a t o r s . This i s allowed f o r with a pion-decay co r r e c t i o n - f a c t o r given by mL dec T c y / E 2 _ m 2 c l t where m, E and T are the pion mass, energy and mean-life; L i s the 42 spectrometer decay-length. Pions decaying i n the l a s t 5 cm before wire chamber 5 w i l l not be cut i n the a n a l y s i s : the muon p o s i t i o n coordinates w i l l be l i t t l e changed from t h e i r undecayed values. So L was taken to be 2.35 m which i s 5 cm less than the spectrometer path-length from Bl to wire chamber 5. 3.9 S o l i d Angle The spectrometer s o l i d angle used was AJ2 = 16 msr. This was determined from a REVMOC c a l c u l a t i o n . Pion tracks were randomly generated from the target area at a l l angles on the spectrometer side of the target. The s o l i d angle was taken as 2TT X proportion of tracks which reached E3. Another method due to Barnett [33] was used to check t h i s value. An e l a s t i c s c a t t e r i n g run was analysed, c a l c u l a t i n g the polar angle the pions made with the spectrometer axis. A polar angle ty was determined for which no pions within the cone defined by ty c o l l i d e with the walls etc. of the spectrometer. Then the t o t a l number of good events N t, and the number of events w i t h i n t h i s cone were found f o r the run. The s o l i d angle was calculated from Afi = ^ x 2TT(1-COSIJ;) 2TT(1-COSI|>) i s the s o l i d angle of a cone with polar angle ty. Outer regions of the target l i e o f f the spectrometer axis: t h i s was allowed for i n the c a l c u l a t i o n . The two methods give the same value, within e r r o r s . Also, the s o l i d angle i s a constant factor i n the cross section c a l c u l a t i o n s ; so any error can be absorbed i n the normalization. 43 3.10 Cross Sect ion Ca l cu la t ions A l l data needed for the d i f f e r e n t i a l cross sections can now be cal c u l a t e d , and used to obtain the cross sections from H = ° (3 7) N T T , p dec where N Q = — i s the number of pions that scattered into the spectrometer wc with the required 6. The 1 8 0 e l a s t i c cross-sections were compared to those i n [3] and [33], to obtain a normalization factor f o r the DCX cross-sections. The e l a s t i c r e s u l t s were very s i m i l a r ( F i g . 4.1) and so the normalization factor was taken to be 1. 3.11 E r r o r Ana lys i s The cross-sections were calculated from Equation 3.7. Hence the f r a c t i o n a l error i n the cross section i s Errors that a f f e c t a l l points equally are absorbed i n the normalization. For example the err o r i n s o l i d angle, Afi, i s not included. ( i ) E r r o r i n N Q From Equation 3.4, 44 r <SNg"|2 r 6N - i 2 r f i E i 2 r 6 i T d e c i 6E I2 The error i n N p i s the main contribution to error i n da/dfi. For DCX cross sections, a l l others are i n s i g n i f i c a n t ; f o r e l a s t i c they make a small contribution. N p i s estimated by making a background subtraction N^ from the t o t a l N t: N = N - N, p t b a 2(N p) = a 2(N t) + a 2(N b) and N^ i s estimated from N = N + N, t p b a 2 ( N t ) = a 2(N p) + cr 2(N b) 6(N p) = °(N p) = Va 2(N ) + 2a 2(N b) = /N + 2N, P b For e l a s t i c runs t h i s error was m u l t i p l i e d by x per degree of freedom, to allow for the fac t that the f i t was not i d e a l . E r r o r i n E i s mainly due to the random nature of whether a wire wc chamber f i r e s or not. Assuming a binomial d i s t r i b u t i o n the f r a c t i o n a l error was found to be less than 0.005. Er r o r i n t r ^ was the same for a l l points, so was not included. I t i s mainly due to uncertainty In the decay length L: i f 6L = ± 5cm then 45 = ± 0.007. dec ( i i ) Error i n $ From Equation 3.5, For 90° and 120°, 6B1-B2 i s n e g l i g i b l e . But at smaller angles B1-B2 was estimated from the 2 muon telescopes. Using the 2 c a l i b r a t i o n s - at 90° and 120° - gives 4 estimates f o r B1*B2. The f r a c t i o n a l deviation of these from the mean was used to estimate t h a t ^ ~ — ^ was ±0.01. Sir,, / I T , was B1-B2 fr a c frac estimated using binomial s t a t i s t i c s ; i t was t y p i c a l l y 0.5 %. ( i i i ) E r r o r i n u From Equations 3.6a and b, • [Wt + 9 59)2 <se in "dl-s) E r r o r i n u(0) was the same f o r a l l points and so was omitted. It was l a r g e l y due to error i n target-holder dimensions; the f r a c t i o n a l error was estimated to be 0.01. E r r o r i n 9 was ± 1° mainly due to uncertainty i n alignment of the t Q t target-angle scale. Maximum error i s at 9 = 60°; then ^ f l l x = 0.033 46 ( i v ) E r r o r i n Spectrometer Angle A theodolite survey [34] showed the spectrometer-angle scale was 2.0° out. T h i s was c o r r e c t e d f o r . It i s estimated 60 = ± 0.1°. This i s not sp shown on the graphs. (v) Angle-independent Er r o r The normalization process reduces the angle-independent error. I t i s estimated to be 12 % from F i g . 4.1. 47 4 R E S U L T S Table 4.1 and F i g . 4.1 show the pion e l a s t i c s c a ttering cross sections for 1 8 0 . The beam energy i s given as 48.3 MeV. This i s the average pion energy at the centre of the target, a f t e r allowing for the energy loss i n the target and windows. Also shown i n F i g . 4.1 are the r e s u l t s of other measurements due to Barnett [33] and Tacik [3]. The current r e s u l t s l i e between the two sets of data; thus no normalization was applied to either the e l a s t i c or DCX data. There i s a ± 12% normalization error because of the uncertainty i n the previously measured sets of data. The points at 30° and 40° represent new data points f o r the 1 8 0 e l a s t i c cross section. Table 4.2 shows the d i f f e r e n t i a l cross sections f o r pion DCX to the DIAS for 1 8 0 . F i g . 4.2 shows that the cross section i s a maximum at forward angles and decreases monotonically as the angle increases to 120°. This forward peaking i s very d i f f e r e n t to the SCX cross section f o r say 1 5N, which has a minimum at forward angles. The 90° point i s the average of runs with the spectrometer both to the l e f t of the beam and to the r i g h t . The data i n Table 4.2 i s cor r e c t . The data published i n [35] contains some errors: ( i ) The cross section at 120° has been revised downwards 7%, as a r e s u l t of a p p l y i n g target cuts more c a r e f u l l y , ( i i ) 6 ^ has been reduced by 4° for the nominal angles 20°,30°,40° and 50° - for these points the spectrometer was to the l e f t of the beam where the 2° scale error should be subtracted [34]; i n [35] i t was added. The 120° point was measured on the r i g h t , so adding the 2° i s cor r e c t . ( i i i ) The 90° point i s the average of runs to the l e f t and r i g h t and so has been l e f t at 90° (The decimal arises from transforming to the CM frame). F i g . 4.3 shows both the 1 8 0 and 1 4 C DCX d i f f e r e n t i a l cross sections. They have the same size (within 20%) and shape. The s i m i l a r i t y suggests that 48 Table 4.1 Experimental D i f f e r e n t i a l Cross sections f o r the Reaction 1 8 0 ( i r + , i r + ) 1 8 0 (g.s.) at 48.3 MeV. 6CM d a / d f lCM (degree) (mb/sr) 28.3 19.22 ± 0.85 38.4 12.0 ± 0.52 48.5 6.34 ± 0.25 88.6 5.70 ± 0.23 92.7 6.50 ± 0.27 122.6 6.06 ± 0.32 49 F i g . 4.1 E l a s t i c cross sections used for normalization. C i r c l e s - this experiment. Squares - [ 3 ] . Triangles -The curve i s to guide the eye. [ 3 3 ] . 50 Table 4.2 Experimental D i f f e r e n t i a l Cross-sections f or the Reaction 1 80(TT+,Tr~) 1 8Ne (DIAS) at 48.3 MeV. da/dn C M (wb/sr) 18.2 3.97 ± 1.07 28.3 3.81 ± 0.51 38.4 3.40 ± 0.46 48.5 1.73 ± 0.23 90.6 0.81 ± 0.20 122.6 0.57 ± 0.11 Errors include a l l angle-dependent sources but not the o v e r a l l normalization error of 12 %. 9 CM (degree) Estimated 0° cross-section Estimated t o t a l cross-section 4.7 ± 0.5 yb/sr. 16.2 ± 1.2 ub/sr. 51 F i g . 4.2 DCX cross sections to the DIAS f o r 1 8 0 . The curve i s a f i t to the data of the function A exp[X(cos6 - 1)] + B. 52 either nuclear structure e f f e c t s are unimportant, or that they are the same for both 1 8 0 and l l +C. The l a t t e r i s possible, since i n some models the two n u c l e i are s i m i l a r : 1 8 0 has an 1 6 0 core with two extra neutrons, 1>*C has an 1 6 0 core with two holes. However, the 5° e x c i t a t i o n functions for 1 8 0 and 1 J |C d i f f e r i n the range 100 MeV to 180 MeV [36], claimed to be an e f f e c t of nuclear structure. Another possible explanation i s that i f the 2 + excited state gives a large contribution to DCX, then since 1 8 0 and 1'*C both have a low l y i n g 2+ state one would expect s i m i l a r cross sections. One theory predicts an A ~ 1 0 / 3 dependence for DCX cross sections [29]. This i s observed at 164 and 292 MeV ( F i g . 1.3). Such a dependence would give 1 8 0 cross sections l e s s than half l 4 C cross-sections; t h i s dependence i s c l e a r l y absent. The curve i n F i g . 4.2 represents a f i t to the data of the convenient function A exp[A(cos6 - 1)] + B varying the parameters A,B,X. This function gives a zero-degree cross section of 4.7 ± 0.5 yb/sr and on i n t e g r a t i o n gives a t o t a l cross section of 16.2 ± 1.2 yb. At backward angles, DCX t r a n s i t i o n s to the f i r s t excited state become important. At 120°, the ground state and f i r s t excited state are equally populated ( F i g . 4.4). This i s s i m i l a r to Seth's r e s u l t s at 164 MeV [11], although at 50 MeV the excited state i s not well populated u n t i l larger angles. The ease of t r a n s i s t i o n s to non-DIAS states i s evidence of mechanisms other than the simple s e q u e n t i a l one v i a the IAS. Comparing DCX and SCX e x c i t a t i o n functions further contrasts the two reactions. Over the energy range 50 MeV to 164 MeV: • The 0° DCX cross sections f o r 1 8 0 and lkC f a l l by a f a c t o r 3.5; whilst the 0° SCX cross sections r i s e by f a ctors 300 f o r 1 8 0 and 340 for lhC. (We have assumed the SCX 0° cross section f o r 1 8 0 i s roughly the same as 53 I 1 I 100 120 140 160 eo 0„, (degrees) O f F i g . 4.3 DCX cross sections f or 1 8 0 ( c i r c l e s ) compared to those f o r 1 4 C (squares) from [8]. The curve i s as i n F i g 4.2. •O a 1.89 V* 48.3 MeV 20 F i g . 4.4 6 Histogram for DCX at 122' 54 that of 1 1 +C [37]). Also, the 0° cross section for the SCX reaction ir~p + ir°n r i s e s by a f a c t o r of more than 2000. • The t o t a l DCX cross sections for 1 8 0 and 1 4 C f a l l by a factor 10; whilst the t o t a l cross section f o r 7r-p •*• ir°n r i s e s by a f a c t o r 10. These r e s u l t s provide further evidence that the DCX mechanism i s more complicated than the simple sequential one alone. The s i m i l a r i t y of the 1 8 0 and 1 L fC data i s very s t r i k i n g , e s p e c i a l l y since nuclear structure e f f e c t s are important i n many DCX c a l c u l a t i o n s (e.g. [19]). Further 50 MeV data on other n u c l e i , e.g. 2 eMg w i l l be very useful to determine the importance of nuclear structure e f f e c t s . 55 5 THEORY 5.1 INTRODUCTION Many models ex i s t for c a l c u l a t i n g charge exchange reactions. This s e c t i o n c o n t a i n s d i s c u s s i o n of a s e l e c t i o n from these, namely: Coupled-channels o p t i c a l p o t e n t i a l c a l c u l a t i o n s [6,38,39]; the six-quark c l u s t e r mechanism proposed by M i l l e r [16 ; recent c a l c u l a t i o n s by Jennings showing the e f f e c t of short range co r r e l a t i o n s [22]; and A-hole model c a l c u l a t i o n s by Karapiperis and Kobayashi [19,20,21]. Important c a l c u l a t i o n s not d i s c u s s e d i n c l u d e : Glauber theory c a l c u l a t i o n s , which use the eikonal approximation and are not expected to be v a l i d at 50 MeV; the multiple s c a t t e r i n g c a l c u l a t i o n s of Kaufman et a l . with the f i x e d scatterer approximation [40]; and the e f f e c t s of meson exchange currents (MEC). Oset et a l . [41] have calculated meson exchange current e f f e c t s such as F i g . 5.1 to be small enough to be treated as a c o r r e c t i o n to other models. + -F i g . 5.1 A meson exchange current contribution to DCX. Understanding the l i t e r a t u r e requires a reasonable knowledge of s c a t t e r i n g theory. The text by Taylor [42] gives a very readable de r i v a t i o n 56 of the main r e s u l t s for n o n - r e l a t i v i s t i c s c attering theory. Generalizations to the r e l a t i v i s t i c case are usually straightforward, such as replacing p r o j e c t i l e reduced mass m by CM energy E, and using the Klein-Gordon equation rather than the Schrodinger equation. Some of the c e n t r a l r e s u l t s are l i s t e d below. We consider a s c a t t e r i n g system with Hamiltonian H = H° + V, where H° contains the k i n e t i c energy operators and V contains the i n t e r a c t i o n . Such a system has a t r a n s i t i o n operator T which obeys a Lipmann-Schwinger equation T(z) = V + VG°(z)T ...(5.1) where G°(z) = (z - H 0 ) - 1 . The matrix elements of T between i n i t i a l |i> and f i n a l |f> states of the system are r e l a t e d to the amplitude f by f = (2ir) 2E<f | T | ±> ...(5.2) where E i s the t o t a l CM energy of the p r o j e c t i l e . The cross section for the process | i> •»• | f > i s ^ = | f | 2 -..(5.3) The Lipmann-Schwinger e q u a t i o n p r o v i d e s the b a s i s f o r many approximations, including the Born approximation which includes only the term V on the r i g h t hand side of Equation 5.1. In Tr-nucleus s c a t t e r i n g , the i n i t i a l and f i n a l states often contain d i f f e r e n t p a r t i c l e s . In the s c a t t e r i n g discussed here the target nucleus i s l e f t i n an analogue state , and the i r + changes charge. The d i f f e r e n t combinations of p a r t i c l e s are known as channels; u s u a l l y there i s an i n f i n i t e number of d i s t i n c t channels. This s i t u a t i o n can often be dealt with using either the d i s t o r t e d wave Born approximation (DWBA) or o p t i c a l p o t e n t i a l theory, both described i n [42]. One of the major goals of tr-nucleus theory i s to account for ir-nucleus reactions i n terms of ir-nucleon reactions. The usual s t a r t i n g point i s the 57 multiple scattering s e r i e s , see e.g. Hiifner's a r t i c l e [43] or [1]. This r e l a t e s the u-nucleus t r a n s i t i o n operator to a ser i e s with terms representing the s c a t t e r i n g from 1,2,3...nucleons. Various approximations then allow forms f o r the o p t i c a l p o t e n t i a l to be derived, with parameters which can be rel a t e d to ir-nucleon reactions. This i s the approach of DCX o p t i c a l p o t e n t i a l c a l c u l a t i o n s by Johnson and S i c i l i a n o , M i l l e r and Spencer, and others. 5.2 Optical-Potential Model Calculations The o p t i c a l p o t e n t i a l model i s the most frequently used method for describing pion-nucleus s c a t t e r i n g . It has s u c c e s s f u l l y been applied to e l a s t i c and i n e l a s t i c s c a t t e r i n g [3,4,44] and more recently has been applied to charge exchange reactions [6,39,38]. See [1] for an introduction to the ir-nucleus e l a s t i c o p t i c a l p o t e n t i a l . Note well that i n p r i n c i p l e the o p t i c a l p o t e n t i a l accounts exactly f or contributions from a l l excited states [42]; i t Is i n f i n d i n g approximate forms for the p o t e n t i a l , e.g. from multiple s c a t t e r i n g theory, that excited-state contributions can be l o s t . For charge exchange reactions on T=l n u c l e i to the IAS and DIAS, three channels must be kept e x p l i c i t l y ; these are 1) i r + + target nucleus 2) -rr° + IAS 3) TT " + DIAS The g e n e r a l i z a t i o n from one e x p l i c i t state to three i s straightforward; L i u gives the g e n e r a l i z a t i o n to two channels needed to account for e l a s t i c and SCX reactions [45]. The cross sections can then be calculated by solving the r e s u l t i n g coupled-channel Klein-Gordon equations e.g.[6]: 58 (-V2 + 2EV C - k 2)i|> = 2E I U J . ct a / T a £ afi g p Q where V i s the Coulomb p o t e n t i a l i n channel a, U _ i s the p a r t of the o p t i c a l p o t e n t i a l causing t r a n s i t i o n s from state 3 to o, E and k^ are the pion t o t a l energy and i t s momentum i n the f i n a l state, and i s the one-body sc a t t e r i n g wave function i n channel a. The boundary conditions are [42] exp(ik .r) <J> (r) 6 „exp(ik .r) + f „ a - aB F V -$ - aB r + a hence f ag» the s c a t t e r i n g amplitude from channel B to a, can be determined. The most general form of an i s o s p i n - i n v a r i a n t o p t i c a l p o t e n t i a l i s U = U Q + U^.T + U2(4>_.T)2 where and T_ are the pion and nuclear i s o s p i n . U Q, U l f and U 2 are referred to as the i s o s c a l a r , isovector and isotensor terms. The U^, i = 0,1,2, can be r e l a t e d to the ir-nucleon phase s h i f t s by making the 'density expansion': U = I U ( j ) i L. i J where the j 1 " * 1 element represents a s c a t t e r i n g i n which j nucleons act. U^*^ (2) i s c a l l e d the f i r s t - o r d e r o p t i c a l p o t e n t i a l , U the second etc. M i l l e r i n [6] gives an example of simple sequential model ca l c u l a t i o n s along these l i n e s ( F i g . 5.2). He considers only i s o s c a l a r and isovector terms. F i r s t he derives forms f o r the f i r s t order o p t i c a l p o t e n t i a l i n terms of the ir-nucleon phase s h i f t s . The method uses the multiple scattering s e r i e s , with the 'coherent' approximation [43]. This r e s t r i c t s the nucleus to the ground state - i . e . the IAS - between scatterings off successive nucleons. At 50 MeV t h i s leads to small, forward dipped DCX predictions i n models f i t t e d to SCX measurements [15], because the SCX cross section i s small and forward dipped. This suggests there i s a large DCX contribution from non-analogue states. M i l l e r also considers second-order terms, but s t i l l omits isotensor terms. / P P TT / IT :+ / / / n n F i g . 5.2 The simple s e q u e n t i a l mechanism f o r DCX.The i n t e r m e d i a t e L i u has shown the importance of the second-order o p t i c a l p o t e n t i a l d i s t r i b u t i o n at 164 MeV, and obtains excellent agreement with the data. Previous c a l c u l a t i o n s without isotensor terms wrongly predicted the p o s i t i o n of the minimum i n the angular d i s t r i b u t i o n . Liu's model f a i l s to predict the 50 MeV data. Johnson and S i c i l i a n o are developing an o p t i c a l model for charge exchange reactions. They are working towards a comprehensive model which w i l l describe charge exchange at 50 MeV as well as at 164 MeV [46]. However so far most of t h e i r work i s at 164 MeV. Johnson begins with f i r s t - o r d e r o p t i c a l p o t e n t i a l c a l c u l a t i o n s i n [47]. He relates the A dependence of cross sections to 'geometric' properties of the nucleus: a r a d i u s parameter R, the r a t i o of v alence neutron density to t o t a l nucleon d e n s i t y at R, and d i f u s e n e s s parameters. In p a r t i c u l a r he explains the state i s the IAS. The v e r t i c e s involve only U x. including isotensor terms [39]. He calculates the 1 8 0 DCX angular 60 (N-Z)(N-Z-1)A 1 0 / 3 dependence of DCX cross sections at 164 MeV (Fig 1.3). He predicts DCX to be a s e n s i t i v e probe of neutron-proton density d i f f e r e n c e s . Whilst the r e l a t i v e A dependence i s c o r r e c t l y predicted absolute values are not - presumably due to the omission of second order terms. In [48] Johnson and S i c i l i a n o f i n d a very general form for the second-order o p t i c a l p o t e n t i a l . It contains 4 complex parameters to be f i t t e d to the data. The form i s derived by considering various Feynman-like graphs. By using t h i s t h e o r e t i c a l motivation they expect the f i t t e d values of the parameters w i l l give information ( i ) on the structure of the target, and ( i i ) on two nucleon and other dynamical e f f e c t s of the pion-nucleus i n t e r a c t i o n . DCX i s l i k e l y to give much information on second-order terms i n the o p t i c a l p o t e n t i a l ; such information i s d i f f i c u l t to obtain by other means. Data at 50 MeV may be e s p e c i a l l y important since DCX contributions from f i r s t - o r d e r terms are small. Johnson and S i c i l i a n o have not yet published r e s u l t s of any second-order o p t i c a l p o t e n t i a l c a l c u l a t i o n s at 50 MeV. A comprehensive model, capable of predictions at 50 MeV as well as higher energies i s needed to r e a l i z e the f u l l p o t e n t i a l of DCX measurements. 5.3 The Six-Quark Cluster Mechanism The presence of six-quark c l u s t e r s i n nu c l e i has been suggested as an explanation of the EMC e f f e c t [17] and the magnetic moments of the A=3 system [18]. M i l l e r i n 1984 suggested they are also responsible for the forward peaking of 50 MeV DCX cross sections [16]. The presence of six-quark c l u s t e r s i n n u c l e i would introduce quark degrees of freedom into the nuclear wave function i n addi t i o n to baryons and mesons. Six-quark c l u s t e r s can form when two nucleons come close together. Most of the time two nucleons i n a nucleus are well separated and behave as 61 independent p a r t i c l e s . But when the distance between t h e i r centres drops below some c r i t i c a l distance r Q , they may behave as one p a r t i c l e : t h i s i s c a l l e d a six-quark c l u s t e r ( F i g . 5.3). 2 nucleons 6q c l u s t e r F i g . 5.3 Formation of Six-Quark Clusters The value of r Q i s constrained. If r Q i s greater than 1.2 fm, nucleon-nucleon s c a t t e r i n g data would be hard to explain. If r Q Is less than 0.7 fm, the p r o b a b i l i t y of six-quark c l u s t e r formation i s too small to give a s i g n i f i c a n t e f f e c t . M i l l e r uses r 0 - l fm: at th i s distance, the edge of one nucleon i s at the centre of the other; 30% of the nucleon volumes overlap. M i l l e r c a l c u l a t e s the probablity of six-quark c l u s t e r formation as follows. For DIAS t r a n s i t i o n s the core nucleons are P a u l i blocked, so only the 2 valence neutrons matter. He assumes when the distance between these neutrons i s l e s s than r Q a six-quark c l u s t e r forms, with wave function whilst f o r r > r Q the neutrons have the usual two-neutron wave function \b ( R , r ) . R i s the p o s i t i o n f o r the c e n t r e of the two neutron system T n n — ' — ' — v J r e l a t i v e to the nucleus centre; r_ i s the vector from R to one of the neutrons. Then the p r o b a b i l i t y of f i n d i n g a six-quark c l u s t e r at R_ i s - / d 3 r l * 6 q ( - ' £ ) | 2 e ( r o ~ r ) - J d * r l ^ n n ( R . £ ) l 2 e C r 0 - r ) The second equality follows from conservation of p r o b a b i l i t y current. He 62 uses a product of two p - s h e l l harmonic-oscillator wave functions f o r \i> with nn b = 1.66 fm f o r 1 1 +C. The t o t a l probablity that the two neutrons w i l l be found i n a six-quark c l u s t e r i s then P6q = ' d 3 R " 0 ' 0 6 The simplest six-quark c l u s t e r contribution to the DCX amplitude i s shown i n F i g . 5.4. When the f i n a l - s t a t e six-quark c l u s t e r s p l i t s i t w i l l form two protons. u ' d / / / / / (a) (b) F i g . 5.4 Simplest six-quark c l u s t e r contribution to DCX (a) uncrossed (b) crossed terms. M i l l e r uses products of single-quark wave functions with complete s p a t i a l symmetry (6-Symmetry) f o r the six-quark wave functions. The single-quark wave functions are the lowest energy o r b i t a l s of the MIT clu s t e r model. The two valence neutrons have spin S=0, i s o s p i n T=l and^Tg 3 -1. The i n i t i a l six-quark state, |i>, must have the same. S i m i l a r l y the f i n a l |f> six-quark state must have S=0, T=l, T 3= +1 i . e |i> = |6q, S-0, T=l, T 3= -1> |f> - |6q, S-0, T=l, T 3= +1> 63 The pion absorption operator i s an a x i a l vector (see Equation 5. ). So fo r pion absorption or emission, AS = ±1. Then the assumed symmetry of the s p a t i a l wave function requires T even. Hence the only possible intermediate states are Im^  = |6q, S=l, T=0> |m2> = |6q, S=l, T=2> Mulders and Thomas [49] have calculated the energies of these states as E-^ = 290 MeV and E 2 = 600 MeV above the mass of two nucleons. The tr-nucleus p o t e n t i a l V can be s p l i t into two parts, V = Vj+ V 2« Vj i s chosen to be the TT i n t e r a c t i o n with the 1 6 0 core and the non charge-exchange i n t e r a c t i o n with the valence neutrons. Then V 2 i s the charge-exchange p o t e n t i a l operator for the valence nucleons. Its matrix elements are determined by requiring p a r t i a l conservation of the axial-vector current (PCAC) [1]. They are 3 6 <m|V2|i> = i f y u ( k R 6 ) ( 2 E ) _ 1 / 2 <m| I o^.k T+|i> ...(5.4) a=l where E and _k are the pion t o t a l energy and momentum; Rfe i s the radius of a s i x - q u a r k c l u s t e r taken to be 1.3 fm [50]; f i s the TTN coupling constant; Q Qand T+ are P a u l i spin and i s o s p i n operators; and u(kR 6) = 3 j 1 ( k R & ) / k R 6 i s a form f a c t o r to allow f o r the f i n i t e s i z e of the six-quark c l u s t e r . The T matrix elements between target and DIAS states are given by the Born s e r i e s : - — < f | T | i > = <f | v|i> + <f|VG°V|i> + ... \ = <f|V 2G°V 2|i> + ... where terms with le s s than two factors V 2 are zero because two charge exchanges are needed. M i l l e r uses the plane wave approximation (PWA), i . e . 64 he ignores higher order terms. Inserting 1 = £|m><m| gives m <f|T|i> = I <f |V 2G°|mXm|V 2|i> m For the uncrossed diagram, G 0 | m . > = J _ _ ' J m whilst f o r the crossed diagram G 0 | m > = 1 I j -E-E J m Adding these and taking into account the p r o b a b i l i t y of fi n d i n g a six-quark c l u s t e r gives 2E <f|T|i> = P (qj I <f |V2|m><m|V2|i> 4 m E z-E 2 m where Pg^C^) i s the Fourier transform of Pg^QO. So f a r no allowance has been made for absorption i . e . TTNN -*- NN. When th i s i s allowed f o r the model gives the curve i n F i g . 5.5 f o r lhC [50]. The model i s seen to reproduce forward angle DCX s c a t t e r i n g well although the forward peaking i s too pronounced. At other angles, more conventional mechanisms can account f o r DCX since the p ( T r +,TT°)n amplitude i s la r g e r . The six-quark mechanism involves only the valence neutrons. This makes the predicted d i s t r i b u t i o n f o r 1 8 0 very much the same as f o r 1 1 +C. 65 F i g . 5.5 Comparison of t h e o r e t i c a l predictions with measured 50 MeV 1 8 0 DCX data. (a) S o l i d l i n e : Six-quark c l u s t e r contribution f o r 1 4 C . (b) Dotted l i n e : Jennings and de Takacsy's p r e d i c t i o n s . (c) Dashed l i n e : A-hole model c a l c u l a t i o n . 66 The PWA neglects higher-than-V 2G°V 2 terms i n the Born s e r i e s . Multiple s c a t t e r i n g terms such as V 1G°V 2G°V 2, representing three scatterings may be expected to be large. Such terms could be accounted for by using d i s t o r t e d waves to represent s c a t t e r i n g to a l l orders by V j . However there i s evidence that such corrections are small [51] at 50 MeV. The six-quark c l u s t e r s do not contribute to DCX at higher energies. This r e s u l t s from the rapid decay of the intermediate states |m> into two nucleons and a TT above E = 100 MeV. The same input gives a six-quark c l u s t e r contribution to SCX which i s roughly 0.3 times the c o n t r i b u t i o n to DCX. M i l l e r claims t h i s i s a r e s u l t of interference between terms with d i f f e r e n t intermediate states. Six-quark c l u s t e r s i n v o l v i n g a core nucleon and a valence neutron can also contribute to SCX; but the Wigner-Eckart theorem implies summing over a l l such pairs gives zero provided the core has equal numbers of neutrons and protons. When M i l l e r ' s paper [16] was published, only the data for 50 MeV 1 4 C DCX of Navon [13] was a v a i l a b l e . This suggested the 0° cross section was as high as 12 ub/sr. M i l l e r discusses several possible contributions to DCX. He shows none can produce DCX 0° cross sections as high as 12 ub/sr whilst keeping SCX small and forward dipped. Leitch's measurements [8] showed the 0° cross section was only 3.9 ± 0.5 ub/sr, s i m i l a r to the 1 5N SCX 50 MeV cross section. In view of t h i s , M i l l e r concluded that other mechanisms could explain DCX, but that the six-quark c l u s t e r mechanism could not be ruled out [52]. 5.4 E f f e c t of Short Range Cor re l a t i ons Recent preliminary c a l c u l a t i o n s by Jennings and de Takacsy have reproduced forward peaking. In t h e i r model short range c o r r e l a t i o n s between the valence neutrons cause the backward DCX cross section to be small. The 67 model i s at an early stage of development; so far only a simple model has been used which requires rather a high closure energy to f i t the data. The importance i n nuclear physics of nucleon-nucleon c o r r e l a t i o n s has been stressed by many authors, including Ericson and Ericson [53], and Eisenberg et a l . [54]. Correlations are expected to have large e f f e c t s on DCX because the r e a c t i o n must involve at least two nucleons. Correlations can be viewed as a r i s i n g from a and co meson exchange. This does not contribute to e l a s t i c s c a t t e r i n g , where the exchanged p a r t i c l e must carry charge. For DCX though, the exchanged p a r t i c l e need only transfer energy and momentum. F i g . 5.6 i l l u s t r a t e s the mechanisms dealt with. The exchanged p a r t i c l e s are not treated e x p l i c i t l y . Instead t h e i r e f f e c t i s included i n the wave function of the two valence neutrons by using a Gaussian dependence on the r e l a t i v e coordinate. / -/ a , i u / a , a ) / 7T+/ / • V F i g . 5.6 The mechanisms considered by Jennings and de Takacsy. A s i m i l a r treatment to Section 5.3 gives 68 2E T = I 5 _ <f |V|m><m|v|i> m E 2 - E 2 m where now the intermediate states are shell-model excitations of 1 8 F . Using the closure approximation r l m X m l 1 , ~ . 2, J L 0 — — , where E i s the closure energy, gives E-E E-E m o p T - <f | VV | i> E 2-E The p o t e n t i a l operator V producing charge exchange i s analogous to M i l l e r ' s but has operators defined on nucleons rather than quarks, and i s summed over the valence neutrons: 2 V - - i f ( 2 E ) - 1 / 2 T a .k T+ L , — a — a a=l The states are taken to be |i> = U r l t r 2 ) |S=0, T=l, T 3= -1> |f> = * ( r l f r 2 ) |S=0, T - l , T 3= -1> with the s p a t i a l wave function 4 » ( r l f r 2 ) = C 1F 1(-N;3/ 2;2bR 2) exp(-bR 2 - J r 2 ) where ^F^ i s a confluent hypergeometric s e r i e s , C i s a normalization c o n s t a n t , R = 1 / 2 ( £ 1 + £ 2 ) > JL = Ll~L2' F o r 1 8°» N i s u s u a l l y taken to be 2 whilst f o r lhC i t i s 1. In the s h e l l model the parameter b i s usually taken to be 0.33 f m - 2 for 1 8 0 and the c o r r e l a t i o n parameter a = b. 69 These wave functions give T 1 2 2E f 2 k 2cos8 exp ("lr-2)Vz> (2TT)3 E E 2 - E 2 where k f ' z q2/8b, £ = k j [- k f , K = / 2 ( k i + k f ) and P N i s a polynomial of degree 2N. For N 1 - i 3 z + 44 15 The f a c t o r 1 3- depends on the wave f u n c t i o n n o r m a l i z a t i o n ; w i t h t h i s (2TT) normalization Equations 5.2 and 5.3 give the cross section. The sum over two neutrons gives one of the factors 2 i n the numerator. The closure energy can be treated as a free parameter. A value E « 7 MeV f i t s the data best. This i s a l i t t l e high i n view of the energy l e v e l s of 1 8 F ( F i g . 5.7). If the low l y i n g states are the main contributors to DCX one would expect a smaller closure energy. This i s an i n d i c a t i o n the model i s an o v e r s i m p l i f i c a t i o n . The above input gives the curve shown i n F i g . 5.5. The strong forward peaking a r i s e s l a r g e l y from the cos0 factor i n T. The cross section i s small a t back angles ( l a r g e q) because of the f a c t o r e . The forward c r o s s section decreases r a p i d l y with energy because of the factor exp(-k 2/2a), i n agreement with the data. The c o n t r i b u t i o n of t h i s mechanism to the SCX t r a n s i t i o n operation i s a fac t o r 2 smaller. Assuming no complications from interference, t h i s gives a fac t o r 4 smaller i n the cross section. The factor 2 a r i s e s from the form of the operator ^.J_, see e.g. [55]. Thus t h i s simple model with conventional mechanisms can explain the 70 data. Exotic mechanisms such as six-quark c l u s t e r s seem unnecessary. 1+, 0 (1.70 MeV) 0+, 1 (1.04 MeV) i 8 F e - s - 1+, 0 F i g . 5.7 1 8 F energy l e v e l diagram. 5.5 The A-hole Model The A-hole model successfully accounts for e l a s t i c , i n e l a s t i c and SCX Tr-nucleus s c a t t e r i n g near resonance energy [56]. Karapiperis and Kobayashi have recently applied the model to DCX reactions [19,20,21]. They reproduce data for 1 4 C , 1 6 0 and 1 8 0 from 50 MeV to 164 MeV with reasonable success, except for 1 8 0 above 100 MeV. For an introduction to the A-hole model, see [4] and references therein. The main improvement of the A-hole model over standard o p t i c a l p o t e n t i a l models i s i n the treatment of the nuclear state between scatterings i n multiple s c a t t e r i n g theory. At resonance energy, Tr-nucleus scattering i s dominated by the A(1232) resonance. In the A-hole model t h i s resonance i s treated as a A - p a r t i c l e coupled to a nucleon hole i n the nucleus; the e f f e c t s of the propagation of t h i s A-hole through the nucleus i s allowed for e x p l i c i t l y . In standard o p t i c a l models these e f f e c t s are omitted by making various approximations, e.g. the ' f a c t o r i z a t i o n approximation' [4]. At 50 MeV resonant A production i s very small. There are some problems 71 with the A-hole model for e l a s t i c s c attering at this energy [58], and these p e r s i s t f o r DCX. But Karapiperis and Kobayashi claim DCX presents no new problems for the A-hole model. The A-hole model c a l c u l a t i o n s are detailed and f a i r l y complicated. They are b r i e f l y summarized below. The r e s u l t s are also summarized with emphasis on 1 8 0 . The t r a n s i t i o n operator for DCX i s s p l i t into two terms, the sequential and the A-hole mechanisms ( F i g . 5.8) where F i g . 5.8 The basic DCX mechanisms i n the A-hole model, (a) Sequential (b)A-hole. The matrix elements of the sequential DCX t r a n s i t i o n operator are then where 72 The operator V allows for nucleon motion v i a the form factor V(K 2) and by using the ir-nucleon r e l a t i v e momentum K: V = f V(K 2) K . O T + V(K 2) = (1 + K2/a2)~l i s the CX part of the non-resonant TT-N transition operator. Karapiperis and Kobayashi only include the s-wave part. The coupling f and parameter a are f i t t e d to the TT +-proton p-wave phase-shifts. The matrix elements of the A-N part of the transition operator are <f |T.„|i> = <f .distortedlvV^t .„G.„VIi,distorted> 1 AN 1 1 AN AN AN 1 Karapiperis and Kobayashi choose the following zero-range form for the A-N interaction, t.„: AN <ll^\tLN\L^> - « ( r i - r N ) 6 ( r A - r N ) 6 ( R ' - R ) I V g TPgP T b ,T where R i s the CM c o o r d i n a t e of the A-N system and _r^>_r^ are r e l a t i v e c o o r d i n a t e s ; Pg and P^ , are p r o j e c t o r s onto the AN s t a t e w i t h spin S and i s o s p i n T; and V g ^ are complex, energy dependent parameters. R e s t r i c t i n g t ^ N to this form i s an o v e r s i m p l i f i c a t i o n ; more c a r e f u l treatment may improve quantitative p r e d i c t i o n s . The parameters Vg^ , are rela t e d to each other. The cont r i b u t i o n to the AN i n t e r a c t i o n from S=2 terms i s suppressed, so that are small; and the r e l a t i o n between the T=l and T=2 amplitudes give further r e s t r i c t i o n s . This leaves e f f e c t i v e l y one complex parameter f o r the model, 6v = v x l ~ v 1 2 . Note that the sequential part has no free parameters f o r DCX; the parameters are fix e d by e l a s t i c s c a t t e r i n g . The propagator i n c l u d e s s e v e r a l terms, such as the k i n e t i c and binding energies of the A, the Hamiltonian of the hole nucleus, a correction for A-decay to P a u l i forbidden states, and a 'spreading p o t e n t i a l ' . The l a s t 73 term i s a phenomenological terra to allow for absorption. This i s the only non-microscopic part of the model. The parameters of the spreading p o t e n t i a l are allowed to vary from nucleus to nucleus and are energy-dependent. The parameters were f i t t e d to Tf-nucleus e l a s t i c s c a t t e r i n g data, and included s-wave repulsive terms. The d i s t o r t e d i n i t i a l and f i n a l states are obtained using the f u l l A-hole e l a s t i c o p t i c a l p o t e n t i a l . The i n i t i a l , intermediate and f i n a l nuclear states used were harmonic o s c i l l a t o r states. For 1 8 0 and 1 6 0 the model of Zuker, Buck and McGrory [57] was used. This has contributions from the 0 p 1 / 2 s h e l l i n a d d i t i o n to 0 d 5 / 2 + l s 1 / 2 « This allows core nucleons to contribute to DCX; and so the amplitudes f o r 1 6 0 can be calculated with the same methods as 1 8 0 . Karapiperis and Kobayashi have made ca l c u l a t i o n s for the n u c l e i 1 4 C , 1 6 0 and 1 8 0 . The c a l c u l a t i o n s include SCX and DCX angular d i s t r i b u t i o n s at 50 MeV and 164 MeV, and e x c i t a t i o n functions. They obtain reasonable ( t y p i c a l l y within a f a c t o r ~ 2) agreement with much of the data. The main exceptions are ( i ) f o r the 1 8 0 angular d i s t r i b u t i o n at 164 MeV for which no value 6v gives agreement. 1 6 0 data i s reproduced by the model, so the model does not account for nuclear structure differences very w e l l . ( i i ) The 1 8 0 e x c i t a t i o n function which f a i l s to predict the cross section peak between 110 MeV and 140 MeV ( F i g . 1.5). Instead a reduced cross section i s predicted here due to a c a n c e l l a t i o n between a n a l o g u e and n o n - a n a l o g u e i n t e r m e d i a t e - s t a t e contributions, and between the sequential and AN i n t e r a c t i o n s . Karapiperis and Kobayashi f e e l a more detai l e d model could e a s i l y remove t h i s c a n c e l l a t i o n . 74 F i g . 5.5 shows t h e i r 1 8 0 DCX predictions at 50 MeV with 6v = 0.2-2.8i; t h i s gave the closest p r e d i c t i o n . They found the AN i n t e r a c t i o n was not important at 50 MeV, as expected for low energies. So the sequential amplitude can account for 50 MeV DCX data, with no need for exotic mechanisms such as the six-quark c l u s t e r . They f i n d the intermediate analogue-state contributes very l i t t l e to 50 MeV DCX. This i s as expected from SCX data. The main co n t r i b u t i o n to 50 MeV DCX i s from non-analogue intermediate states, and so the s-p c a n c e l l a t i o n i n SCX provides p r a c t i c a l l y no constraint on DCX. They f i n d pion d i s t o r t i o n s important. Using plane waves reduces the 0° l l +C DCX cross section at 50 MeV by a factor 2.4. Core contributions are also important. Using a simple (Px/2^ 2 m O Q e l with DCX only on valence neutrons, reduced the cross section by another factor 2.4. One reason for t h i s large factor i s that the simple model has no 2+ st ate; t h i s state gives a large c o n t r i b u t i o n . SCX i s backward peaked, and the 2+ state Is most e a s i l y excited by back-scattering a 50 MeV pion. Thus a large contribution from two back scatterings may be responsible f o r the foward peaking of DCX. Karapiperis and Kobayashi claim s u f f i c i e n t agreement with data to e s t a b l i s h that multiple s c a t t e r i n g theory i s adequate to describe charge exchange reactions. 75 6 CONCLUSION The angular d i s t r i b u t i o n f o r the DCX reaction to the DIAS f o r 1 8 0 has been measured at 48.3 MeV pion k i n e t i c energy. The 0° d i f f e r e n t i a l cross section i s 4.7 ± 0.5 yb/sr (by ex t r a p o l a t i o n ) . The t o t a l (angle integrated) cross section i s 16.2 ± 1.2 yb. The angular d i s t r i b u t i o n i s forward peaked and very s i m i l a r to that of ll*C. The forward peaking i s not predicted by the simple sequential model. Various proposals have been made f o r the o r i g i n of t h i s forward peaking. These include six-quark c l u s t e r e f f e c t s , short range c o r r e l a t i o n s , and contributions from non-analogue intermediate states. DCX i s expected to provide u s e f u l information f o r ir-nucleus models, e.g. on second-order and isotensor terms i n the o p t i c a l p o t e n t i a l , and on the A-nucleon i n t e r a c t i o n i n the A-hole model. It i s also expected to give information on nuclear structure such as neutron-proton density differences and short-range c o r r e l a t i o n s . Low energy pion (~ 50 MeV) DCX reactions are important f o r understanding the mechanism. The small si z e of the simple sequential contribution makes other mechanisms easier to study. More data i n t h i s energy region i s needed to f i x parameters and help choose between the various models. The recent measurement of the 50 MeV DCX angular d i s t r i b u t i o n f o r 2 6Mg at TRIUMF [23], and the proposals at TRIUMF f o r measurements on 5 6 F e , 31*S, and the e x c i t a t i o n function f or 1 8 0 from 30 MeV to 80 MeV [24], are important steps towards understanding the DCX mechanism. 76 BIBLIOGRAPHY 1 R. Baliau, M. Rho, G. Ripka ( e d i t o r s ) , Nuclear Physics with Heavy Ions and Mesons Vol. II (North Holland Pub. Co. 1977). 2 D.H. Perkins, Introduction to high energy physics 2nd ed. (Addison Wesley 1982). 3 R. Tacik, Ph.D. thesis (University of B r i t i s h Columbia, 1984, unpublished). 4 W. Gyles, Ph.D. thesis (University of B r i t i s h Columbia, 1984, unpublished). 5 K. K. Seth, International Topical Conference on Meson Nuclear Physics, Houston, 1979. 6 G. A. M i l l e r and J . E. Spencer, Ann. Phys. (N.Y.) 100(1976)562. 7 Burman et a l . , Phys. Rec. C 17(1978)1774. 8 M. J . L e i t c h et a l . , Phys. Rev. L e t t . 54(1985)1482. 9 S. J . 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