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Nuclear proton radii from low energy pion scattering Barnett, Bruce M. 1985

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NUCLEAR PROTON RADII FROM LOW ENERGY PION SCATTERING by BRUCE M. BARNETT B.Sc, McGill University, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard November 1985 © Bruce MacLeod Bamett 1985 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 of Physics The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D a t e December 3, 1985 DF.fin/ft-n i i Abstract The subject of this thesis is the study of the use of elastic scattering differential cross section ratios of positive pions on light nuclei in the determination of nuclear proton matter distribution differences and their moments. These are compared with results from electron scattering and muonic X-ray measurements of the charge density differences. The measurements are relative to nuclei whose matter distributions and absolute cross sections are considered as references. The use of the ratio of cross sections, rather than the absolute cross sections themselves, minimises the effects of uncertainties in the understanding of the pion nuclear interaction in our extraction of density difference information; these effects are investigated. A furthur advantage of the technique i s that the measurement of cross section ratios is insensitive to many systematic experimental effects encountered in measurement of absolute cross sections. The ratios of elastic scattering of positive pions at 38.6 MeV and 47.7 MeV on the isotones ^B, 1 2C are presented. Also, cross section ratios at 48.3 MeV and 62.8 MeV on the nuclei 1 2C, 1 60 and 1 80 are presented. The measured cross sections do not rival the quality of the cross section ratios, but are also presented. The RMS radius differences extracted from the pion elastic scattering cross section ratios at low energy are consistent (within a standard deviation) with the results of other methods. The proton matter radius (N+Zr+) differences which we obtain are as follows: i i i ( U r + _ 1 2 r + ) = _.Q64 (28) fm ( l t r + _ 1 2 r + ) = .098 (44) fm ( 1 6 r + _ 1 4 r + ) = # 1 5 9 ( 4 4 ) f m ( 1 8 r + _ 1 6 r + ) = . 0 8 4 ( 7 ) f m ( l l B , 1 2 C ) ( 1 2 G > m N > 1 6 0 ) {-.032(33) [ALK83] } { .074(21) [SCH75] } { .180(22) [SCH75] } { .077 (5) [MIS79] } The er rors r e f l e c t s t a t i s t i c a l and, to a large extent , the systemat ic u n c e r t a i n t i e s i n the q u a n t i t i e s . The best e l e c t r o n s c a t t e r i n g r e s u l t s are shown w i th in braces . A n a l y s i s of the 1 2 C , ^ B experiment i n terms of RMS radius d i f f e r e n c e s i n d i c a t e s systematic u n c e r t a i n t i e s of about the same order as the s t a t i s t i c a l u n c e r t a i n t i e s . In p a r t i c u l a r , choice of o p t i c a l p o t e n t i a l , densi ty form and o p t i c a l parameter set i s of l i m i t e d importance. Analyses of the 1 8 0 , 1 6 0 experiments i n terms of F o u r i e r Besse l and novel Four ier Laguerre proton matter ( r a d i a l ) dens i ty d i f f e r e n c e s (App(r))agree with p r e c i s i o n "model independent" [NOR82] e l e c t r o n s c a t t e r i n g r e s u l t s , i n the region i n which the pion can be s e n s i t i v e to the nuclear proton matter d i s t r i b u t i o n s . (The p h y s i c a l l i m i t s enforce r ? 1.5 fm.) The e f f e c t s of o p t i c a l parameter uncer ta in t i es are d i s c u s s e d . S i m i l a r analyses of the 1 2 C , 1 4 N , 1 6 0 experiments i n d i c a t e a s h i f t i n the proton matter densi ty d i s t r i b u t i o n of 1 4 N towards the center of that nucleus , r e l a t i v e to that suggested by e l e c t r o n s c a t t e r i n g experiments [SCH75] and at about the same radius suggested by S_elf -Consistent S ingle P a r t i c l e P o t e n t i a l (SCSPP) c a l c u l a t i o n s [H0D85]. The analyses of these e l e c t r o n s c a t t e r i n g experiments were not model independent, though, and may w e l l generate d e n s i t i e s which are incor rec t at small radius (high momentum t r a n s f e r : r < 2 fm.) We present these r e s u l t s as testimony to the a b i l i t y of the T f + to i v probe App(r) r e l i a b l y . The o p t i c a l parameter s e n s i t i v i t y i s minimal, e s p e c i a l l y when cross section f i t t i n g i s used to determine the reference nucleus o p t i c a l parameters. This work provides corroboration f o r the analogous tr" neutron density measurements of [JOH79, GYL85]. Table of Contents Abstract i i Table of Contents v List of Figures x List of Tables xx i Acknowledgements xxv i Chapter I Introduction 1 1.1 In t roduct ion 1 1.2 T r a d i t i o n a l Matter Probes 1 1.3 Low Energy Pion S c a t t e r i n g : The I s o r a t i o Method 3 1.4 The Pion Nuclear In te rac t ion 5 1.4.1 The Pion Nucleon In te rac t ion : Currents 5 1.4.2 The Pion Nuclear In te rac t ion 9 1.5 Matter D i s t r i b u t i o n S e n s i t i v i t y of the Rat io 12 1.6 Previous I sora t io Experiments 14 1.7 Current Object ives 16 1.8 Summary 16 Chapter II Experimental Details 18 2.1 In t roduct ion 18 2.1.1 Machine D e t a i l s 18 2.2 1 2 C a r b o n , n B o r o n (CB) Experiment 20 2.2.1 Experimental Setup 25 2.2.2 Sca t te r ing Telescopes 25 2.2.2 .1 Operat ing C h a r a c t e r i s t i c s 28 2.2.3 E l e c t r o n i c s 33 2.2.4 Targets 36 2.3 1 2 C , 1 6 0 , 1 8 0 (CNO 2Experiments 37 2.3.1 The QQD Spectrometer 41 v i 2.3.2 E l e c t r o n i c Logic 42 2.3 .3 Targets 44 2.4 Summary 49 Chapter III Analysis 50 3.1 In t roduct ion 50 3.2 1 2 C , n B Experiment 50 3.2.1 Pre l iminary A n a l y s i s : Software Cuts 51 3.2.2 E v a l u a t i o n of Absolute Cross Sect ions 51 3.2.3 Eva lua t ion of Rat ios 66 3.2.4 Resu l ts 68 3.3 CNO2 Experiments 81 3.3.1 Momentum Spectrum Opt imisat ion 81 3.3.2 Software Cuts 82 3.3.2.1 NMR Cuts 82 3.3 .2 .2 Target Traceback Cuts 83 3 .3 .2 .3 TOF Cuts 86 3.3 .2 .4 Muon Cuts 86 3 .3 .2 .5 Energy Spectra 89 3.3.3 Peak F i t t i n g 94 3.3.4 Absolute Cross Sect ion Measurement 96 3.3.4.1 BM1»BM2 Rate Loss 96 3 .3 .4 .2 Spectrometer Of fset Angle 103 3 .3 .4 .3 Average Sca t te r ing Angles 103 3 .3 .4 .4 S o l i d Angle Determination 104 3.3.5 Rat io Evaluat ions and Correct ions 107 3.3.6 Resul ts 108 3.4 Summary 108 Chapter IV Interpretations 134 4.1 In t roduct ion 134 4.2 O p t i c a l P o t e n t i a l Ana lys is 135 v i i 4.2.1 SMC Potentials 137 4.2.1.1 The Potential 137 4.2.1.2 The Calculation 139 4.3 CB Experiment 140 4.3.1 Cross Section Calculations with the SMC Potential 140 4.3.2 Generic Low Energy Pion Scattering 143 4.3.3 Ratio Calculations 146 4.3.4 Sensitivities to Higher Moments 153 4.3.4.1 Modified Density Results 155 4.3.5 Sensitivities to Optical Parameters 160 4.3.6 Miscellaneous Sensitivities 160 4.3.7 Optical Model Dependence of Results 165 4.4 CNO2 Experiments 165 4.4.1 Cross Section Fitting 165 4.4.2 Density Distribution Difference Analysis 170 4.4.3 1 80, 1 60 Experiments 176 4.4.3.1 Matter Distribution Determinations 176 4.4.3.2 Optical Parameter Sensitivities 180 4.4.4 ^ C , 1 ^ , 1 ^ (CNO) Experiments 181 4.4.4.1 Matter Distribution Difference Analysis 181 4.5 Summary 187 Chapter V Discussion and Summary 192 5.1 Overview 192 5.2 Optical Potential Fitting 193 5.3 1 2C, n B Experiment 194 5.4 CNO2 Experiments 197 5.4.1 1 80, 1 60 Experiments 197 5.4.2 CNO Experiments 199 5.5 Summary 202 5.6 Epilogue 203 v i i i List of References 205 Appendix I Subscripts and Superscripts 215 Appendix II Powder Boron Target Mass Measurements 216 Appendix III Decay Kinematics 219 A3.1 Muon Cone Angle 219 A3.2 Beam A t t r i t i o n 221 Appendix IV Core + Valence Matter Distributions 225 A4.1 The D i s t r i b u t i o n s 225 A4.1.1 The MGCV D i s t r i b u t i o n 225 A4.1 .2 The CF D i s t r i b u t i o n 229 A4.1 .3 The CFCV D i s t r i b u t i o n 231 Appendix V A Replacement for Viewfit 234 Appendix VI Spectrometer Transfer Coefficient Optimisation 236 Appendix VII Peak Fitting in the CNO2 Experiments 242 A7.1 In t roduct ion 242 A7.2 The Peak Shape 242 A7.3 Mechanics of F i t t i n g 245 A7.4 C o r r e l a t i o n E r r o r s 245 A7.5 Propagat ion of E r r o r s 247 A7.5.1 Rat io Measurements 249 Appendix VIII Fourier Expansions of the Nuclear Density 255 A8.1 In t roduct ion 255 ix A8.2 Fourier Bessel Analysis 255 A8.3 Fourier Laguerre Expansions 256 A8.3.1 The Basis 256 A8.3.2 Zero Sum Constraint 258 A8.3.3 Derivatives of PFLU) 2 5 8 A8.3.4 Radial Moments 259 A8.3.5 Uncertainties i n ppL( r) 2 6 0 A8.3.6 The Densities 261 Appendix IX Of Charge Densities and Matter Densities 263 A9.1 Convolution of the Proton Form Factor 263 A9.2 Folding Ppp into a Fourier Laguerre Sum 264 A9.3 Folding p p p i n t o a Fourier Bessel Sum 266 A9.4 Folding p p p into the Starting Density P j ( r ) 266 A9.5 Evaluations of $Q 266 A9.5.1 «(»0 f o r the Folded MG Charge Density 267 A9.5.2 <(>0 for the Unfolded FB Charge Density 267 A9.5.3 <|>0 f o r the Folded FB Charge Density 267 A9.5.4 <j>0 f o r the Unfolded FL Charge Density 268 A9.5.5 <|>0 f o r the Folded FL Charge Density 268 A9.6 Unfolding of the Proton Form Factor from E l e c t r o n Scattering Data 268 Appendix X Moments and the Folding Process 274 A10.1 The Folding and the RMS Integrals 274 A10.2 Other Moments v i a the Fourier Transform 275 List of Figures 1.1 A low energy n + scattering experiment (The 15 MeV experiment: [GIL82]) similar in configuration to the carbon-boron T T + ratio experiment described in the text. 4 1.2 Kinematic observables for pions scattering from nuclear material. The pion, of mass m, charge e^, isospin x, momentum nk and angular momentum I scatters from a nucleus with Z protons, N neutrons, total isospin t, total spin J. The protons and neutrons have distributions p+(r) and p_(r), respectively. 6 1.3 The results of i r ~ scattering experiments (29 MeV) on isotopes yield relative neutron r a d i i . 1 2 a _ ( 1 3a_) is the elastic differential scattering cross section for pions on 1 2C ( 1 3C) [GYL79a]. 15 2.1 a) Channel configuration and b) momentum resolution of M13 [0RA81]. 19 2.2 Comparison of fluxes available from meson channels Mil and M13. 21 2.3 M13 TOF spectra at 40 and 50 MeV. 23 2.4 Typical beam profiles at 40 (a and b) and 50 (c and d) MeV, for i r + on M13. a) and c) are horizontal profiles, b) and d) ve r t i c a l . Structure on the vertical profiles is an ar t i f a c t . 24 2.5 Scattering table arrangement, showing TT scattering telescopes and beam monitoring apparatus. 26 6 Peak, height and r e s o l u t i o n f o r S i ( L i ) detectors. 7 Passing r e s o l u t i o n of the S i ( L i ) detectors to pions. 8 Energy spectrum for the Nal detector. 9 L i n e a r i t y of Nal detectors. 10 Contour plot of E vs AE for 128 MeV/c p a r t i c l e s emerging from Ml3. 11 C i r c u i t diagram for CB experiment. 12 Experimental setup showing (a) spectrometer-channel configuration on M13 and (b) enlargement of spectrometer. 13 Pion TOF spectra of beams from (a) M13 at 50 MeV and (b) Mil at 65 MeV. 14 QQD spectrometer e l e c t r o n i c l o g i c . 15 Beamspot p r o f i l e at the target frame at 65 MeV and 120 degrees (RUN1116). Contours show increases i n i n t e n s i t y of 10% of the maximum. 1 Uncut s c a t t e r p l o t for (run28, ARMl). Pions incident on 1 2 C at 47.7 MeV as scattered though a lab angle of 90°. 2 Uncut AE histogram for (run28, ARMl). AE cuts applie to E histogram i n figure 3.3 are shown. x i i 3.3 Energy histogram fo r (run28, ARM1) a f te r cut on AE. Pions i n s i d e the cuts are e l a s t i c a l l y s c a t t e r e d . 55 3.4 AE histogram as i n f i g u r e 3 .2 , but cut only on the f i n a l E spectrum. This shows the AE cut of that f igure to have adequate width and cen te r ing . 56 3.5 Ca lcu la ted pion losses while stopping i n a Nal de tec tor : s o l i d curve shows losses from r e a c t i o n s ; broken curve shows those from pion decay. 63 3.6 Muon decay c o n t r i b u t i o n to te lescope e f f i c i e n c y . 65 3.7 Exc i ted s ta tes of s tab le boron and carbon i s o t o p e s . 67 3.8 1 2 C e l a s t i c d i f f e r e n t i a l cross s e c t i o n at 38.6 MeV. Curves are from o p t i c a l model (SMC79) c a l c u l a t i o n s with parameter sets 1 (broken) and l a ( s o l i d ) . 70 3.9 1 2 C e l a s t i c d i f f e r e n t i a l cross sec t ion at 47.7 MeV. Curves are from o p t i c a l model (SMC79) c a l c u l a t i o n s with parameter sets 1 (broken) and lb ( s o l i d ) . 71 3.10 Ratio of cross sect ions of TT+ on 1 2 C and U B at 38.6 MeV. O p t i c a l c a l c u l a t i o n s use parameter set 2. Centra l curve uses best f i t value of 1 1 r + . 79 3.11 Rat io of cross sec t ions of TT+ on 1 2 C and at 47.7 MeV. O p t i c a l c a l c u l a t i o n s use parameter set 2. Cent ra l curve uses best f i t value of 1 1 r + . 80 3.12 Diagram of the QQD spectrometer, showing coordinate d e f i n i t i o n s . 84 3.13 T y p i c a l d e l t a d i f f e r e n c e , DDIF, spectrum [RUN220, 50 MeV; 1 8 0 , 70°]. 3.14 T y p i c a l , angle c o r r e l a t i o n spectrum, ANGL, [RUN220, 50 MeV; 1 8 0 , 70°]. 3.15 Density p l o t i l l u s t r a t i n g the c o r r e l a t i o n between the DDIF and ANGL spec t r a of f i g u r e s 3.13 and 3.14. 3.16 T y p i c a l energy spectrum 48.3 MeV. w + on 1 2C at 80°. [RUN007] 3.17 T y p i c a l energy spectrum 62.8 MeV. i r + on 1 2 C at 120°. [RUN1110] 3.18 Nuclear l e v e l s observed i n 1 2 C , ^N, 1 6 0 , and 1 8 0 . 3.19 Peak f i t t i n g r e s u l t s from T T + 1 8 0 spectrum at 62.8 MeV, 110°. [RUN1109] 3.20 Rate c o r r e c t i o n due to m u l t i p l e pions per beam b u r s t . The range of the c o r r e c t i o n found i n these experiments i s shown as a shaded area. 3.21 T y p i c a l observed s o l i d angle of spectrometer as a f u n c t i o n of T Q. [RUN1228 , 62.8 MeV; 1 6 0 , 50°] 3.22 E l a s t i c d i f f e r e n t i a l cross s e c t i o n s of T T + i n c i d e n t on 1 2 C and 1 6 0 at 48.3 MeV. C a l c u l a t i o n s use d e n s i t i e s derived from e l e c t r o n s c a t t e r i n g (FL) and the SMC81 p o t e n t i a l . S o l i d , Set E50; broken, Set Ef50f. 3.23 E l a s t i c d i f f e r e n t i a l cross sect ions of T C + i nc iden t on 1 2 C and 1 6 0 at 62.8 MeV. C a l c u l a t i o n s use d e n s i t i e s der ived from e l e c t r o n s c a t t e r i n g (FL) and the SMC81 p o t e n t i a l . S o l i d , Set EBLE65f; long dashes, Set EC65f (Set E065f f o r 1 6 0 c a l c u l a t i o n ) ; short dashes, Set E65; and d o t s , Set EBLE65co. 3.24 Rat ios of e l a s t i c cross sect ions fo r TT+ at 48.3 MeV: lkc/12a. C a l c u l a t i o n uses Set Ef50f i n an SMC81 p o t e n t i a l and a FL parameter isat ion of a model independent e l e c t r o n sca t te r ing density for 1 2 C . The l t f N proton dens i ty i s a best f i t FL form ( c f . f i g u r e 4 .15 ) . 3.25 Rat ios of e l a s t i c cross sect ions for T T + at 48.3 MeV: 16a/lka. C a l c u l a t i o n uses Set Ef50f i n an SMC81 p o t e n t i a l and FL parameter isat ions of the model independent e l e c t r o n sca t te r ing dens i t i es of 1 6 0 and 1 2 C . The 1**N proton densi ty i s a best f i t FL form ( c f . f i g u r e 4 .16 ) . 3.26 Rat ios of e l a s t i c cross sect ions for it + at 48.3 MeV: 1 8 o 7 1 6 a . C a l c u l a t i o n uses Set EIM50 i n an SMC81 p o t e n t i a l and a MG form for the 1 6 0 matter d e n s i t i e s . The 1 8 0 proton densi ty i s a best f i t FL form ( c f . f i g u r e 4 .13) . 3.27 Ratios of e l a s t i c cross sect ions fo r TT+ at 62.8 MeV: lka/l2o. C a l c u l a t i o n uses Set EC65f i n an SMC81 p o t e n t i a l and a FL parameter isat ion of a model independent e l e c t r o n s c a t t e r i n g densi ty for 1 2 C . The l l *N proton densi ty i s a best f i t FL form ( c f . f i g u r e 4 .17 ) . 3.28 Rat ios of e l a s t i c cross sect ions for T T + at 62.8 MeV: 1 6 a / 1 4 t a . C a l c u l a t i o n uses Set E065f i n an SMC81 p o t e n t i a l and FL parameter isat ions of the model independent e l e c t r o n s c a t t e r i n g d e n s i t i e s of 1 6 0 and 1 2 C . The 1 1 + N proton densi ty i s a best f i t FL form ( c f . f i g u r e 4 .18 ) . 3.29 Rat ios of e l a s t i c cross sect ions fo r T T + at 62.8 MeV: 1 8 a / 1 6 a . C a l c u l a t i o n uses Set EIM65 i n an SMC81 p o t e n t i a l and a MG form for the 1 6 0 matter d e n s i t i e s . The 1 8 0 proton dens i ty i s a best f i t FL form ( c f . f igure 4 .14) . 4.1 The 40 MeV Los Alamos T T + , 1 2 C e l a s t i c cross s e c t i o n data [BLE79]. O p t i c a l c a l c u l a t i o n s use parameter set 1 (dashed curve) and 2 ( s o l i d curve ) . 4.2 The 49.9 MeV Los Alamos T T + , 1 2 C e l a s t i c cross s e c t i o n data [MOI78]. O p t i c a l c a l c u l a t i o n s use parameter set 1 (dashed curve) and 2 ( s o l i d curve ) . 4.3 O p t i c a l model c a l c u l a t i o n s (parameter set 2) contrast the natures of charged pion sca t te r ing from 1 2 C at 43.1 MeV. Character of Coulomb-nuclear in te r fe rence d i s t i n g u i s h e s pion charge s t a t e s . 4.4 O p t i c a l model c a l c u l a t i o n s (parameter set 2) of s c a t t e r i n g of p o s i t i v e pions on 1 2 C (dashed curve) and U B at 43.1 MeV. Not ice the r e l a t i v e magnitudes and the s h i f t i n the minimum. 4.5 Rat io of cross sec t ions i n f igure 4 .4 . 4.6 x values generated from equation 4.24. Solid curves are f i t s to equation 4.25. Dashed lines indicate unity increases in x2 from values at the respective minima. 4.7 Contour plot for MGCV density at 38.6 MeV. RAD is llr+ and RCOP is the core radius. 4.8 Contour plot for MGCV density at 47.7 Mev. RAD i s x lr+ and RCOP is the core radius. 4.9 Contour plot for Re(c Q) at 38.6 MeV. RAD is llr+ and RECS i s Re(c Q). 4.10 Contour plot for Re(c Q) at 47.7 MeV. RAD is n r + and RECS is Re(c 0). 4.11 Elastic differential cross sections of TT+ on 1 60 at 62.8 MeV and 48.3 MeV showing SMC81 potential calculations with parameter sets E (E50 and E65; broken curves) and the fitte d sets EIM (EIM50 and EIM65). Matter densities are of the MG form. 4.12 Elastic differential cross sections at 65 MeV for Tf + on 1 2 c as measured by Blecher et a l . Calculations with various parameter sets are shown: Solid, Set EBLE65F; Long dashed, Set EC65F; Short dashed, Set E65; Dotted, Set EBLE65co. Matter densities are of the FL form. x v i i 4.13 Proton matter dens i ty d i f f e rences ( 1 8 p p ( r ) - 1 6 p p ( r ) ) der ived from 48.3 MeV TT+ r a t i o s ( c f . f i g u r e 3 .26) . (a) V a r i a t i o n i n der ived Ap(r) with +10% v a r i a t i o n s (except as ind ica ted ) i n o p t i c a l parameters from Set EIM50 va lues: Broken curve (+5%), Imt>0; Dotted curve , ImB 0; i ) , Reb Q ; i i i ) , Imc Q; i v ) , ImC 0; i i ) , those remaining. (b) Best f i t FL dens i ty wi th Set EIM50 o p t i c a l parameters. E r r o r envelope inc ludes completeness e r r o r . E l e c t r o n s c a t t e r i n g matter d e n s i t i e s are shown f o r comparison. 177 4.14 Same as f i g u r e 4.13 at 62.8 MeV ( c f . f i g u r e 3 .29) . (a) V a r i a t i o n i n der ived Ap(r) with +10% v a r i a t i o n s i n o p t i c a l parameters from Set E65 va lues : i ) , ImBQ; i i ) , Imc Q; i v ) , X; i i i ) , those remaining, (b) Best f i t FB densi ty with Set E65 o p t i c a l parameters. E r r o r envelope inc ludes completeness e r r o r . E l e c t r o n s c a t t e r i n g matter d e n s i t i e s are shown fo r comparison. 178 4.15 Proton matter dens i ty d i f f e rences ( l l*Pp ( r)- 1 2Pp ( r ) ) der ived from 48.3 MeV i r + , 1 ' T N / 1 2 C r a t i o s ( c f . f i g u r e 3 .24) . Best f i t FL densi ty with Set Ef50f o p t i c a l parameters. E r r o r envelope i s s t a t i s t i c a l . Dashed curve i s d i f f e r e n c e of model independent e l e c t r o n s c a t t e r i n g der ived proton matter dens i ty i n 1 2 C and the best a v a i l a b l e MG densi ty for 1 I +N. Also shown are SCSPP c a l c u l a t i o n s with a ' s tandard ' nuclear p o t e n t i a l developed i n the  k0Ca reg ion [HOD85]. 183 4.16 Same as f i g u r e 4.15 f o r ( l l * p p ( r ) - 1 2 p p ( r ) ) der ived from 48.3 MeV T T + , 1 6 0 / 1 u N r a t i o s with Set EF50f ( c f . f i g u r e 3 .25) . 184 x v i i i 4.17 Same as f i g u r e 4.15 f o r ( l l t p p ( r ) - 1 2 P p ( r ) der ived from 62.8 MeV TT +, 1 1 +N/ 1 2C r a t i o s with Set EC65f ( c f . f i g u r e 3.27) . 185 4.18 Same as f i g u r e 4.15 f o r ( 1 * * P p ( r ) - 1 2 p p ( r ) ) der ived from 48.3 MeV T f + , 1 6 0 / l l + N r a t i o s with Set E065f ( c f . f i g u r e 3 .28) . 186 4.19 Rat io c a l c u l a t i o n (SMC81, Set Ef50f) with the best f i t densi ty ( c f . f i g u r e 4.15) and a MG densi ty ( s o l i d ) with the same RMS r a d i u s . In t h i s case use of s o l e l y the MG form would lead to a ser ious overest imat ion of the 1 1 +N RMS r a d i u s . 189 A2.1 Apparatus fo r determining the ^ B target mass th ickness by measurement of e l e c t r o n mul t ip le s c a t t e r i n g e f f e c t s . 217 A3.1 D e f i n i t i o n of the kinematic observables in the LABoratory (LAB) frame, and the Center of Momentum frame (CM). 220 A3.2 A p e n c i l beam of pions inc ident upon a plane c i r c u l a r detector perpendicular to the beam. 222 A3.3 CM angle cosines corresponding to pions of LAB energy 43 MeV decaying i n t o a cone with a vertex angle equal to one h a l f of the muon cone angle . Decays i n s i d e the shaded regions enter t h i s cone. 223 A4.1 Various dens i ty d i s t r i b u t i o n s P+(r) evaluated f o r f i v e nucleons with < r 2 > 1 / 2 = 2.25 fm. 227 x ix A4.2 QD (dashed) and < r 2 > 1 / 2 as c a l c u l a t e d for the Fermi d i s t r i b u t i o n as a f u n c t i o n of c / t . 232 A6.1 Spectrometer c o e f f i c i e n t determination a lgor i thm. 240 A6.2 Improvement of QQD r e s o l u t i o n : 48.3 MeV Energy Spectrum, a) With t ransport c o e f f i c i e n t s and no muon c u t s , b) With opt imised c o e f f i c i e n t s and no muon c u t s , c) With optimised c o e f f i c i e n t s and muon cuts on DDIF and ANGL. 241 A7.1 a) Covariance supermatrix constructed i n peak, f i t t i n g to the energy spect ra from the QQD. b) The c o r r e l a t i o n between GROUP 1 v a r i a b l e s k and I, c r ^ , i s wr i t ten ' k £ ' i n the matr ix . The E ' s are the usual e r ro r matr ices with f ree v a r i a b l e s u , h , and f . A7.2 Reduction of supermatrix for a 'group' with two 'subgroups' corresponding to s c a t t e r i n g from the same angle and t a r g e t . A8.1 a) and b) The f i r s t s i x (zero norm) FL d e n s i t i e s . The t o t a l number of l o c a l minima and maxima on a given curve i s the order of the d e n s i t y , c) and d) Dens i t i es of a) and b) fo lded with the proton form f a c t o r . [a 0 normal isat ion] A9.1 1 6 0 and 1 2 C charge d e n s i t i e s , a) and b) The s o l i d curves are fo lded modif ied Gaussian d e n s i t i e s and the broken curves are the model independent d e n s i t i e s of [NOR82] and [CAR80]. c) and d) FL components represent ing the d i f f e r e n c e s between the d i s t r i b u t i o n s i n a) and b ) . [a 0 normal isat ion] 271 250 252 262 XX A9.2 Proton matter ( s o l i d ) and charge (short dash) dens i ty d i f f e r e n c e s between 18>1(>o. The unfo ld ing was c a r r i e d out as descr ibed i n sec t ion A9 .6 . A lso shown i s the charge dens i ty d i f f e r e n c e with MG forms with the RMS charge r a d i i of 1 8 0 and 1 6 0 ( long dash) . The dotted curve i s the charge densi ty from a c a l c u l a t i o n of Brown et a l . [BR079a, BR079b]. 272 A9.3 Matter d e n s i t i e s (a and b) and charge d e n s i t i e s (c and d) corresponding to the f i r s t 6 FL d e n s i t i e s , constra ined to zero RMS i n t e g r a l and zero norm. [a 0 normal isat ion] 273 < List of Tables 2.1 Determination of Beam Energies at the Target Center 2.2 Iso top ic Composition of Scat ter ing Targets 2.3 CNO2 Experiment Target Summary 2.4 Isotopic Composition of Target Ma te r ia l 3.1 Meaning of Symbols used i n the C a l c u l a t i o n of E l a s t i c D i f f e r e n t i a l Sca t te r ing Cross Sect ions 3.2 Angle Averaging and CM Transformations for Pions at 43.1 MeV. Angles Weighted by TT+ Cross Sect ions are Denoted by CM"1" ^"^ 3.3 Symbols used i n Pion Decay Cor rec t ion 3.4 S o l i d Angle Data f o r Scat te r ing Telescopes 3.5 Quant i t ies Involved i n C a l c u l a t i o n of Rat io C o r r e c t i o n Factors 3.6 E l a s t i c D i f f e r e n t i a l Cross Sect ions for TT+ on 1 2 C f o r Lab Energy 38.6 MeV 3.7 E l a s t i c D i f f e r e n t i a l Cross Sect ions for u + on 1 2 C f o r Lab Energy 47.7 MeV 3.8 Rat io of TT+ E l a s t i c Cross Sect ion for 7t + on 1 2 C to that on n B at Lab Energy 38.6 MeV xx i 22 38 46 48 57 59 60 62 69 72 73 75 X X I I 3.9 Ratio of n + Elastic Cross Section for T T + on 1 2C to that on n B at Lab Energy 47.7 MeV 76 3.10 Correction Factors for Ratios at Lab Energy 38.6 MeV 77 3.11 Correction Factors for Ratios at Lab Energy 47.7 MeV 78 3.12 Values of Target Traceback Coefficients Defined in Equation 3.18 85 3.13 Ratio of Inelastic (4.44 MeV) Cross Sections to Elastic for Tf+ on 1 2C at 48.3 MeV 98 3.14 Ratio of Inelastic (4.44 MeV) Cross Sections to Elastic for TT+ on 1 2C at 62.8 MeV 99 3.15 Meaning of Symbols used in the Calculation of Elastic Differential Scattering of T T + on 1 2C at 48.3 MeV and 62.8 MeV 100 3.16 Differential Cross Sections for Elastic Scattering of T f + on 1 2C at 48.3 MeV 110 3.17 Differential Cross Sections for Elastic Scattering of TT+ on 11+N at 48.3 MeV 111 3.18 Differential Cross Sections for Elastic Scattering of T f + on 1 60 at 48.3 MeV 112 3.19 Differential Cross Sections for Elastic Scattering of T f + on 1 80 at 48.3 MeV 113 3.20 Differential Cross Sections for Elastic Scattering of T f + on 1 2C at 62.8 MeV 114 3.21 D i f f e r e n t i a l Cross Sections f o r E l a s t i c Scattering of TT + on 1 1 + N at 62.8 MeV 3.22 D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of i r + on 1 6 0 at 62.8 MeV 3.23 D i f f e r e n t i a l Cross Sections for E l a s t i c Scattering of TT + on 1 8 0 at 62.8 MeV 3.24 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c S c a t t e r i n g of TT+ on 1 4 f N and 1 2 C at 48.3 MeV 3.25 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of TT+ on 1 6 0 and 1 2 C at 48.3 MeV 3.26 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of Tr+ on 1 6 0 and 1 4 N at 48.3 MeV 3.27 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of TT+ on 1 8 0 and 1 6 0 at 48.3 MeV 3.28 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of TT + on 1 H N and 1 2 C at 62.8 MeV 3.29 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c Scattering of TT+ on 1 6 0 and 1 2 C at 62.8 MeV 3.30 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c S c a t t e r i n g of TT+ on 1 6 0 and l l *N at 62.8 MeV 3.31 D i f f e r e n t i a l Cross Section Ratios of E l a s t i c S c a t t e r i n g of TT+ on 1 8 0 and 1 6 0 at 62.8 MeV X X I V 4.1 Var ious O p t i c a l Parameter S e t s . 1 2 C , X 1 B Experiment. SMC79 P o t e n t i a l 142 4.2 RMS Matter R a d i i of n B from O p t i c a l Model C a l c u l a t i o n s . Uncer ta in t i es are Shown i n P a r e n t h e s i s . (Units are fm) 154 4.3 Resul ts of Tests fo r S e n s i t i v i t y to D e t a i l s of the Matter D i s t r i b u t i o n . 1 2 C , X 1 B Experiment. (Uni ts are fm) 156 4.4 Dependence of Measured 1 1 f on the Assumed Core Radius . (Units are am per percent change) 159 4.5 S e n s i t i v i t i e s of  l l i to O p t i c a l Parameters. (Units are am per percent change) 161 4.6 Miscel laneous Parameter Dependencies. 1 2 C , 1 1 B Experiment 164 4.7 Dev ia t ion of l x f from the P o t e n t i a l Averaged Means i n Colorado, LT and SMC79 Ana lyses . n f i s the Average over O p t i c a l Parameter Sets 0 and 2. (Units are am) 166 4.8 O p t i c a l Parameter Sets (SMC81). 1 8 0 , 1 6 0 Experiments 169 4.9 O p t i c a l Parameter Sets (SMC81). 1 2 C , 1 4 N , 1 6 0 Experiments 171 4.10 Four ie r Laguerre Parameter isat ions of Reference D e n s i t i e s 175 X X V 4.11 Proton Matter Distribution Differences: 1 80- 1 60 179 4.12 Proton Matter Distribution Differences: 1 2C, l l +N, 1 60 Experiment 188 5.1 Measured Values of 1 1 r + - 1 2 r + for n B 196 5.2 Proton Matter Distribution Differences: 1 80- 1 60 198 5.3 Proton Matter Distribution Differences: 1 2C, l l +N, 1 60 Experiment 200 A4.1 Summary of Characteristics of Matter Density Distributions 226 A4.2 Solutions to Equation A4.16 for 2 through 8 Nucleons 233 A6.1 Proliferation of Spectometer Transfer Coefficients 237 A7.1 Definition of Peak. Fitting Parameters 244 A9.1 Nucleon Form Factor Parameterisation 265 xxvi Acknowledgement s I t i s d i f f i c u l t to put pen to paper to compose these l i n e s . During the course of t h i s project many collaborators and friends have taken time to contribute expertise or encouragement towards i t s completion; I am somewhat reluctant to thank those who most r e a d i l y come to mind l e s t I neglect others deserving of mention. The project i t s e l f , of course, would not have been possible without the support of TRIUMF and NSERC. TRIUMF provides an i n t e l l e c t u a l l y nourishing environment i n which to work and study. NSERC makes t h i s p r a c t i c a l . The s t a f f at TRIUMF I thank for t h i s environment and for the p r o v i s i o n of resources (such as the AES word processing and VAX 11/780 computer systems, and much tec h n i c a l support) without which the production of t h i s thesis would have been more tedious by f a r . My thanks also to those people who helped to provide a smooth in t e r f a c e to the UBC physics department. During the course of the work, I have been p r i v i l e g e d to collaborate with many people. I thank the numerous researchers, post doctoral fellows, graduate students, technicians and summer students with whom I have had the pleasure to work. To several of these I owe p a r t i c u l a r thanks. Randy Sobie Roman Tacik, Sig Martin and Chris Wiedner contributed much through t h e i r developmental work on the QQD spectrometer. To B i l l Gyles I express my gratitude f o r h i s humour, c r e a t i v i t y and invaluable i n s i g h t . I thank also David G i l l . Dave i s a dynamic i n d i v i d u a l who has always been both interested i n and a v a i l a b l e for consultation. He continues to play a key role i n the Piscat Group's experimental endeavours. F i n a l l y , l e t me mention Byron Jennings i n gratitude for his readiness to discuss physics (nuclear or otherwise). xxv i i I have not forgotten my f r i e n d and research supervisor Dick Johnson. He i s deserving of s p e c i a l thanks. I have enjoyed the p r i v i l e g e of working with him and am g r a t e f u l f o r having had the benefit of h i s guidance and experience. Here also my thanks to the members of my thesis advisory committee f o r t h e i r time, contributions and i n t e r e s t i n t h i s p r o ject. Many personal friends deserve mention. People such as B i l l Gyles, J u l i Brosing, Wally Friesen and Roland P i e r r o t (and t h e i r respective better halves: L i z Hewetson, Keith Lecomte, Irma Friesen and Sharon P i e r r o t ) helped to maintain my perspective on l i f e . Pat B e l l i s a dear and much loved f r i e n d who along with her family ( e s p e c i a l l y Jean B e l l ) has provided the l o c a l s t a b i l i t y afforded by family r e l a t i o n s h i p s . At l a s t , l e t me express my sincere and loving thanks to my own family: Don, Linda and, i n p a r t i c u l a r , my mother and father. They have always provided encouragement and a supportive atmosphere. To them, my h e a r t f e l t love, respect and thanks. x x v i i i Through f a i t h we understand that the worlds were framed by the word of God, so that things which are seen were not made of things which do appear. Hebrews 11:3 1 Chapter I Introduction 1.1 Introduction One of the t r a d i t i o n a l i n t e r e s t s of nuclear physics i s the study of nuclear s i z e and shape. Rutherford's alpha s c a t t e r i n g paper [RUTH] of 1911 was a landmark contribution. The p o i n t - l i k e nucleus that he proposed was an innovation that explained the s t a t i s t i c s of large angle alpha s c a t t e r i n g . He remarked that: Considering the evidence as a whole, i t seems simplest to suppose that the atom contains a c e n t r a l charge d i s t r i b u t e d through a very small volume... . . . i t should be possible from a close study of the nature of the d e f l e c t i o n to form some idea of the c o n s t i t u t i o n of the atom to produce the e f f e c t s observed. This l a s t observation has formed the basis of many of the attempts at quantifying the term ' d i s t r i b u t e d ' . 1.2 Traditional Matter Probes There are two t r a d i t i o n a l methods of measuring nuclear s i z e . The f i r s t of these uses the electromagnetic probes; t h i s c l a s s i f i c a t i o n includes p a r t i c l e s such as muons and electrons. Electron s c a t t e r i n g [DON75, FRI75] i s a r e f i n e d and tested technique which u t i l i s e s w e ll understood dynamics. E l a s t i c s c a t t e r i n g of electrons y i e l d s precise information about charged 2 nuclear matter through measurement of the electromagnetic form f a c t o r , but i s c l e a r l y i n s e n s i t i v e to neutral matter. Attempts at measuring nuclear valence neutron de n s i t i e s from information gleaned from magnetic backscattering [LI70, DON73] have met with some success [SIC77, PLA79]. A major disadvantage of electron scattering as a technique i s the low mass of the p a r t i c l e . This necessitates the i n c l u s i o n of numerous tedious corrections such as those for r a d i a t i v e e f f e c t s . The muon's heavier mass redeems i t , somewhat, on t h i s point. The study of emission spectra from muonic atoms provides information about the RMS (Root M_ean Square) r a d i i of the corresponding n u c l e i . Small differences i n the r a d i i determined with these probes ( f o r example, i n the case of 1 2 C : ( < r 2 > 1 / 2 ^ - < r 2 > 1 / 2 e ) = .0129 ± .0053) [RUC82] ) are often a t t r i b u t e d to QED corrections not properly accounted f o r i n the muonic atom analyses. The second broad class of nuclear probes contains those which interact strongly with nuclear matter [TH081]. Included i n t h i s class are the protons and the alpha p a r t i c l e s . Protons have the disadvantage that they are s e n s i t i v e , i n essence, only to the t o t a l matter density (p++p_) of t h e i r s c a t t e r e r . (See appendix I for a discussion of l a b e l l i n g conventions.) Alpha p a r t i c l e s , on the other hand, absorb strongly at the nuclear surface so that they tend to be i n s e n s i t i v e to the d e t a i l of the nuclear i n t e r i o r . Furthermore, the composite nature of alphas makes them somewhat d i f f i c u l t to handle from t h e o r e t i c a l f i r s t p r i n c i p l e s . In recent years, the copious quantities of pions a v a i l a b l e from high-flux meson f a c t o r i e s have made possible the extensive use of pions (and 3 the muons ref e r r e d to e a r l i e r ) i n nuclear s i z e measurements. At energies near the A-resonance (1^=180 MeV) comparisons of TT + and i r ~ e l a s t i c s c a t t e r i n g have been used to extract proton and neutron matter r a d i i [JAN78] by using the strongly d i f f r a c t i v e nature of the cross sections. The strong absorption of pions at these energies, however, pr o h i b i t s d i r e c t probing of the nuclear i n t e r i o r . These pions cannot d i r e c t l y measure the lowest moments of the matter d i s t r i b u t i o n s . 1 .3 Low Energy Pion Scattering: The Isoratio Method. A major stumbling block i n pion-nuclear physics i s found i n the fa c t that the i n t e r a c t i o n p o t e n t i a l i s not w e l l 'understood'. One can, at best, use knowledge of pionic atoms [BAC70] and of free pion-nucleon (TTN) s c a t t e r i n g [ROW78] i n a semi-phenomenological o p t i c a l p o t e n t i a l i n the hope of describing pion-nuclear s c a t t e r i n g processes. A problem e x i s t s , though, i n the fa c t that ambiguities and unc e r t a i n t i e s i n parameters may tend to obscure the e f f e c t s of bona f i d e nuclear structure. The PISCATtering group at TRIUMF has been u t i l i z i n g a method which i s o l a t e s nuclear s i z e e f f e c t s , while avoiding many of the e f f e c t s that can a r i s e from experimental and t h e o r e t i c a l systematics. This i s done by considering the r a t i o s of cross sections, rather than the cross sections themselves ( f i g u r e 1.1). The method makes use of the low energy i s o s p i n dependence of the TTN e l a s t i c s c a t t e r i n g amplitude [ERI70]. As w i l l be seen, near can c e l l a t i o n s i n the pion-nucleon p-wave amplitude r e s u l t i n a Tr -n (T r + p ) amplitude s i g n i f i c a n t l y larger than the Tr'p ( i r + n ) amplitude. This may be understood by examining the free pion nucleon s c a t t e r i n g amplitude [ERI70]: 4 1. Alignment L o c a t i o n 7 . P a s s i n g S c i n t i l l a t o r C1 13. S c a t t e r i n g T a b l e 2 . AlIgnment Marks 8. Ion Chamber 14 . P a s s i n g Counter S1(L1)1 3. Bending Magnet B2 9. S t o p p i n g Counter NaI1 15. Beam Counter S1»S2 4 . C a b l e T r a y 10. NIM B i n S t a t i o n 16. T a r g e t L o c a t i o n 5. C a b l i n g 11 . Quadrupole 07 17 . Beam Veto (15 MeV) € . C y c l o t r o n S h i e l d i n g 12 . S c a t t e r i n g Arm ARM0 18. To C o u n t i n g Room Figure 1.1 A low energy i r + s c a t t e r i n g experiment (The 15 MeV experiment: [GIL82]) s i m i l a r i n configuration to the carbon-boron TT + r a t i o experiment described i n the text. 5 f(9) = b 0 + biCTfr) + ( c 0 + C j a t T ) ) ^ ' ) (1.1) where one neglects a small spin dependent term. irN sc a t t e r i n g lengths determine the parameters: ( i s o s c a l a r s-wave) (1.2) (isovector s-wave) (1«3) ( i s o s c a l a r p-wave) .(1.4) (isovector p-wave) (1«5) (TT+p.rr-n) (1.6) (TT-p.Tr+n) (1.7) Some of the relevant dynamical quantities are shown i n figure 1.2. 1.4. The Pion Nuclear Interaction Let us consider now, i n a h e u r i s t i c way, the o r i g i n of the form of the sc a t t e r i n g amplitudes 1.2 through 1.5 and t h e i r incorporation into our study of nuclear s i z e s . We w i l l see the manner i n which the nuclear structure and nuclear i n t e r a c t i o n are convoluted to a r r i v e at a pion-nuclear o p t i c a l p o t e n t i a l . The s t a r t i n g point i n our discussion i s of necessity the TT nucleon (TTN) i n t e r a c t i o n , as we assume the nucleon to be the basic b u i l d i n g block on which our i n t e r a c t i o n occurs. 1.4.1 The Pion Nucleon Interaction: Currents R e c a l l that i n the formulation of c l a s s i c a l perturbation theory for a charged p a r t i c l e , e, with an electromagnetic f i e l d characterised by a vector p o t e n t i a l A, the i n t e r a c t i o n i s introduced to a free system by making b Q = -0.005 fm b : = -0.13 fm c 0 = 0.64 fm 3 c : = 0.43 fm 3 where: T»T = +1 -1 Figure 1.2 Kinematic observables fo r pions s c a t t e r i n g from nuclear m a t e r i a l . The p i o n , of mass m, charge e^, i s o s p i n T , momentum nk and angular momentum I sca t te rs from a nucleus with Z protons, N neutrons, t o t a l i s o s p i n t , t o t a l sp in J . The protons and neutrons have d i s t r i b u t i o n s p+(r) and p_(r) , r e s p e c t i v e l y . 7 minimal s u b s t i t u t i o n : _p_ -> £ - eA i n the Lagrangian L=T-V, where T i s the k i n e t i c energy and V the p o t e n t i a l energy of the system. This i s just a statement of how the electromagnetic f i e l d a f f e c t s a p a r t i c l e ' s motion. Note that p_ f ° r a charged p a r t i c l e i s proportional to i t s charge current j_. Since £ appears i n T q u a d r a t i c a l l y , the i n t e r a c t i o n p o t e n t i a l V £ n t = L - L 0 has terms proportional to e(j»A) and e 2(A«A). E l a s t i c s c a t t e r i n g of an electron from a nucleon, neglecting the complications of spin, i s e s s e n t i a l l y the i n t e r a c t i o n of a non-massive p a r t i c l e ( j ) with the s t a t i c vector p o t e n t i a l A created by a massive p a r t i c l e . This A i s a c t u a l l y that of a v i r t u a l photon whose exchange mediates the e l e c t r o s t a t i c force. Note the emphasis here, that the exchanged quantum i s not the probe. Inte r a c t i o n of a free photon with a charged spinless p a r t i c l e i s known as 'Compton Sc a t t e r i n g 1 , and i s described by the term e2(Af»A-f) combined with the c l a s s i c a l second order perturbation theory term of the form (2i#A_f H J i * A i ) » w i t h A t n a t 0 1 t n e free photon. Minimal Coupling i n QED has proven to be a successful premise, so one might expect the i n t e r a c t i o n between pions and nucleons to also appear as a c u r r e n t - f i e l d i n t e r a c t i o n . Yukawa described spinless nucleon-nucleon s c a t t e r i n g as the exchange of v i r t u a l pions: an i n t e r a c t i o n that looks l i k e a "j]»A" and i s a case analogous to the e l a s t i c s c a t t e r i n g of an electron from a nucleon. Pion nucleon s c a t t e r i n g i s a process analogous to Compton sc a t t e r i n g , then, and i s a second order process. We write the pion f i e l d as <j> (ignoring the e f f e c t s of i s o s p i n ) . Since <J> i s to be written as an expansion i n terms of creation and a n n i h i l a t i o n operators and appropriate plane waves, the 8 operator form of _j should include a gradient (Ve~^^* r « v e l o c i t y « current). We acknowledge now that the nucleon has spin; the simplest appropriate current operator i s then: (£ #V) [EIS80]. In a second order i n t e r a c t i o n , we w i l l a r r i v e at a s c a t t e r i n g amplitude l i k e : (o»k)(£»k')=k«k'+!£•(kxk') where k and k' are the i n i t i a l and f i n a l pion momenta. Notice that k»k'is proportional to the cosine of the s c a t t e r i n g angle and i s hence, i n the usual terminology, a p-wave i n t e r a c t i o n . The s c a t t e r i n g amplitude i s , to f i r s t order, just the c l a s s i c a l perturbation theory matrix element: <«i'f|v(r)|'i'i> = / V(r) exp(i£«r) dr = V( a) (1.8) where V(r) i s the i n t e r a c t i o n p o t e n t i a l , ^ ± y f are the i n i t i a l and f i n a l states, and aj=k_-k'. Hence the pion nucleon s c a t t e r i n g amplitude i s just the momentum space p o t e n t i a l and has the form: f(9) = b(E) + c(E)k«k' + i d(E) (o»kxk'), (1.9) where we have added an s-wave term b(E), the i n c l u s i o n of which i s required by a proper r e l a t i v i s t i c treatment [BJ065]. To a r r i v e at a f i n a l form for the TTN s c a t t e r i n g amplitude, we must now deal with the e f f e c t s of i s o s p i n . The symmetry between proton-proton and neutron-neutron strong i n t e r a c t i o n i s incorporated into nuclear physics by the introduction of the concept of the i s o t o p i c spin of a p a r t i c l e . Protons and neutrons are considered (apart from electromagnetic n i c e t i e s ) to be i d e n t i c a l p a r t i c l e s , both with an i s o t o p i c spin 1/2 and d i f f e r i n g only i n the p r o j e c t i o n (T 3=±l/2) of that spin. S i m i l a r l y , the pion i s a p a r t i c l e of i s o s p i n 1, with three states: 9 Tf+ : T3=+1 (1.10) TT° : T 3= 0 (1.11) IT" : x 3 = - l (1.12) We know that under the strong i n t e r a c t i o n , i s o s p i n and i t s p r o j e c t i o n are conserved q u a n t i t i e s . Our s c a t t e r i n g amplitude must r e f l e c t t h i s f a c t . Using standard angular momentum cou p l i n g r u l e s [EDM74], the TTN system may have t o t a l i s o s p i n of 3/2 or 1/2; i t i s easy to show that f o r t o t a l i s o s p i n I_==T+j_/2 where j_ i s the i s o s p i n operator f o r a meson and J_ that f o r the nucleon: Ql/2 = (1-T»T)/3 (1.13) Q3 / 2 = (2+T«T)/3 (1.14) p r o j e c t out s t a t e s w i t h t o t a l i s o s p i n 1/2 and 3/2 r e s p e c t i v e l y . We may then w r i t e the i s o s p i n dependent amplitude as: f - f° -f^T«r) (1.15) where f° i s a symmetric, i s o s c a l a r part and f 1 i s antisymmetric and i s o v e c t o r . Combining t h i s w i t h equation 1.9 we a r r i v e a t : f(q)= b Q(E) + c0(E)k»k* + ( b ^ E ) + c j (E) ( kfk/ ) )T*r (1-16) + i ( d 0 + d j T ^ ^ k x k ' 1.4.2 Pion Nuclear Interaction The d i s c u s s i o n u n t i l now has considered only the i n t e r a c t i o n of a s i n g l e pion on a s i n g l e nucleon. In a nucleus ( f i g u r e 1.2) one might expect the pion to i n t e r a c t through a p o t e n t i a l which i s j u s t the sum of p o t e n t i a l s due to i n d i v i d u a l nucleons. The nucleus i s t y p i c a l l y s p i n saturated (the i n d i v i d u a l nucleon spins couple to give approximately net zero spin) so we ignore the presence of the nucleon spin function. For e l a s t i c s c a t t e r i n g the i n i t i a l and f i n a l nuclear states must be i d e n t i c a l . Including pion-nucleon t o t a l s p i n - i s o s p i n functions, X J F » i » f equation 1.8 i s written: f(6) = <¥ f| xt fV(r) X i|l' i>, (1.17) so we write the pion nuclear p o t e n t i a l as a f o l d i n g of the coordinate space density with the coordinate space nuclear p o t e n t i a l : U(r) = / I x t V (|r-r'|) p (r«) x ± dr' (1.18) n summed over the nucleon's allowable eigenvalues of T 3 ( i e : n = ± 1). Given that V(r) and p ( r ) are well behaved [MES58] th i s means that the momentum space Tv-nuclear p o t e n t i a l i s ju s t : U(q) = E X f t V(q) p (q) X ± (1.19) Tl where V(q) and p(q) are the Fourier transforms of V(r) and p ( r ) , r e s p e c t i v e l y . Now, f o r a charged pion of charge incident upon a nucleon of is o s p i n p r o j e c t i o n n/2 (n = +1 for protons and -1 for neutrons) the TTN i s o s p i n function i s : 1 _ | e n > - - { |e +n | | 3/2,3e „/2 > + |e -n | |3/2,e /2 > / /3 ' T t 2 Tt 1 Tl ' ' TT + /2" (e -n) I 1/2, e 12 > / / I } (1-20) Tf 1 TT This i s obtained from the standard Clebsch Gordan decomposition of the mixed i s o s p i n TTN states | TT 1 N > in t o i s o s p i n 3/2 and 1/2 components. 11 Performing the sum i n equation 1.19, we a r r i v e at: 1 U(q) = - ( ( £ i+ l) 2 + (e f f- 1)2/3 + 2 ^ - l) 2/3 ) p +U Q 4 1 + - ( (e f f- l ) 2 + Uv+ D 2/3 + 2U + D 2/3 ) p-U0 4 1 + - ( {e + l ) 2 + (£it- 1)2/3 - 4 ( E i i - D 2/3 ) p +U x 4 1 + - ( (e^- l ) 2 + Uv+ D 2/3 - 4( E 7 r+ 1)2/3 ) p-Uj [ U 0 (p+(q)+p-(q)) + U L (p+(q)-p-(q)) ] (1.21) where: u o = C b o + c o i i • i i , ^ » a n d U x = (bi+Cikfk') To a r r i v e at equation 1.21, the values: T«T - 1 (1=3/2) T»T_ = -2 (1=1/2) have been used. It i s customary to make the d e f i n i t i o n s : p(q) = p+(q) + P-(q), 6p(q) = p_(q) - p+(q) which gives: Consider f o r a moment the i s o s c a l a r part of t h i s expression: U 0(q) = ( b Q + c 0k«k ,)p(q). (1.22) (1.23) (1.24) (1.25) (1-26) (1.27) U(q) = ( b Q + c 0k«k')p(q) - ( b x + Cjkpk')6p(q)e i f (1.28) (1.29) Then U 0(q)=(b 0 + c 0k«k')/p(r)exp(iq»r) dr (1-30) = (b 0/exp(-ik'«£)p(r)exp(ik«r) dr; -cJexp(-ik ,»£)(V«p(r)V)exp(ik«r) d_r (1.31) In the second term, the r e s u l t of a p a r t i a l i n t e g r a t i o n , each gradient i s understood to act on a l l functions of £ to the r i g h t of i t . The configuration space p o t e n t i a l i s then obtained by removing the plane wave factors i n the integrand, so that: V Q p t ( r ) = b Q p ( r ) - c 0Vep(r)V , (1.32) for the i s o s c a l a r term. Hence incl u d i n g the e x p l i c i t isovector terms of equation 1.23 we have a r r i v e d at: V o p t = b 0 p ( r ) " c 0 V ° P ( r ) V " b l E 7 r 6 p ( r ) + c i e 7 TV«6p(r)v, (1.33) which i s the K i s s l i n g e r form [KIS55] f o r the pion nuclear o p t i c a l p o t e n t i a l . It i s t h i s p o t e n t i a l which, when modified by some second order corrections, allows the separation of the i n t e r a c t i o n (characterised by b g j C Q . b j j C j ) from the density information (p(r),6p(r)) i n the nucleus. 1.5 Matter Distribution Sensitivity of the Ratio The Born approximation cross section i s proportional to the square of the momentum space p o t e n t i a l (equations 1.17 and 1.19). Ignoring the i s o s p i n s e n s i t i v i t y of that p o t e n t i a l , f o r the moment, we can see that the r a t i o of pion e l a s t i c s c a t t e r i n g cross sections on d i f f e r e n t n u c l e i , one of which i s considered as a reference, i s : (V(q)p(q)) 2 p 2(q) R = = (1.34) ( V ( q ) p ( q ) ) 2 r e f P 2 ( q ) r e f From t h i s , one expects a r a t i o of cross sections to be i n s e n s i t i v e to the s i n g l e nucleon p o t e n t i a l d e t a i l s . The i s o s p i n d e t a i l s , though, mix p+ p_ and V, so that the f a c t o r i s a t i o n of the p o t e n t i a l i s not t o t a l . We choose ^ k - k ' , so the F o u r i e r transform of the nuclear d e n s i t y i s : p(q) = p(k-k') <* /" s i n ( q r ) q - 1 p(r) r dr 0 - /" r 2 { I ( ( " ) n / (2n+l)!)p(r) } dr 0 n=0 <* 1 - q 2<r 2>/6 + q'*<r1»>/120 - • • • (1.35) = 1 - q 2<r 2>/6 (dimensionless) (1.36) Equation 1.36 assumes, of course, that the c o r r e c t i o n terms of equation 1.35 are s m a l l : q 2 <r't> « 20 <r 2> (1.37) To o b t a i n some idea of the c o n s t r a i n t , we use a hard sphere d i s t r i b u t i o n w i t h moments: <r n> = ^ (1.38) and ra d i u s R = 1.45 A 1 / 3 (1.39) The r e s u l t i s : k = A _ 1 / 3 (1.40) or: T = 25 MeV (1.41) u Equation 1.18 gives the r e s u l t i n g c o n s t r a i n t : the s e n s i t i v i t y of the r a t i o i s exclusive to the RMS radius only at low energy (a r e s u l t which Is not s u r p r i s i n g ) . If one considers the r a t i o of the low energy e l a s t i c s c a t t e r i n g of p o s i t i v e (negative) pions from isotones (isotopes), then i t i s apparent that the r a t i o i s p r i m a r i l y dependent upon the RMS r a d i i of the proton (neutron) d i s t r i b u t i o n s . In p a r t i c u l a r , the r e l a t i v e RMS r a d i i of the d i s t r i b u t i o n s determine the r a t i o normalisation. Measuring the r a t i o then determines the r e l a t i v e s i z e . The question a r i s e s as to whether the s e n s i t i v i t y i n i t s e l f allows absolute measure of the nuclear matter r a d i i . I t i s the large degree to which the ir nuclear p o t e n t i a l f actors i n equation 1.34 that implies i n s e n s i t i v i t y to the precise nature of that p o t e n t i a l . Furthermore, given nuclear si z e information f o r the reference from some independent source, the p o t e n t i a l V(q) may be optimised to describe the reference nucleus and hence n u c l e i with nearly the same structure. Attempts to extract absolute nuclear size d i r e c t l y from the pion s c a t t e r i n g e l a s t i c cross sections would s u f f e r on both counts. 1.6 Previous Isoratio Experiments In "Neutron Radii Determinations from the Ratio of TT" E l a s t i c S cattering from 1 2C> 1 3C and 16,18 0- [JOH79,GYL79] the i s o r a t i o method has been used to determine the sizes of 1 3 C and 1 8 0 r e l a t i v e to 1 2 C and 1 6 0 re s p e c t i v e l y . The neutron radius was found to be e s s e n t i a l l y independent of p o t e n t i a l model and i n s e n s i t i v e to uncertainties i n the associated proton d i s t r i b u t i o n s . In f i g u r e 1.3 the r e s u l t s of t h i s experiment f o r the carbon case at 29 MeV are i l l u s t r a t e d , along with an o p t i c a l model c a l c u l a t i o n 15 b' CM b' O 1.9 1.8 1.7 -1.6 -1.5 1.4 1.3 1.2 I.I -1.0 -20 40 60 80 100 120 140 160 e C M . Figure 1.3 The r e s u l t s of TT~ s c a t t e r i n g experiments (29 MeV) on isotopes y i e l d r e l a t i v e neutron r a d i i . 12, a- ( 1 3o_) i s the e l a s t i c d i f f e r e n t i a l s c a t t e r i n g cross section for pions on 1 2 C ( 1 3C) [GYL79a]. [GYL79a]. The value of A r _ = 1 3 r _ - 1 2 r _ was found to be 0 . 0 4 ± 0 . 0 3 fm. Notice the c h a r a c t e r i s t i c s t r u c t u r e occur ing near 9 0 ° ; t h i s r e s u l t s from a s h i f t between isotopes of an s -p par t ia l -wave in te r fe rence minimum. More recent experiments with negative pions on isotopes of magnesium and s u l f u r [GYL85] have extended the technique in to a region where d i f f r a c t i v e e f f e c t s must be considered c a r e f u l l y i n the e x t r a c t i o n of r e l a t i v e s i z e in format ion , but where comparison with the r e s u l t s of nuclear s t ruc ture c a l c u l a t i o n s such as those using modif ied Hartree Fock techniques become more tenable . 1.7 Current Objectives In order to v e r i f y the r e l i a b i l i t y of the i s o r a t i o method i n determining nuclear s i z e , a s e r i e s of experiments were proposed to measure the r e l a t i v e proton matter s i z e s of 1 1 B , 1 ' + N , 1 8 0 r e l a t i v e to those of I 2 C and i 6 0 with p o s i t i v e p ion r a t i o s [BAR80, BAR85]. The main i n t e r e s t i n the method, of course , had o r i g i n a l l y been i n i t s a b i l i t y to measure neutron r a d i i , but i t soon became apparent that proton radius measurement was v i a b l e as w e l l . Furthermore, r e s u l t s fo r the proton d i s t r i b u t i o n s of n u c l e i are immediately comparable to the more common e lec t ron s c a t t e r i n g r e s u l t s prev ious ly mentioned. The remainder of th is thes is i s devoted to a d e t a i l e d d e s c r i p t i o n of these experiments and the i r a n a l y s i s . 1.8 Summary T h i s , the in t roductory chapter , s tar ted with an overview of the T T + r a t i o experiment, i n l i g h t of the h is to ry of nuclear s i z e measurement. The p ion-nuc lear o p t i c a l p o t e n t i a l , which i s the bas is of e x t r a c t i o n of matter 17 d i s t r i b u t i o n d i f f e r e n c e in format ion from the r a t i o of TT + e l a s t i c d i f f e r e n t i a l cross s e c t i o n s , was examined. Background f o r the v a l i d i t y of the method was d iscussed i n terms of the i s o s p i n dependence of the p o t e n t i a l and the enhanced s e n s i t i v i t y of p o s i t i v e pions to nuclear proton matter d i s t r i b u t i o n s . In Chapter I I , a d i s c u s s i o n of the experimental setups used to c o l l e c t e l a s t i c s c a t t e r i n g data on f i v e n u c l e i at a s e l e c t i o n of three energies i s g iven . Chapter III then d e t a i l s the ana lys is of the data sets and subsequently the e x t r a c t i o n and c o r r e c t i o n of r a t i o s and cross s e c t i o n s . Chapter IV fo l lows with d e s c r i p t i o n of the o p t i c a l p o t e n t i a l a n a l y s i s of the r a t i o data to ext ract r e l a t i v e nuclear densi ty in format ion . D i s c u s s i o n about p o s s i b l e weaknesses i n the analyses induced by the l i m i t a t i o n s of the o p t i c a l p o t e n t i a l i s i n c l u d e d . Chapter V provides overview and summary of these experiments. Comparison of the r e s u l t s of these experiments to those obtained through other methods i s made. Chapter II Experimental Details 2.1 Introduction The experiments included i n t h i s work f a l l n a t u r a l l y i n t o two ca t e g o r i e s : 1 2 C , n B and 1 2 C , 1 1 +N, 1 6 0 , 1 8 0 experiments. This d i v i s i o n i s suggested by d i f f e r e n c e s i n the experimental techniques and i n the times at which the experiments were performed. In t h i s chapter, we w i l l d i s c u s s the experimental d e t a i l s : apparatus, hardware and f a c i l i t i e s , which were used to c o l l e c t the data. A f t e r d e a l i n g w i t h some general d e t a i l s common to a l l of the experiments, we w i l l d e s c ribe the ^ 2C,^B experiment which was the f i r s t generation T T + r a t i o experiment. The r e s u l t s of t h i s experiment have been summarised i n [BAR80]. Fol l o w i n g t h i s , we w i l l proceed to describe the 1 2 C , 1 4 N , 1 6 0 , 1 8 0 experiments, as summarised i n [BAR85]. 2.1.1 Machine Details A l l of the measurements were performed using the low energy i r + beams a v a i l a b l e from the TRIUMF 500 MeV isochronous c y c l o t r o n [RIC63] located on the campus of The U n i v e r s i t y of B r i t i s h Columbia. P r e s e n t l y , two channels, designated M i l [STI80] and M13 [0RA81], provide s u i t a b l e beams. These channels both are loc a t e d at the T l production target on the 1A beamline i n the meson h a l l at TRIUMF. M13 provides pions of energy up to 50 MeV. This channel ( f i g u r e 2.1a) i s 9.5 meters i n length and provides f l u x e s of about 5 x 10 5 p a r t i c l e s per second ( a c h r o m a t i c a l l y tuned) f o r a momentum r e s o l u t i o n of 1% ( s e l e c t e d by >86 5 t * CALCULATED VALUE o MEASUREMENT WITH 2mm CARBON o MEASUREMENT WITH Icm CARBON 5 10 15 F l HORIZONTAL S L I T WIDTH cm Figure 2.1 a) Channel configuration and b) Momentum re s o l u t i o n of M13 [0RA81]. the s e t t i n g of momentum l i m i t i n g s l i t s at F l and F2, f i g u r e 2.1b) per 10 uA primary beam current. M i l , on the other hand, provides pions of energy greater than about 65 MeV, with fluxes dropping ra p i d l y as energy i s decreased. M i l has a forward takeoff (at an acute angle to the primary beam) while the lower energy M13 has a backward takeoff. Figure 2.2 compares the a v a i l a b l e fluxes on the two channels as a function of energy. In ad d i t i o n to pions, muons and electrons (of both charges) are produced at T l . The secondary channels select p a r t i c l e s of given momentum and charge, so that muons and electrons are also transported to the secondary target l o c a t i o n . Such p a r t i c l e s , however, have d i f f e r i n g masses, hence v e l o c i t i e s , and so require d i f f e r e n t amounts of time to t r a v e l from the production target to the secondary target l o c a t i o n . 2.2 l 2Carbon, 1 1Boron (CB) Experiment The 12C_, 11B_ (CB) experiments were performed e x c l u s i v e l y on the M13 channel. Data were taken at two energies. The f i n a l quoted energy i s the energy of pions momentum selected at the second bending element, B2, but allows for the energy l o s t i n a r r i v i n g at the target center (table 2.1). The difference i n Bl and B2 momenta i s caused by the presence of ~0.2 g/cm2 polyethylene inserted at F l to eliminate proton contamination from the beam. Time Of F l i g h t (TOF) spectra are shown i n figure 2.3. These spectra were recorded on a Lecroy 4001 qvt with a p l a s t i c s c i n t i l l a t o r located at the target p o s i t i o n and show p a r t i c l e contaminations of Tf: y : e=348:15:1 at 38.6 MeV and 2500:62:4 at 47.7 MeV. Beam p r o f i l e s f o r each channel tune, recorded with a s i n g l e M u l t i Wire Proportional Chamber (MWPC) at the target p o s i t i o n , are shown i n fig u r e 2.4. These chambers consist of anode and 10J CM E o Q. < < 10* w -i r-/ M i l J M13 J 1 i • i 5 0 100 150 2 0 0 2 5 0 3 0 0 M O M E N T U M (MeV / c ) Figure 2.2 Comparison of fluxes a v a i l a b l e from meson channels M i l and M13. Table 2.1 Determination of Beam Energies at the Target Center Data Set 50 MeV 40 MeV Bl Momentum (MeV/c) 126.90 112.91 B2 Momentum (MeV/c) 126.12 112.26 Pion KE, post B2 (MeV) 48.54 39.54 Pion KE, target center (MeV) 47.7 38.6 Uncertainty i n pion KE (target center) (MeV) ± 0.23 ± 0.15 23 I 1 = 10 n 8 38.6 MeV J u . 47.7 MeV Figure 2.3 M13 TOF spec t r a at 40 and 50 MeV. 24 o. b. c. d. •—• • I cm Figure 2.4 Typical beam profiles at 40 (a and b) and 50 (c and d) MeV, for TT+ on M13. a) and c) are horizontal profiles, b) and d) vertical. Structure on the vertical profiles is an artifact. cathode planes of wires w i t h a p o t e n t i a l of s e v e r a l k i l o v o l t s placed between them. When a p a r t i c l e passes between the planes, a gas avalanche i s created and induces s i g n a l s on a delay l i n e attached to planes of sense wires i n each d i r e c t i o n . The d i f f e r e n c e of the times of a r r i v a l of the r e s u l t i n g pulses at the ends of these l i n e s i s p r o p o r t i o n a l to the distance along that l i n e at which the event occurred. Beam divergence through these chambers was estimated to be ~2° [GYL80, ORA79]. 2.2.1 Experimental Setup A s c a t t e r i n g t a b l e (see f i g u r e s 1.1 and 2.5) was placed w i t h i t s center l o c a t e d 1.35 meters from the l a s t quadrupole (Q7). The t a b l e was a l i g n e d beneath the s c a t t e r i n g plane w i t h a t h e o d o l i t e placed ~5 meters downstream, the t h e o d o l i t e being placed w i t h the use of benchmarks on Q7. Beam f l u x e s were monitored upstream of the target by an i o n i z a t i o n chamber [SH079, SH079a, SH081] and downstream by two 1.5 mm t h i c k by 10 cm square s c i n t i l l a t i o n counters set i n c o i n c i d e n c e . 2.2.2 Scattering Telescopes Scattered pions were detected i n two three-element stopping telescopes (designated ARMO and ARM1), i d e n t i c a l i n c o n f i g u r a t i o n to one another, but s l i g h t l y d i f f e r e n t i n s o l i d angle. The f i r s t element of each telescope was a 1 mm t h i c k s c i n t i l l a t i o n passing counter (NE102) [NUC80], 5 cm In diameter, l o c a t e d 15 cm from the target a x i s . These counters, designed to l i m i t the f i e l d of view of the s c a t t e r i n g telescopes to the immediate v i c i n i t y of the t a r g e t , d i d not a f f e c t the telescope s o l i d angles. 26 2 5 C M B E A M L I N E M I 3 \ S C A T T E R I N G T A B L E Figure 2.5 Scattering table arrangement, showing IT s c a t t e r i n g telescopes and beam monitoring apparatus. The second element of each telescope was a 2 mm thick Kevex [KEV80] l i t h i u m - d r i f t e d s i l i c o n ( S i ( L i ) ) detector with an act i v e area of 1250 mm2. This detector acted as the s o l i d angle d e f i n i t i o n f o r the telescope and provided a AE s i g n a l for the scattered p a r t i c l e s . I t was capable of about 0.5% r e s o l u t i o n when used as a stopping counter f o r low energy p a r t i c l e s , but exhibited ~30% FWHM i n t h i s configuration. The f i n a l stopping element was a 3-inch diameter by 4-inch deep Harshaw sodium iodide (Nal(TZ)) c r y s t a l [HAR80]. This c r y s t a l was capable of stopping up to 90 MeV pions or 175 MeV protons. I t s phototube was connected to a preamplifier which provided an anode s i g n a l as well as a sig n a l from the t h i r d dynode. The dynode s i g n a l ( r e f e r r e d to as E) was used to provide an energy signal with better l i n e a r i t y than that a v a i l a b l e from the anode. The stopping telescopes were aligned i n t h e i r mountings using the theodolite. The S i ( L i ) s were mounted on aluminium, providing some l a t e r a l s h i e l d i n g from stray r a d i a t i o n . P l a s t i c , 20 mg/cm2 thick on either side of the c r y s t a l mount, maintained a l i g h t - f r e e environment. The Nal crystal-phototube assemblies were provided by the manufacturers with mu-metal magnetic s h i e l d i n g and .048 cm thick (.025 at the entrance window) aluminium cans. A d d i t i o n a l lead s h i e l d i n g was placed around the c r y s t a l to provide some protection from stray charged and neutral p a r t i c l e s without obstructing the entrance window. The front passing counters were wrapped i n 20 mg/cm2 p l a s t i c . They were coupled to the standard TRIUMF phototube assemblies, which provide magnetic and l i g h t s h i e l d i n g . 2 . 2 . 2 . 1 Operating Characteristics Some discussion of the operating c h a r a c t e r i s t i c s of the TT+ stopping telescopes i s u s e f u l . As noted, the S i ( L i ) passing r e s o l u t i o n i s decreased markedly over i t s stopping r e s o l u t i o n . Figure 2.6 shows peak height and re s o l u t i o n f or these counters i n the stopping mode for ARMO as obtained f o r an 2 < + 1Am alpha source. It can be seen that the detector reaches depletion well before the operating point. Contrast t h i s r e s o l u t i o n with that of e l a s t i c a l l y scattered 47.7 MeV pions from 1 2 C , about 30% FWHM (figur e 2.7). The shape of the AE spectrum hints that Landau straggling [LAN44] i n pion energy loss to the S i ( L i ) i s the cause of t h i s poor r e s o l u t i o n . In f a c t , we are well within the Landau regime [RIT61]; the expected FWHM i s about 33%, i n good agreement with our r e s u l t . Figure 2.8 shows a t y p i c a l energy spectrum for a Nal element at 47.7 MeV. Resolution i s 1.5 MeV FWHM with the channel's momentum s e l e c t i o n s l i t s set at 10.7 mm. The states suggested assume the 4.44 MeV separation between the ground (g.s.) and f i r s t excited states of carbon. We notice the presence of sc a t t e r i n g from higher e x c i t a t i o n states, a l b e i t with poorer s t a t i s t i c s . Figure 2.9 shows an attempt to extract from the 38.6 MeV data l i n e a r i t y Information for the Nals. With the exception of the points marked 56 ( i . e . : run 56), a cl e a r trend with angle can be seen. This Is probably the e f f e c t of re s i d u a l magnetic f i e l d s on the phototubes [ENG52]. (Subsequent experiments with these telescopes such as the one shown i n fig u r e 1.1 [GIL82] incorporated a .64 cm thick soft i r o n cylinder around the Nals to avoid gain changes r e s u l t i n g from fringe f i e l d s of the beam l i n e elements. These cy l i n d e r s also provided a d d i t i o n a l r a d i a t i o n s h i e l d i n g for those detectors.) The excited state gains are about 0.8% per MeV larger Figure 2.6 Peak height and r e s o l u t i o n for S i ( L i ) detectors. 30 200 A in ZD o o 200 400 600 CHANNEL NUMBER 800 Figure 2.7 Passing r e s o l u t i o n of the S i ( L i ) detectors to pions. w a n> n OQ to •a n o rt l-t g o II rt ET (D 25 a. o •1 OQ c n> 00 200 A 150 A c/> i -I 100 4 o u 50 J gg ID ID O O O ) I + I — o m •n + o I I I I CM O A E ARM I 420 440 CHANNEL NUMBER 460 480 32 : * t run 56 • • A ARM 0 (G.S.) ARM 0 (4.44 MeV) ARM I ( G.S.) ARM I (4.44 MeV) 1 1 1 1 30 60 90 120 150 180 0LAB Figure 2.9 L i n e a r i t y of Nal detectors. than the g.s. gains, i n d i c a t i n g a n o n l i n e a r i t y of that order. Scattering telescope e f f i c i e n c i e s are discussed i n chapter 3. Composite information from the stopping telescopes may be displayed i n the form of E-AE s c a t t e r p l o t s . Figure 2.10 shows a contour p l o t of the sc a t t e r p l o t seen i n f i g u r e 3.1. Although the o v e r a l l s t a t i s t i c s are not good, several features are apparent. These include proton and deuteron bands, the TT+ e l a s t i c peak, TT+ reactions i n the Nal c r y s t a l , a decay muon cone and positrons r e s u l t i n g from muon decay (there i s no TOF cut here). 2.2.3 Electronics Preamplification f o r signals from the Nal was provided i n t h e i r phototube bases. The S i ( L i ) s were used i n conjunction with Canberra 970D preamplifiers, which also provided bias through 110 r e s i s t o r s . These units provided i s o l a t e d energy and timing outputs. Ion Chamber (IC) current was d i g i t i s e d i n the experimental area. A l l signals were then passed to an adjacent counting room v i a standard 50ft (RG-58/U) BNC cables. Figure 2.11 i s a c i r c u i t diagram f or the TT+ r a t i o experiment. The fro n t end consists of three parts: ARMO, ARMl, and BEAM MONITORS. The monitor stream includes input from the SI and S2 counters, the IC, and a capa c i t i v e probe located i n the primary beam l i n e . This probe detects bursts of protons as they move to the pion production target and produces a waveform re f e r r e d to here as RF. IC counts are scaled on a CAMAC [BAR69] s c a l e r . The SI and S2 sig n a l s , placed i n discriminators, define a coincidence b i t (referred to as 1«2 or S1«S2). For each telescope, signals follow two paths. The analogue paths amplify and attenuate energy signals before reaching i n t e g r a t i n g ADCs. Time (INHIBIT) C SCALERS"") VISUAL SCALERS (ENABLE) LATCH) (8T0>) (•ENJD D D s i OR GATE AND GATE ( o . INVERSION ) ACTIVE ELEMENT TYPE C - CONST. FRACT. DISC. 0 - DELAY AMP E - FANOUT. ANALOGUE F - FANOUT, LOGIC G - GATE GENERATOR L - LOWER LEVEL DISC P - PREAMP S - SPECTROSOPY AMP T - TIMING FILTER AMP U - UPDATING DISC CAMAC MODULE O CZI2 PATTERN UNIT > 2249W ADC 0 TDC t» SCALER CAMAC INPUT CHANNELS ATTENUATOR VISUAL SCALER INPUT CURRENT INTEGRATOR 50 a TERMINATOR 36 constants are selected to provide moderate r i s e times and short t a i l s . The timing paths proceed through Timing F i l t e r Amplifiers (TFA) and then through Constant F r a c t i o n Discriminators (CFD) f o r the c r y s t a l detectors; the passing counter signals are passed d i r e c t l y into l e v e l d i s c r i m i n a t o r s . Timing signals then define one threefold coincidence b i t for each telescope. The l o g i c l e v e l s are f a s t NIM. Both telescope b i t s are given the time signature of the RF i n an a d d i t i o n a l coincidence u n i t . This coincidence i s wide enough to accept any of the beam related I T ' S , U'S, and e's. A 1*2 b i t i s time sampled at a l e v e l of ~.005% and i s also given the RF time signature. Any of these three b i t s w i l l a c t i v a t e a Master Strobe (MS) which i n h i b i t s further strobes and s c a l i n g , s t a r t s CAMAC TDCs, sends a 700 nS gate to the ADCs, and signals the computer with a LAM (Look At Me) to i n i t i a t e CAMAC reads. Timing and energy information, c o l l e c t e d by a Data General Nova based data a c q u i s i t i o n [FEE79] system, i s placed on 800 BPI magnetic tape for l a t e r a n a l y s i s . 2.2.4 Targets Targets were mounted at the table center with t h e i r normals displaced i n the s c a t t e r i n g plane from the incident beam d i r e c t i o n . The amount of angular displacement was selected to optimise the e f f e c t i v e target thickness at d i f f e r e n t angles. A technique s i m i l a r to that used i n e a r l i e r r a t i o experiments [GYL79a] was used. Three targets, n B , 1 2 C , and EMPTY (MT), were placed i n the pion beam during successive runs for each p a r t i c u l a r angle. In t h i s way, no point i n the r a t i o suffered from the e f f e c t s of long term v a r i a t i o n s i n the machine performance. The targets were contained i n i d e n t i c a l low mass styrofoam-plastic containers. The MT target provided information f o r a background subtraction. The boron target was composed of powdered natural boron. This was v e r i f i e d by an assay performed by the Chalk River Nuclear Laboratories of Atomic Energy of Canada Limited [0LI80]. As t h i s target was powder and rather large i n area, thickness measurement required s p e c i a l a t t e n t i o n . A method was devised [GYL80] to use the multiple s c a t t e r i n g properties of electrons to measure the boron target thickness as a function of p o s i t i o n . This method i s described i n some d e t a i l i n appendix I I . The resultant data was incorporated with beam p r o f i l e information to a r r i v e at a weighted, mean target thickness, of 377.6±3.0 mg/cm2. The carbon target material was a plate of reactor grade graphite with a uniform thickness 332.6±1.0 mg/cm2. This target data i s summarised i n table 2.2. 2.3 1 2 C , 1 H N , 1 6 0 , 1 8 0 (CNO 2) Experiments The experiments on 1 2 £ , 1 4 tN, 160_, 1 80_, which w i l l forthwith be referred to as the CNO2 experiments, were performed at each of two energies. Data was c o l l e c t e d on Ml3 with a 50 MeV tune. The 65 MeV measurements were made on M i l , as M13 was not at the time able to provide adequate fluxes of the higher energy pions due to l i m i t a t i o n s i n the quadrupoles. The quoted energies, 48.3 MeV and 62.8 MeV, are derived from NMR measurements of the bending f i e l d of channel dipoles, M13B1 and M11B1, but allow for the mean energy l o s t i n t r a v e l l i n g to the sca t t e r i n g target centers. Beam constituents were sampled continuously during these experiments with beam monitors BMl and BM2 (see figure 2.12) straddling the target p o s i t i o n . The contaminations were t y p i c a l l y Tr:u:e = (261:13:4) on M13 at 48.3 MeV and Tr:y:e = (154:24:17) on M i l at 62.8 MeV (fi g u r e 2.13). Proton Table 2.2 Isotopic Composition of Scattering Targets Target Isotopes % Mass (mg/cm2) Type boron U B 80.8 377.6±3.0 powder 1 0 B 19.2 carbon 1 2 C 98.89 332.6±1.0 plate 1 3 C 1.11 b) Figure 2.12 Experimental setup showing (a) spectrometer-channel configuration on M13 and (b) enlargement of spectrometer. o 9 4 to > Z LU 0 25 50 75 100 BEAM T O F (channels) [50 MeV. RUN 7] 4 I — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — r BEAM T O F (channels) [65 MeV, RUN 11] Figure 2.13 Pion TOF spectra of beams from (a) M13 at 50 MeV and (b) M il at 65 MeV. contamination i n the beam was eliminated with polyethylene absorber placed at the momentum dispersed f o c i of the beamlines. 2.3.1 The QQD Spectrometer The scattered pions were detected with the use of the QQD magnetic spectrometer placed at the end of the secondary (Mil or M13) beamline. The QQD spectrometer [SOB84] subtends a s o l i d angle of about 17 msr, and i s constructed with three magnetic elements: two quadrupoles, providing v e r t i c a l and ho r i z o n t a l focussing, and a dipole to provide momentum analysis ( f i g u r e 2.12). The spectrometer i s p h y s i c a l l y capable of spanning the angular range between 90° and -137°. A rate l i m i t a t i o n i n the front end of the spectrometer currently l i m i t s u seful angles to those greater than about 25°. At angles less than about 30° d i r e c t beam and beam rela t e d muons scatter into the entrance to the spectrometer, causing voltage breakdown and loss of e f f i c i e n c y i n the MWPCs located there. These MWPCs, designated WC1 and WC3, measure the t r a j e c t o r y of p a r t i c l e s en route, and allow ray traceback to the target plane. At the back end of the spectrometer, larger chambers WC4 and WC5 measure the e x i t t r a j e c t o r y of p a r t i c l e s e x i t i n g the device, thereby defining the p a r t i c l e s ' momenta. Counters BM1, E l , E2 and E3, when f i r i n g together, s i g n a l a sc a t t e r i n g event. Beam fluxes were measured by s c i n t i l l a t o r counters BM1 and BM2, placed i n coincidence. In addition, r e l a t i v e f l u x monitoring was provided by a two element muon counter [WAD76], yl»u.2, placed upstream of the target. This telescope was aligned above and at angle of about 9° to the beam l i n e to detect muons decaying from beam pions, and hence i n d i r e c t l y measure the incoming pion f l u x . The angle chosen i s well less than the maximum kinematically allowed for n->u decay (17.9° at 50 MeV and 15.3° at 65 MeV; see appendix III for a discussion of muon decay kinematics). L a s t l y , the f l u x was measured by a Rate Reduction Monitor (RRM) located downstream of the target. This monitor was constructed by placing disks of s c i n t i l l a t o r i n t o a p l e x i g l a s lightguide so that only 10% of the detector area was s e n s i t i v e to the passage of charged p a r t i c l e s . 2.3.2 Electronic Logic Figure 2.14 summarises the e l e c t r o n i c l o g i c used to define and record an event. There were input signals from each of the s c i n t i l l a t o r s i n f i g u r e 2.12b. The three E counters, along with a BM1 event, defined a spectrometer event. E l and E2 were large rectangular counters with light-guides and phototubes at e i t h e r end; t h i s construction increased detection e f f i c i e n c y and minimised the e f f e c t s of l i g h t attenuation on energy measurement. Signals from these counters passed through mean timers to give a precise time signature f o r each event. This was also the case with E3, which i n the 65 MeV experiment had the configuration shown. In the 50 MeV experiment, a s i n g l e p h o t o t u b e - s c i n t i l l a t o r was used i n place of the E3 shown here. The output of the El«E2«E3 coincidence was anded with BM1 to create a LAM, s t a r t TDCs and open ADC gates. LAM s i g n a l l e d the data a c q u i s i t i o n computer to i n i t i a t e an interrupt sequence. Subsequently, "stops" a r r i v e d at the CAMAC TDCs, giving event time information and MWPC p o s i t i o n a l information. The computer then read the CAMAC ADCs and TDCs. Various beam and event r e l a t e d quantities such as BM1»BM2 and yl«u2 were scaled; those sca l e r s were gated o f f during the i n t e r r u p t s e r v i c i n g process. There was a second source of LAMs. At regular i n t e r v a l s BM1»BM2 was E1R E1L E2R E2L r E3R< E3R- 12 E3L* E3L--00 -00 -co ^_ -00 -00 -00 -00 -00 Enable (Monuol) LAM ADCAOC Stort MWPC xx iv) _ <xjx iy o D E> • o Discriminator AND OR Mean Gate Logical Timer Generator not 00 CAMAC ADC pi Scalers L J (Visual and CAMAC) V CZ1? Pattern Unit O CAMAC TDC Stop Linear Output t T1 Capocitive Probe Figure 2.14 QQD spectrometer e l e c t r o n i c logics 44 allowed to generate a "beam sample" event strobe. The timing of such events, r e l a t i v e to protons a r r i v i n g at target 1AT1, was recorded to allow monitoring of the beam contaminants (u,e) i n the pion beam. The sample i n t e r v a l was selected to give adequate (not overwhelming) s t a t i s t i c s . The type of event generating the strobe was recorded as a b i t - f l a g i n a C212 b i t pattern u n i t . The data a c q u i s i t i o n system DA, [MIL84] ran on a PDP11/34 operating under RSX-llM. This system serviced the CAMAC interrupts and managed a data buffer of approximate length 8K bytes. When t h i s buffer was f i l l e d with event information a tape write followed by a CAMAC scaler read was i n i t i a t e d . The data was written at 1600 BPI i n a n t i c i p a t i o n of o f f - l i n e a n a l y s i s . On-line data monitoring and analysis was performed with MULTI [FER79], running concurrently. 2.3.3 Targets Targets were mounted on a target ladder [GYL84] placed within a scat t e r i n g chamber at the front end of the spectrometer. The target angles were chosen to minimise energy straggling e f f e c t s on the energy resolution; i n p a r t i c u l a r , the target normal angles were chosen to bisect the spectrometer measurement angles. The targets were placed i n the beam during successive runs. Measurement times were kept short and multiple runs performed, as necessary, to minimise the e f f e c t s of long term systematics on the r e s u l t s . Between runs at a given angle s e t t i n g , only the v e r t i c a l target ladder p o s i t i o n was changed, and t h i s without access to the experimental area, again avoiding the p o s s i b i l i t y of e f f e c t s due to systematic changes (eg. r e s u l t i n g from magnet hysteresis) i n the channel tune. The target c h a r a c t e r i s t i c s are summarised i n table 2.3. Each was contained i n an i d e n t i c a l frame. The target support frames were aluminum. The windows were composed of 50 urn Kapton glued under tension; t h i s provided the r i g i d i t y required to maintain uniformity across the target plane. Aluminium f o i l of thickness 12.5 ym was placed over the Kapton to prevent evaporation from the water targets and i s o t o p i c exchange i n the case of the 1 8 0 target. Figure 2.15 shows the dimensions of the target frames i n r e l a t i o n to the beamspot p r o f i l e . The 1 6 0 and 1 8 0 targets were made from water combined 1.5% by weight with agar; the r e s u l t i n g gel Is s e l f supporting. The 1 8 0 composition i s an important quantity, the value a f f e c t s the cross section r a t i o normalisation. The composition was measured independently by three laboratories to assure consistent determinations. Those compositions are given i n table 2.4 for comparison. The 1 L fN target was constructed from ammonium azide (NHtfN3) [OBE66], a white c r y s t a l l i n e powder. The 1 2 C was reactor grade plate graphite. The a c t u a l construction techniques used i n the target manufacture var i e d from target to target. Agar i s a white powder when purchased and w i l l d i s s o l v e i n water which has been heated to near b o i l i n g . As the mixture i s allowed to cool, the g e l becomes more viscous. This gel must be placed into the target frame before i t s o l i d i f i e s and while i t i s s t i l l warm enough to flow. Care must be taken, on the other hand, to not in s e r t material into a frame while too hot, l e s t window d i s t o r t i o n should occur. The water targets were manufactured by f i r s t placing windows on the frame and subsequently f i l l i n g through a small hole i n the frame. This hole 46 Table 2.3 CNO2 Experiment Target Summary Target Mass (mg/cm2) Type 18 0 361±3 (355)t Enriched H 2 1 80 1.5% agar gel 16 0 317±3 (306) Natural, demineralised d i s t i l l e d H 20. 1.5% agar gel 1<*N 249±3 (248) NH^Nj, ammonium azide. Natural i s o t o p i c content. White c r y s t a l l i n e powder. 1 2 C 333±1 Reactor grade, graphite p l a t e . Natural i s o t o p i c composition. MT = 17 I d e n t i c a l frame and windows to those of above t Numbers i n parenthesis pertain to 62.8 MeV experiment. o I 1 (i:g*l) 31IJ0dd A I E Figure 2.15 Beamspot p r o f i l e at the target frame at 65 MeV and 120 degrees (RUN1116). Contours show increases i n i n t e n s i t y of 10% of the maximum. Table 2.4 Isotopic Composition of 1 80 Target Material AECL LASL MPI H 2 1 80 95.0 95.3 93.8 H 2 1 7 0 2.57 2.8 — H 2 1 60 2.52 1.9 6.2 + .1 1.2 Methods: AECL: D i r e c t reduction with chlorine t r i p e n t o -f l u o r i d e . Gas mass spectroscopy.[CR083] LASL: Chemical E q u i l i b r a t i o n with C0 2. Gas mass spectroscopy. [CAP83] MPI: High r e s o l u t i o n mass spectroscopy of water vapour. [WIE83] was plugged with epoxy a f t e r f i l l i n g . The powder target, on the other hand, was completed window by window, the f i l l i n g being placed into the frame l i k e that of a sandwich. Pressure was applied to the material before mounting the second window to ensure uniformity i n the target density and near-closest packing. The plate graphite target was the easiest target to assemble; the material was glued by i t s edges to a frame and windows mounted subsequently. 2.4 Summary The experimental techniques used i n the TT+ r a t i o experiments have been described. These experiments were performed on two of the meson channels located at the TRIUMF f a c i l i t y . The channels provided low energy pion beams of high q u a l i t y and f l u x , and of a convenient time structure. The detection apparatus and setups have been described, along with t h e i r use. Two types of detector were used: the TT+ stopping telescope and a 3 element magnetic spectrometer. The experiments thereby covered a s i g n i f i c a n t range i n measurement technique. F i n a l l y , the design and manufacture of the s c a t t e r i n g targets has been discussed. The target q u a l i t y i s an important feature of any s c a t t e r i n g experiment, as a cross section measurement varies inversely with the target thickness. 50 Chapter III Analysis 3.1 Introduction The t h e o r e t i c a l and experimental background to the TT+ r a t i o experiments which are the focus of t h i s work have been presented. In t h i s chapter, facets of the analysis other than those concerned with the i n t e r p r e t a t i o n of the experiment are described and questions concerning corrections to the data and measurement e f f i c i e n c i e s are also dealt with. The evaluation of s c a t t e r i n g cross sections i s described as i s that of t h e i r r a t i o s . P a r t i c u l a r care Is taken i n the evaluation of the cross section r a t i o s . These are used i n chapter IV to provide a measure of the proton matter density differences between n u c l e i . As i n the l a s t chapter, the CB and CNO2 experiments are dealt with i n d i v i d u a l l y . 3.2 1 2C, 1 1B Experiment The analysis of the experimental data proceeded i n several steps. The data tapes were transported from the experimental s i t e to the Univ e r s i t y of B r i t i s h Columbia Computing Center. A f t e r t r a n s f e r r i n g the data to higher density tapes, i t was translated to a format compatible with the Amdahl based system and placed i n f i l e s stored on a d d i t i o n a l tapes. The bulk of the analysis was subsequently performed with FIOWA [HAY79] (Fortran Input Output Working Arrays), a package which allows f l e x i b l e one and two dimensional binning of data. 3.2.1 Preliminary Analysis: Software Cuts Various software cuts were applied to eliminate i n e l a s t i c events and background from the a n a l y s i s . The TOF signature of an event i s one such cut; i t allows the elim i n a t i o n of sc a t t e r i n g events associated with beam p a r t i c l e s of pion momentum but non-pion mass. This technique i s very valuable with f a s t s c i n t i l l a t o r s . The o v e r a l l beam composition for TT+ ( f i g u r e 2.3) was such that TOF cuts were non-essential. TOF cuts were used i n the ARMO analysis only. More important i n the analysis were P a r t i c l e I D e n t i f i c a t i o n (PID) cuts u t i l i s i n g the Bethe-Bloch [BET53, NOR79, HEC69] formula f o r s p e c i f i c energy l o s s . Such cuts eliminate p a r t i c l e s of mass Ml<M7r<M2 and are of the fun c t i o n a l form f(E,AE,£)=0 , where SL i s the passing counter thickness. For example, i n the regime i n which: dE -cE° (3.1) dx (that i s , p a r t i c l e energy« minimum i o n i z a t i o n energy), i t follows that: (E+AE) 1" 0 1 - E 1 " 0 1 = c(l-a)£ (3.2) In the present experiment AE did not change r a d i c a l l y across the width i n E of the e l a s t i c pion peak, so that a cut on the LHS of equation 3.2 would have offered no advantages over cutt i n g on AE alone. In pra c t i c e , then, the PID cut was i n the form of a AE cut applied through the use of computer software. A second, or E cut, was then applied by hand to the r e s u l t i n g energy (E) histograms. Care was taken that the cuts were the same for each of the targets and that there was no v a r i a t i o n i n t h e i r e f f i c i e n c y with angle. In p a r t i c u l a r , the l o c a t i o n of the cut allowed f o r kinematic d i f f e r e n c e s i n p a r t i c l e s scattered from d i f f e r e n t targets at d i f f e r e n t angles. Figures 3.1 through 3.4 show a progression of computer generated binned outputs. In fig u r e 3.1 we see an uncut E-AE s c a t t e r p l o t f o r the 47.7 MeV TT+, 1 2 C data at a lab angle of 90°. Figure 3.2 i s the accompanying AE histogram. A f t e r a AE cut, figu r e 3.3 r e s u l t s , t h i s l a s t histogram having been cut by hand to obtain the number of events i n the e l a s t i c pion peak. A AE histogram with no cuts but the f i n a l E cuts i s shown i n f i g u r e 3.4 and v e r i f i e s that the AE cut shown i n figure 3.2 eliminates few e l a s t i c a l l y scattered pions. 3.2.2 Evaluation of Absolute Cross Sections The low energy T T + , 1 2 C e l a s t i c d i f f e r e n t i a l cross sections have been measured by various groups, so an extensive data set e x i s t s at 40 MeV and 50 MeV [AJZ80, AJZ82]. The data have disagreements, p a r t i c u l a r l y at backward angles, but the general features are understood. The following expression was used to c a l c u l a t e absolute cross sections from the experimental quantities of t h i s experiment: , N . N cos* J ,£A , da r c,b mt - i / - c,b c,b /-> i\ a ( e ) = d a " ( T3— ~ T T M NQ • s AQ T . ) ( 3 - 3 ) c,b mt u ° c , b Table 3.1 gives the meanings of the symbols used, but some explanation i s necessary. An e f f e c t i v e Jacobian allowed f o r the deviation of the s c a t t e r i n g experiment from a point geometry, and folded such information i n v i a an a o o c rt O cu • rt rr (TJ H XI < O rt CO Ml CO O o H Co rr rr l-t > (0 c m r| 3 n> N> 95 i a. 00 c H rt n =r 3 o to c • 0Q I-* rr z Co • § H 1 Co H-cr O 3 CO CO 0 0Q H* i — 1 3 CD O O (X Hi (T> 3 VO rt O o O • 3 o o o o o o o o o o o o 5 o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O St - fa M fa fat U 1 • U I ss; * ft I ui in i tfl -* W I a w • Si M -w -A -S: ss: 8 s I -* w •» — --- MUM : 8 I M M * ft ft w u * M -ft M til at u W * « U >J •" !•«» • • • t u g • at fat — -» UD-iaAIIV'JI i * k fa> hi hi ft a> M *• (M — •J - fat Ot M fat fa) — M ~> k U M * * ft M U y M » -• M -> — — M -» — M U > - ~ M a ~ M M W f t M f t M -«.«-. ^ fthl9JM«IM«>MM M fal M W — -» * -J • W * M M W — - M — - i M V M -» l f « « f l M 4 M U U N " U ^ M m ft -i » I I I I U W ' < I M M U * - - -• M — M ft MWU ^ ' M U UM M W ft W) » U A « ' M » . hl<*-* -> M •> — 0tM i M M fal ft M U M M — fat M - » - • - • 0»M — - h i u M — - ^ — - - w — -t-uia -— M -* M -MtlU M — fa kJ - -. — - . _ U U A U » » a o -_ _ J - o M -JQ«I f » n u i < .> M - » - » - • M — PO - -» u u u u o ^ » • ft >l I M ft -* M — — M — — M — U U U WW M IO -* * J - — - . k i > o t t - e . w * i J w -. -» M u - > > o u t j k u u c r > w -« M -» -- —— •* «J w » w M " U u u t a ' t i --* -• M — ff» U » - » "J l» * * -» M »J U ft M ~ — U U K> M A U U 1 M M M t l fat M — - - M •» U U M U I U ^ . k • — M M — — — -* W U - kj a k> M - - - - - - W O fal * n * - M ft -* M M M fat — W — .> -. — 'J -— a u M u M M U U ' i i ft -*MW->fal-»Mfal - . . .MM — 91 » tt 9> B> IB 9> 01 -J 91 91 fat M - it fa Ul - 9l U * « e w «t fa J ot -• - - ui -• — -» fal --J <J" w u -• fal -• Ul — I. at M hi (, M tfl M • Ut fa) «t -» u -» • -» M fat Ut M M K • — • • f a t - W - — MM-> -> M W JkM^^MMM M — M — m M - m -• — M m — 58-' — A » M Is? i » fal -* - - w w IMM — -> M -> ^ M -» - . W W * * * * fal fal J ^ k • u u -» M » -> u " w ~ — ^ w SM M Ut ft m A fl W i • W i l l s • • at i • ft I M «H 1 CB Ul I (H 91 -> ft M at fa> ^ M M M 41 M ft JWfa|UI9l9iai(B fa) fai M a M *i « • MfalVftftftft *h)UI<|tfa|9ia>M l th l « 9 l * l » U I 9 l j m k at « ' m u kft»>UIUI>4fal-* u s u u m v o i K - fat at ft -j _ fflUII»-J—OIM-W O f a l f t f t f a l - h ^ * M Si 5 n -< at o — ft m 8* ho O oo~SSS:°!SS:iS3J!!S58l3?2SSS3XtS28S;Si8S;8 = 2SSS5S8S;S8SSS53S;;38JSSSSS2S3 — X X n x CM x ( £ X X CD x x X x Oft x x 0} X * X X 10 X - X X X eo x x W X X X X - x x x X X - x x x : x x x ) X X X If X X } X X X ) X X X ( X X X I X X X C X X X ) X X X x x x • x x x 10 X X V X X X X X X I X X X • X X f» X X CM X X X X X X X IP X X X 0) X X CO X X X r - x x *> X X X 0 X X X 03 X X X X X X X X X X X « X X X X X X X 0) X X X X X X X X X X X X X X X X X X X ' X X X X X X X X X X X X X X X X X X X X 01 X X X X X X X X X X X X X X X X X X X ' X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X- X X X ' X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ' X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X tgggggg-L X X 0) X X i - x x x — x x x x x x x x x n x x x x x x x x x x a t x x x x x x x x x x x x x ( P X X X X X X X X X X X X X X X X X a t x x x x x x x x x x x x x x x x x x C D X X X X X X X X X X X X X X X X X X K X X X X X X r i x x x x x x x x x x x x x x x x x x x x x x x x x x x C O X X X X X X X X X X X X X X X X X X X X X X X X X X X — x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x M T X X X X X X X X X P- X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X CM X X X r> x X X X X (0 X X X m x r* x ( X X X — X X m x x X CM X X CO X «0 X X CD X } X X X 0) X 10 X CO X X x x x X X X x x x x x x x x x x x x X X X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X X x x x X X X X X X X X X X X X X X X X CO X X X X X X X X X X X X X X X X X X X X X X X ' X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ' X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X PTST CO X X n x x ggggggggggg X X X X X X X X * X X X CD x x X X X X X X x x x x x x x x x X X X X X X X X ' X X X X X X X X X X X X (0 X X X I*- X X X x x x x x x X X X X V) X X X X X X X i n x x 01 * CM (0 ( - CM m ' pt n ( • Ci « i • Ci CM i ' CM — i ) CM i _> o ! CD 0) ( OB CO I at i 0) (0 ( 0> kf) I 0> V < 0) r> C 0) CM i a> n t CO f x i co — i I P I , 10 CD i to t (0 (0 < « m < ID Mr < (0 n t IP f l ( (0 — ( (0 O < *» ot ( m co ( m r- < cn u> < tn cn < CO n < m r> t m CM t to — ( p» I o f - I r> i p ( n co ( m M> < « o < n c * < p» — < o ( CM ( CM CO < CM f - ( CM IP I CM m ( CM ^ < CM n < CM CM < CM — < 0 « IP •» - 5 ! - 8 : - 8 8 t — r» n : =§s — ft CM = R8 O CO — — CO — CM CO — — CM — * CM — r» (0 — MT CM — CM O — m » — (0 10 — * (0 — r> CM — * (0 — m CM — (0 * m co — (0 «r — m CM * • MT O — m v — o CM — V ) O — * CO — tf> o — r> w > — — *- « u? > co i - - o = 88 r* w • W W — r» ip — O * — CM * — to — n CM — - tp — *> CM < CM »- Mf CM IP O - C M r» CD CM 0) V M O « Mf CM (P c» eo CO f - CM m CD co n UJ t- U> <P K M P - I P CD CM Mf CO O n ID <o CM O MT — 10 U? CM M O i O co • CD CO t- o — CO CM CO O - 8 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o > o o o o o o o o o 6 o 6 o 6 o o o o o o o o o o Q d 6 o o o 6 o o o o b b o o o ' n n n n n n n n n t O O 3 Z »- V) Figure 3.2 Uncut AE histogram for histogram (run28, ARM1). AE cuts applied to E i n figure 3.3 are shown. ft X n x tt x n x n x • C» X et x x » x ft X X w o n to • tt * t tt 10 Mt • « <r • Mt MT to PS Mt t CM Mt • -- M t 0 Q MJ t> CA Mt »* • M» r- r-t M l M r« W * f- — -8 CM X CM X r» x x x x x x n x x x x x x x x x x x ttXXXXXXXXXXXXXXXX ttxxxxxxxxxxxxxxxxxxxxxxxxxx X X X X X X X X X X X X X X X X X X X X X X X X X - X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X K X X X K X X X X X X X X X X X X X X X X X X X X X X X X X X r - x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X X X X X X X X X X X X X X X X X X X X K X X X X X t » X X X X X X X K X X X X X X X X X X n x x x x x x x x x x x x x x C M X X X X X X X X X X X f - X X X X X X X X X O X X x x x X X X X X X X X X X X X X X X X X X X X X X X X X X X x « X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X x « CM X X X ft X X m x r - x * 10 tt MT IP Mt t tfn Ml IP CM MT LP — - M T <P Q MT tt a M) tt CO Mt tt I -Mt tt IP MT tt co •MJ tt MT M m PI MT tt CM Mr tt — -MT tt O MT * ft M M « w MT r -Mf MT <0 tt MT MT Mt MT Mt o Mt MT CM Mt Mr — - M t Mt O n rt a MT n co ^ r> e-M O IC Mt ft tt * ft Mt Mt P ) P ) MT PI CM IP * 0) CM — CM P ) — tt p) - E g -8 tt X X P) X x x x CM X X X CM X X X CM X X X • tt X X CO X Mt CM r -Mt CM IP Mt CM tt Mt CM Mt M f l D Mt CM CM Six!' — CO — 01 — o> — 0» — tt — o > CM - M 8 S P) 0) CO P) 0) f -P) 0) IP P) 01 tt r» cn Mr P i 0) n i P> 0» CM P) 0> — - P ) ft o n co co P) « p-i PJ CO IP P> CO tt PI CO Mt PI CO P) n co CM P) CO - f t CO O ft r* ft pj r- to p> r~ f~ P> r- IP P) r» in ft f» Mt ft f". « ft f* CM - o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o * O M t t t r t u > O M r e r M t p O M r a r M U 9 0 M t B C M t t O M t t t r « t t O M t t t C M t t O M t t t C M t t O M t O ^ ft CM CM — »- o c n f t c o c o p - i p i p t t t t ^ n n c M C M * ' 0 0 f t f t t t P - ^ - i P t P t t M t M r n m c M « - * -- " C M C M C M C M C M C M C M — — — — — Figure 3.3 Energy histogram f or (run28, ARM1) a f t e r cut on AE. Pions inside the cuts are e l a s t i c a l l y scattered. to x n x x cn x x r* x x x (O X X X X X P> X X f- X P* X P> X ai ID ( OJ o c ee cn f CD CD ( CD m < CD V ( co n ( X X X X P - X X X X X X X tfixxxxxxxxx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x n x x x x x x x x x x x x x x x x x x x x x x x x x c o x x x x x x x x x x x x x x x x x x x x x x x x r - x x x x x x x x x x x x x x x x x x x x x x x n x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x W X X X X x x x x x x x x x x x x x x x x x x X X X x x x x x x x x x x x x x x x P» X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X f» X P3 X CO X -P* X X r t x P> X X X Ct X X X P> X X X P) X X X X X X X X X X X X X X X X X X X -X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X -X X X X X X X X X X X X P» X X r- x n x cn x p- O ( ID 0) < ID CO ( ID r» < ID ID < ID m ( ID *r ( ID P> ( ID W i ID ' m co ( m r- 1 tn ID < B Y in < If) * < m P I i cn r t ( — \ O i * T 0) ( P> O ) i n CD < P) V , p> p> i PI Pt < p» — ( P) o < ct cn j rt co i « r rt ' rt p> ( rt rt ( rt — ( rt O < 0) < « - CD ( — r»- < — IP < o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o < • s o « c D N ( D O « c D r t u > 0 « c o r « i D O * c o r i i p o « c o r i i D O M C D N i D O M r c D C M i P O v c o r i ( D O O M . r ) N c < - - o c ^ 0 ) « « ^ » ^ p l n l l l « ^ ^ ^ N w • • o o » 0 > » ^ ^ u » « 0 D t t n B N • • » 3 — Z U O 3 Z ^  I Figure 3.4 AE histogram as i n spectrum. This shows figur e 3.2, but cut only on the f i n a l E the AE cut of that fi g u r e to have adequa width and centering. Table 3.1 Meaning of Symbols Used In the Calculation of Elastic Differential Scattering Cross Sections SYMBOL MEANING N ( ) Counts i n () e l a s t i c peak l O IC counts f o r () run 4> Angle of target normal to beam d i r e c t i o n J()(6) E f f e c t i v e Jacobian dft /dn* for () 5 Number of IC counts per pion at target Fr a c t i o n of pions reaching ARM within cuts * F r a c t i o n of scattered pions surviving to ARM Point s o l i d angle presented by ARM da Center of Mass (CM) d i f f e r e n t i a l s o l i d angle dn* LABoratory (LAB) d i f f e r e n t i a l s o l i d angle 9 E f f e c t i v e CM sca t t e r i n g angle of ARM a(8) CM d i f f e r e n t i a l s c a t t e r i n g cross section T ( ) Target thickness i n g/cm2 for () A ( ) Atomic weight f o r () N0 Avogadro's number () One of (carbon, _boron, or empty) averaging process. The e f f e c t i v e s c a t t e r i n g angle, i n addition, was weighted by do (SMC79 [STR79, STR80] p o t e n t i a l , parameter set 0). These were computed with a Monte Carlo i n t e g r a t i o n routine and r e s u l t i n changes from the point values of generally l e s s than 0.1° for the angle and 0.001 f o r the Jacobian. Some sample c a l c u l a t i o n s are shown i n table 3.2. (At each s c a t t e r i n g angle the optimum target angle was assumed.) These corrections are small enough to be ignored; point geometry values were used for J(9) and 9. The factor £ was important i n determining an absolute normalisation f o r the cross section. The normalisation was obtained by determining the number of IC counts per p a r t i c l e passing through the S1»S2 counters ( f i g u r e 2.5) and then making appropriate decay corrections to obtain the number of pions incident on the target per IC count. We have: 5 ° r i C= — [ (1-PF)+PF[A(TGT,S1«S2)]J (3.4) PF where: 1 _ P F 0 -1 PF=( 1+ ( ) A(IC,TGT)exp(x 1/x 0) ) (3.5) and the other symbols are explained i n table 3.3. PF 0 (see for example f i g u r e 2.3) i s given i n chapter II for both energies of i n t e r e s t . The factor A allowed for the fa c t that pions decaying i n t o muons a f t e r the l a s t quadrupole had pion TOF and hence contributed to the pion f r a c t i o n i l l e g i t i m a t e l y . I t Is shown i n section A3.2 that, for a p e n c i l beam incident on a c i r c u l a r beam counter at energies such that a cone e x i s t s : Table 3.2 Angle Averaging and CM Transformations for Pions at 43.1 MeV. Angles Weighted by *•+<-) Cross Sections Are Denoted by CM**-) e (degrees) dQ/dQ* (point: J Q ) Lab CM CM+ or J 0 ( e ) J ( e ) 30.00 30.49 30.45 30.25 1.0298 1.0289 40.00 40.63 40.55 40.37 1.0263 1.0255 50.00 50.75 50.65 50.44 1.0220 1.0213 60.00 60.85 60.83 60.64 1.0170 1.0164 70.00 70.92 70.97 71.01 1.0115 1.0111 80.00 80.96 81.01 81.07 1.0057 1.0055 90.00 90.98 90.94 90.97 0.9998 0.9999 100.00 100.96 101.04 101.05 0.9939 0.9936 110.00 110.92 110.96 110.96 0.9882 0.9878 120.00 120.85 120.86 120.85 0.9829 0.9823 130.00 130.75 130.73 130.72 0.9781 0.9775 140.00 140.63 140.59 140.58 0.9740 0.9733 150.00 150.49 150.44 150.42 0.9707 0.9700 Table 3.3 Symbols Used in Pion Decay Correction SYMBOL MEANING P F 0 Observed pion f r a c t i o n at target PF Corrected pion f r a c t i o n at target x0 Decay length of pion at beam energy x l IC to target distance x 2 Target to S1»S2 distance ^0 IC counts per S1»S2 p a r t i c l e A(a,b) Fract i o n of pions at 'a' giving pion signals at 'b' H ( l - e ) ( l - x 2 ) 1 / 2 A = 1 - A / I exp ( C/x) ( : ) dx (3.6) x 2(x 2+B) where A , B , C , E are kinematics-geometry dependent and H(l-e) i s the Heaviside unit step function. This was numerically integrated. The s o l i d angles of the telescopes were determined from the usual expression: fi0=2TT[l-(l+E2)-l/2] (3.7) with 5 being the r a t i o of the S i ( L i ) a c t i v e radius to i t s distance from the target. The C counters do not enter i n t o t h i s s o l i d angle d e f i n i t i o n . Calculations have indicated that fewer than 0.05 % of the pions incident would have scattered from the target and proceeded to a S i ( L i ) without having passed through the respective C counter. The relevant s o l i d angle data i s shown i n table 3.4. The mean active area of the S i ( L i ) s was taken to be 1250 mm2[KEV80]. The r a t i o of the a c t i v e areas of the detectors at t h e i r operating points was then assumed to be proportional to the r a t i o of the squares of t h e i r p h y s i c a l r a d i i . Some f i n a l comments are necessary with regard to the observed e f f i c i e n c y of the s c a t t e r i n g telescopes to pions. We r e c a l l that for pion decay the two step chain predominates: TI -> y -> e (+v's) (3.8) A f r a c t i o n : l - e x p ( - / 0 r a n 8 e ( e c y x ) - 1 d x ) (3.9) of the pions decay while stopping [SAL72] and many of these w i l l have an Incorrect Energy Signature (IES). Figure 3.5 gives t h i s f r a c t i o n as a Table 3.4 Solid Angle Data for Scattering Telescopes Active Target Point S o l i d ARM Radius Distance Angle (cm) (cm) (msr) 0 2.07 45.0 6.64 1 1.92 45.5 5.59 0 10 20 30 40 50 60 INITIAL PION ENERGY (MeV) Figure 3.5 Calculated pion losses while stopping i n a Nal detector: s o l i d curve shows losses from reactions; broken curve shows those from pion decay. function of pion energy, assuming the form of equation 3.1 f o r the stopping power along with appropriate values of a [MEA72]. This was a small con t r i b u t i o n to the o v e r a l l i n e f f i c i e n c y , of the order of 1 %. A more important contribution resulted from the decay of the stopped pion. A 700 ns time gate width to the ADCs allowed most pions to decay only to the muon stage. As a r e s u l t , the e l a s t i c peak a c t u a l l y had an energy 4.1 MeV greater than that kinematically a l l o t t e d to the pion. Any pion-related muons which did decay, however, l e f t behind 52.3 MeV positrons, most of which would not have come to rest i n the Nal. The r e s u l t was the spread i n energy above the pion peak seen i n figure 3.1. The p r o b a b i l i t y that a pion decays only as f a r as a muon during a time gate, t, i s : X [ exp(-X i it)-exp(-X i jt)] (3.10) V TT where ^ y - 1 and X^ - 1 are the mean l i v e s of the muon and pion r e s p e c t i v e l y . Equation 3.10, portrayed g r a p h i c a l l y as a function of t i n figure 3.6, i s the r e s u l t i n g e f f i c i e n c y . The maximum occurs at t=117 ns. A gate width of 700 ns r e s u l t s i n an e f f i c i e n c y of 74%. F i n a l l y , nuclear e l a s t i c scatterings caused some of the pions to scatter out of the Nal before coming to rest, giving IESs f o r these events. The telescope geometries were designed to minimise t h i s e f f e c t . The i n e l a s t i c events which occurred i n the Nal did, however, e f f e c t i v e l y remove some events from the e l a s t i c s c a t t e r i n g peak. I f , following Richard-Serre et a l . [RIC68], we use: o. = (70/32) A 2 / 3 [E 3-12E 2+36E] mb, (3.11) Figure 3.6 Muon decay contribution to telescope e f f i c i e n c y . we f i n d that some 5% (see f i g u r e 3.5) of the pions interacted i n e l a s t i c a l l y before stopping. From the above considerations alone, even before i n e f f i c i e n c i e s i n our cut t i n g procedures, we expect telescope e f f i c i e n c i e s of less than 70%. This i s a disadvantage of these n+ telescopes. 3.2.3 Evaluation of Ratios Given the cross section data, i t i s conceptually simple to c a l c u l a t e the r a t i o s . In t h i s process, decay co r r e c t i o n factors and normalisation constants tend to f a c t o r , so that t h e i r uncertainties do not a f f e c t the u n c e r t a i n t i e s i n the r a t i o . This, i n f a c t , i s a major advantage of doing the experiment as one r a t i o experiment instead of as two separate cross section measurements. Equation 3.3 i s correct for the case of 1 2 C , where system res o l u t i o n allowed easy separation at most angles (and at least high q u a l i t y estimates at a l l angles) of the f i r s t excited states. Figure 3.7 shows, though, that the boron isotopes 1 0 J 1 1*} have low l y i n g excited states which were not completely resolvable. O p t i c a l model c a l c u l a t i o n s were used to estimate the required i n e l a s t i c contributions to the cross sections of these n u c l e i . Furthermore, the natural boron target's high 1 0 B contamination had to be allowed f o r . The net c o r r e c t i o n factors by which the experimental cross sections were m u l t i p l i e d are: ( ( l - f 1 0 ) + f 1 0 * j 1 0 ) L = — (3.12) b [ ( l + g n ) ( l - f 1 0 ) + f 1 0 * k ( 8 ) * ( l + g 1 0 ) ] and: 67 5H > CD >-Q: 2« UJ UJ H OH 3/2" 2* 5/2 * 2 + 3/2" 1/2* - I . | + 1/2" 0 + | +3 + 3/2" 0 + 1/2" 10 B 1 1 B l 2C l 3C Figure 3.7 Excited states of stable boron and carbon isotopes. 6 8 L c = t d - f n J + f ^ ^ ^ ] (3.13) with table 3.5 explaining the meaning of the symbols. The quantity k(6) was not experimentally known and hence an o p t i c a l model estimate was used. The f i n a l expression for the r a t i o i s : 1 2 o ((N / i )-(N J x J ) J T, A D _ _ c c 7 mt mt / y c b c , s b e b Equation 3.14 assumes that the normal angles of the targets are a l l rigourously the same. Differences of order e° would have lead to uncertainties of the order tan(0)*e*Tr/18O° i n the r a t i o s ; a small e f f e c t i n comparison to the s t a t i s t i c a l errors involved. S t a t i s t i c a l errors i n R and o(9) were calculated using the standard expression f o r propagation of errors [BEV69] with uncorrelated v a r i a b l e s X j . For: y=f(x 1,x 2,...,x n ), (3.15) n (6y) 2= I (6x .) 2(3y/8x.) 2 (3.16) j - l 1 where 6XJ i s the uncertainty i n X j . 3.2.4 Results The experimental e l a s t i c d i f f e r e n t i a l cross sections for p o s i t i v e pions on 1 2 C are shown i n figures 3.8 and 3.9 and tabulated i n tables 3.6 and 3.7. The kinematic transformations to CM were performed using standard r e l a t i v i s t i c formulae [BAL61] averaged over target nuclei mass. The errors Table 3.5 Quantities involved in calculation of Ratio Correction Factors SYMBOL MEANING f13 Isotopic abundance of 1 3 C *10 Isotopic abundance of 1 0 B 8 l l Ratio of d i f f e r e n t i a l cross sections, 11 B* «10 Ratio of d i f f e r e n t i a l cross sections, i o B * /IOB k(9) Ratio of d i f f e r e n t i a l cross sections, 1 0 B / U B J10 Ratio of atomic masses, 1°B/HB J13 Ratio of atomic masses, 1 3 C / 1 2 C Lb Correction factor f o r boron Lc Correction factor f o r carbon h(6) Ratio of corre c t i o n factors (Lc/Lb) 70 Figure 3.8 *C e l a s t i c d i f f e r e n t i a l cross section at 38.6 MeV. Curves are from o p t i c a l model (SMC79) ca l c u l a t i o n s with parameter sets 1 (broken) and l a ( s o l i d ) . 71 Figure 3.9 2 C e l a s t i c d i f f e r e n t i a l cross section at 47.7 MeV. Curves are from o p t i c a l model (SMC79) c a l c u l a t i o n s with parameter sets 1 (broken) and l b ( s o l i d ) . Table 3.6 Elastic Differential Cross Section for w+ on 1 2C for Lab Energy 38.6 MeV e (deg cm) 0(6) (mb/sr) 6 (mb/sr) 30.5 4.88 0.45 45.6 3.49 0.14 60.8 2.62 0.24 75.9 3.29 0.09 90.9 5.17 0.12 105.9 6.73 0.19 120.8 7.75 0.28 135.6 8.42 0.21 150.4 7.66 0.26 Table 3.7 Elastic Differential Cross Section for ir + on 1 2C at Lab Energy 47.7 MeV e (deg cm) o-(e) (mb/sr) 6 (mb/sr) 30.5 5.53 0.24 40.6 5.13 0.24 50.7 3.21 0.11 60.8 2.78 0.10 70.9 3.23 0.19 80.9 4.22 0.13 91.0 5.91 0.35 100.9 7.22 0.36 110.9 7.95 0.27 120.8 8.34 0.29 130.7 7.28 0.41 140.6 7.44 0.24 150.4 7.45 0.67 are purely s t a t i s t i c a l except at backward angles, where they r e f l e c t u n certainties i n the 1 2 C * (4.44 MeV) contribution. The cross sections dip below the t h e o r e t i c a l c a l c u l a t i o n s at forward angles, a problem possibly related to higher counting rates near the muon cone. This cone i s at 20.8° for 38.6 MeV pions and 18.4° for 47.7 MeV pions. The backward angle measurements are much larger than previous measurements have Indicated to be the case. The absolute normalisation i s correct to within about 10%. Table 3.8 gives the experimental r a t i o of 1 2 C / l i B e l a s t i c cross section (assuming i s o t o p i c a l l y pure 1 1 B ) , the correction factor h(9), and the corrected r a t i o at 38.6 MeV. The r e s u l t s at 47.7 MeV are tabulated i n table 3.9 The i n e l a s t i c contributions g 1 Q and g l x (see tables 3.10 and 3.11) were calculated f or the excited states at 0.72 MeV and 2.14 MeV res p e c t i v e l y with the coupled channel [EIS74, EIS76] code CHOPIN [R0S79]. The value of the deformation parameter, B 2 , was taken as 0.67 [SWI76] for 1 0 B and 0.25 [KAR69, ASP74] for n B . The factor k(6) i s a strong function of 9 and i s dependent upon the value of the 1 0 B RMS matter radius ( 1 0 r + ) . The corrected r a t i o s use the best a v a i l a b l e experimental value of 2.43 fm [BEE79, 0LI81] for the 1 0 B charge radius, 1 0 r 0 ; h(9) i s calculated here with the SMC p o t e n t i a l , using parameter set 2 (see table 4.1) and values of 1 1 r + r e s u l t i n g from best f i t s to the r a t i o data at the respective energies. We note that for f i t t i n g purposes, h(9) must be allowed to vary with the f i t t i n g parameter, 1 1 r + . The r a t i o s are plotted i n figures 3.10 and 3.11 along with several t h e o r e t i c a l c a l c u l a t i o n s . Table 3.8 Ratio of T T + Elastic Cross Section on 1 2C to that on 1 1B at Lab Energy 38.6 MeV 6 Uncorrected Uncorrected h(6) Corrected Corrected (degrees) Ratio Error Ratio Error 45.6 1.353 0.068 0.983 1.329 0.067 60.8 1.596 0.077 1.000 1.595 0.077 75.9 1.648 0.066 1.034 1.705 0.068 90.9 1.479 0.062 1.036 1.533 0.064 105.9 1.398 0.108 1.032 1.443 0.111 120.8 1.096 0.080 1.030 1.129 0.082 135.6 1.138 0.073 1.029 1.171 0.075 150.5 1.038 0.071 1.028 1.068 0.073 Table 3.9 Ratio of * + Elastic Cross Section on 1 2C to that on nB at Lab Energy 47.7 MeV e Uncorrected Uncorrected h(9) Corrected Corrected (degrees) Ratio Error Ratio Error 30.5 1.083 0.062 0.985 1.067 0.061 40.6 1.134 0.049 0.986 1.118 0.048 50.7 1.115 0.060 0.982 1.094 0.059 60.8 1.299 0.068 0.985 1.279 0.067 70.9 1.583 0.076 1.009 1.597 0.076 80.9 1.426 0.068 1.022 1.458 0.069 91.0 1.432 0.039 1.022 1.464 0.040 100.9 1.400 0.063 1.021 1.429 0.064 120.8 1.232 0.034 1.017 1.253 0.034 130.7 1.024 0.042 1.016 1.040 0.042 Table 3.10 Correction Factors for Ratios at Lab Energy 38.6 MeV e (degrees) k(6) (1+8 1 0) (l+g n> 45.6 0.817 1.0060 1.0015 60.8 0.902 1.0090 1.0024 75.9 1.078 1.0080 1.0024 90.9 1.089 1.0060 1.0018 105.9 1.070 1.0060 1.0019 120.8 1.058 1.0070 1.0023 135.6 1.051 1.0090 1.0029 150.5 1.045 1.0110 1.0035 Table 3.11 Correction Factors for Ratios at Lab Energy 47.7 MeV e (degrees) k(6) (1+810> d + g n ) 30.5 0.831 1.0030 1.0005 40.6 0.836 1.0040 1.0006 50.7 0.811 1.0060 1.0008 60.8 0.826 1.0080 1.0012 70.9 0.948 1.0080 1.0015 80.9 1.015 1.0070 1.0016 91.0 1.019 1.0060 1.0015 100.9 1.010 1.0070 1.0018 120.8 0.988 1.0100 1.0030 130.7 0.981 1.0130 1.0038 150.5 0.966 1.0180 1.0056 T 1 r 80 100 120 B ( C M . ) r 140 T 160 Figure 3.10 Ratio of cross sections of i r + on 1 2 C and 1 X B at 38.6 MeV. O p t i c a l c a l c u l a t i o n s use parameter set 2. Central curve uses best f i t value of 1 1 r + . 1.8 T-^47.7 MeV C r 4 > 2288 2.238 2 188 i i 20 40 60 T 80 — i — 100 —I 120 - I — 140 160 6 (CM.) Figure 3.11 Ratio of cross sections of TT+ on 1 2 C and 13-B at 47.7 MeV. O p t i c a l c a l c u l a t i o n s use parameter set 2. Central curve uses best f i t value of 1 1 r + . 3.3 CNO2 Experiments The data analysis was performed at TRIUMF with a VAX 11/780 operating under VAX/VMS. The data tapes were scanned with MOLLI (Multi £ffline In t e r a c t i v e A n a l y s i s , an i n t e r a c t i v e version of FIOWA) [BEN83] to remove random time background. Only events with the C212 spectrometer f l a g set and a l l four MWPC's f i r i n g were accepted. Information from these events was then written to summary ARRAY f i l e s . The i n e f f i c i e n c i e s of the spectrometer MWPC's were calculated f o r events passing through the ce n t r a l region of the spectrometer's e x i t port from the number of events with only three of four MWPC's f i r i n g . As an example, the i n e f f i c i e n c y , EF2, i n WC2, was : N(WClFeWC2F»WC3FeWC4F) 1 ? N(WC1F«WC3F«WC4F) ' U - l / ; where WC(i)F i s true i f WC(i) f i r e d and N(x) i s the number of events with condition "x" true. The uncertainty i n t h i s i n e f f i c i e n c y i s just the square root of the numerator divided by the denominator i n equation 3.17. These i n e f f i c i e n c i e s , along with a beam contamination spectrum and the accumulated sc a l e r values f o r the run were written at the end of each ARRAY f i l e . 3.3.1 Momentum Spectrum Optimisation The combined e f f e c t s of f i n i t e s i z e of the T l production target, secondary channel achromaticities, energy st r a g g l i n g i n the sca t t e r i n g targets and o p t i c a l imperfections i n the QQD spectrometer l i m i t e d the re s o l u t i o n attained i n these experiments to about 1.6 MeV FWHM. This r e s o l u t i o n was obtained a f t e r extensive e f f o r t was applied at optimising a set of "transfer c o e f f i c i e n t s " which describe the transport of charged p a r t i c l e s through the spectrometer. A computer code was developed (QQD Optimisation of Parameters or QQDMP) to perform this optimisation and owes much conceptually to a less sophisticated code due to W. Gyles [GYL84]. The code has since become a de facto standard [TAC84,R0Z84,WIE84] for others wishing a convenient parametrisation/optimisation scheme for determining spectrometer transfer c o e f f i c i e n t s . D e t a i l s of the procedure are found i n appendix VI; here i t s u f f i c e s to say that the momentum deviation of an event, <5p, from the spectrometer c e n t r a l momentum i s derived v i a the transfer c o e f f i c i e n t s , and that the full-acceptance r e s o l u t i o n of the spectrometer depends upon t h e i r optimisation. 3.3.2 Software Cuts The ARRAY f i l e s were analysed with i n t e r a c t i v e routine QQDANA (QQD  Analysis) which i s a modified version of QQDMP suitable for analysis of complete array f i l e s . QQDANA contains code for displaying data, peak f i t t i n g and i n t e g r a t i n g , as well as f o r setting software cuts (high and low l i m i t s of a v a r i a b l e between which data Is accepted). A number of cuts were placed on relevant quantities i n t h i s a n a l y s i s . 3.3.2.1 NMR Cuts Dipole f i e l d s were ensured to be stable by requiring NMR readings of the B l , B2 and BT (see f i g u r e 2.12a) f i e l d s to vary by less than .25%. In most cases, t h i s constraint did not come into play. By ensuring t h i s s t a b i l i t y , the r e s o l u t i o n of the apparatus was not affected by the spread i n the e x i t chamber coordinates that would have otherwise occurred. 3.3.2.2 Target Traceback Cuts Target cuts eliminated pions seen to o r i g i n a t e from the v i c i n i t y of the target frame. The target p r o j e c t i o n coordinates X Q and Y Q and the angles 6 0 and cj>0 (projections of s c a t t e r i n g angle on the X Q-Z and Y Q-Z planes r e s p e c t i v e l y , f i g u r e 3.12; Z i s the d i r e c t i o n of the spectrometer axis) are derived from the f i r s t order c o e f f i c i e n t s r e l a t i n g them to the values of XI, Y l , X3 and Y3: XI 1 SL 0 0 X 0 X3 A00 A 0 1 0 0 6 0 Y l 0 0 1 £ Y 0 Y3 0 0 B00 B 0 1 • o (3.18) here £=211/1000. S p a t i a l coordinates are i n mm and angular coordinates are i n mrad. Values of a^j and bj_j are given i n table 3.12 The p r o j e c t i o n coordinates are rel a t e d to the l o c a t i o n on the target (^tar> Y t a r ) a t which the s c a t t e r i n g took place. To f i r s t order, Y t a r " Y 0 (3.19) (3.20) X t a r = X o / c o s ( * > where i)r i s the acute angle between the target normal and the spectrometer a x i s . More p r e c i s e l y , however: X = i tar s X Q COS0Q cos(6 0+e i) , and X Q tancfig C O S 9Q sine Y = Y n + = , where cos(e 0+e i) e i - 9 t g t ) + e s P e c - f ] (3.21) (3.22) (3.23) Figure 3.12 Diagram of the QQD spectrometer, showing coordinate d e f i n i t i o n s . Table 3.12 Values of Target Traceback Coefficients defined i n Equation 3.18 C o e f f i c i e n t Energy (MeV) Value Uni ts a0 0 50 1.4 mm/mm a0 1 50 1.0 mm/mrad b0 0 50 .79 mm/ram b01 50 .61 mm/mrad a0 0 65 1.6 mm/mm a01 65 1.2 mm/mrad b0 0 65 .60 mm/m m b01 65 .60 mm/mrad 8 S p e c i s the angle between the spectrometer axis and the beam ( p o s i t i v e angles are taken to be to the r i g h t as the beam sees them). ®tgt * s t n e acute angle between the target normal and the beam d i r e c t i o n . ijj, i s +1 (-1) when the spectrometer i s on the l e f t ( r i g h t ) side of the incident beam. i t i s +1 (-1) f o r transmission ( r e f l e c t i o n ) geometry. i s i s +1 (-1) to set X t a r=+X t a r ( - X t a r ) . Limits were set on X t a r and Y t a r so that only those events o r i g i n a t i n g from i n s i d e of 2.5 mm from the target frame edge were accepted. Figure 2.15 shows the u t i l i t y of these cuts for the 62.8 MeV data at 120° ( 9 s p e c ) . The target outlines are shown over a contour diagram of the traceback of pion s c a t t e r i n g events to t h e i r o r i g i n on the target. 3.3.2.3 TOF Cuts The spectrometer events were required to have the time signature of pions incident on the target. The time signature i s given by the diffe r e n c e i n times of the spectrometer event strobe (timing set on s c i n t i l l a t o r E2) and the cap a c i t i v e probe timing pulse s i g n a l l i n g the a r r i v a l of protons at T l . This cut removes few events, since the nuclear s c a t t e r i n g at a l l measured angles i s much stronger than the Coulomb sc a t t e r i n g which i s responsible for beam leptons s c a t t e r i n g i n t o the spectrometer. 3.3.2.4 Muon Cuts As pions decay i n f l i g h t , some 30% of the pions scattered from a target i n t o the spectrometer w i l l decay before reaching the e x i t chambers. Those that decay before the dipole w i l l have a momentum which precludes t h e i r passage through the c e n t r a l region of the e x i t chambers. Many pions which pass the bender s u c c e s s f u l l y before decay, however, w i l l be observed as legitimate s c a t t e r i n g events. These events harm the r e s o l u t i o n of the energy spectrum. Furthermore, as the pion decays p r e f e r e n t i a l l y at an angle to the pion d i r e c t i o n , the energy calculated f o r a given decay event w i l l not be that of the decaying pion, so that e l a s t i c and i n e l a s t i c events i n the energy spectrum w i l l be intermixed. Two types of cuts were used to remove decay muon events from the f i n a l energy spectrum. Both of these were cuts which demanded consistency i n the observed WC4 and WC5 coordinates. The f i r s t cut, DDIF, or Delta Difference cut, demanded that the percent v a r i a t i o n of momentum from the spectrometer c e n t r a l momentum (100*6p/p="6") as determined from WC4 should be equal to that from WC5, within some cut l i m i t s . In the present case, -1.5<6lf-65<1.5 (3.24) A t y p i c a l DDIF spectrum i s shown i n Figure 3.13. The second muon cut applied was an Angle consistency cut [GYL84]. This cut used the spectrometer transport c o e f f i c i e n t s to determine the angle at which a given event's ray was expected to pass through the spectrometer e x i t region. R e c a l l that X4=X4(Xl,Yl,X3,Y3,6 £ f). 6^  i s determined from t h i s r e l a t i o n , so we cannot use i t to determine the expected value of X4. We have, however a s i m i l a r r e l a t i o n f o r X5: X5=X5(XI,Y1,X3,Y3,65) which i s solved to give 6 5. The expected coordinates at the e x i t may be calculated from: 88 o CM CM 3 CL < H V ^ ( 001 x ) S1N3A3 Figure 3.13 T y p i c a l d e l t a difference, DDIF, spectrum [RUN220, 50 MeV; 1 8 0 , 70°]. 89 X4*=X4(X1,Y1,X3,Y3,65) Y4'=Y4(X1,Y1,X3,Y3,65) X S ' - X S C X I . Y I . X S . Y S ^ ) Y5'=Y5(X1,Y1,X3,Y3,61+) (3.25) (3.26) (3.27) (3.28) These give an expected polar angle of ex i t from the spectrometer. The diff e r e n c e between t h i s angle and the experimentally observed angle as calculated from X4, Y4, X5 and Y5 i s the consistency angle, ANGL. This angle i s normally expected to be about 1.2° (50 MeV) which i s the RMS polar multiple s c a t t e r i n g angle due to materials (windows, a i r , etc.) within the spectrometer; muon decays contribute a large angle t a i l to the observed spectrum. Figure 3.14 i s a histogram of ANGL t y p i c a l to these experiments, upon which a t y p i c a l cut has been shown. Figure 3.15 shows a density plot demonstrating that the DDIF and ANGL spectra are strongly c o r r e l a t e d . Together they contribute to an improvement i n the spectrometer r e s o l u t i o n . (See fi g u r e A6.2 i n appendix VI.) 3.3.2.5 Energy Spectra Figures 3.16 and 3.17 show energy spectra at 48.3 MeV and 62.8 MeV re s p e c t i v e l y . Figure 3.16, with the spectrometer at 80° to the incident beam shows small i n e l a s t i c contributions to the sc a t t e r i n g , t y p i c a l of l i g h t elements at low incoming pion energy. Figure 3.16 i s taken with the spectrometer at 120° for 62.8 MeV pions on 1 2 C . Here there i s a large contribution from i n e l a s t i c s c atterings, i n p a r t i c u l a r the 4.44 MeV sta t e . 90 Figure 3.14 T y p i c a l , angle c o r r e l a t i o n spectrum, ANGL, [RUN220, 50 MeV; 1 8 0 , 70°]. 91 L O CM O CM O CM CM 3 w O V < - 1 m m ro m ( d / d V % ) s o - +Q F i g u r e 3.15 Density p l o t i l l u s t r a t i n g the c o r r e l a t i o n between the DDIF and ANGL spectra of figures 3.13 and 3.14. X ) o CB ID 3 n CO ID ft) o ft 1 e 5 -p-oo 2 n < o 3 N> n Co rt 00 O S3 O o 00 e n to co 600 ' I 1 1 • ' | ' ' ' ' | i i I I | i i I I | i I I i JI 400 co 200 -im. 25 30 35 40 45 50 55 ENERGY ( MeV ) [RUN 7] NJ o o o o o o LO O LO O LO CM CM *~ SING A3 Figure 3.17 T y p i c a l energy spectrum 62.8 MeV. TT + on 1 2 C at 120°. [RUN1110] 3.3.3 Peak F i t t i n g As the combined channel-spectrometer r e s o l u t i o n i n the CNO2 experiments was 1.6 MeV, peak f i t t i n g techniques were employed to remove the i n e l a s t i c s c a t t e r i n g contributions from the e l a s t i c s c a t t e r i n g spectra. Figure 3.18 shows the low l y i n g nuclear energy l e v e l s i n 1 2 C , l i +N, 1 6 0 and 1 8 0 [LED78]. The separation of the lowest l y i n g excited state i n 1 6 0 from the ground state i s not d i f f i c u l t . The cross sections i n t h i s case were evaluated d i r e c t l y from the energy spectra number of binned counts i n the e l a s t i c region. The 2 + state i n 1 8 0 , however, i s expected to exh i b i t a large e x c i t a t i o n i n our energy regime which, combined with the e l a s t i c cross section's decrease at large angles, required a c a r e f u l subtraction ( e s p e c i a l l y at 65 MeV where the i n e l a s t i c s c a t t e r i n g at 120° i s 35% of the e l a s t i c i n strength). The nitrogen state at 2.31 MeV i s also d i f f i c u l t to resolve, but i t s e x c i t a t i o n i s suppressed because of the p a r i t y and angular momentum of the state (figure 3.18). The use of peak f i t t i n g i n these r a t i o experiments requires consistency to avoid systematic e f f e c t s i n the r a t i o s , and c a r e f u l error analysis to allow f o r c o r r e l a t i o n s i n the f i t t i n g due to peak shape systematics. For t h i s reason, although the 1 6 0 cross section was determined without peak f i t t i n g and the 1 2 C cross section determination only used peak f i t t i n g as a c o r r e c t i v e technique, the evaluation of the r a t i o s i n a l l cases u t i l i s e d a peak f i t t i n g protocol. The d e t a i l s of the peak f i t t i n g a n a l y s i s , as implemented i n f o r t r a n routines QQDANA and GSRATIO are given i n appendix VII. An example of the re s u l t s with the 62.8 MeV data set f o r the 1 8 0 target i s shown i n figure 8 6 \ -4 h o 2 h 0 Figure 3.18 Nuclear l e v e l s observed i n 1 2 C , 14*N, 1 6 0 , and 1 8 0 . 96 3.19. There are two states contributing to the peak at around 4 MeV excitation energy, but the exact relative amplitudes of the two i s not important as their effect on the area correction to the elastic peak is small. The f i r s t excited state has a significant amplitude. The sum of the contributions from the various scatterings is also shown. Tables 3.13 and 3.14 l i s t the fitted strengths, of the f i r s t excited state relative to the elastic scattering for TT + on 1 2C at the two energies. These, combined with the cross section data of tables 3.16 and 3.20 give inelastic cross sections for the scattering 1 2C(ir +,ir +') 1 2C*C* . In each of the other cases, the peak f i t t i n g was used only in the correction of the elastic scattering data. 3.3.4 Absolute Cross Section Measurement The peak f i t t i n g used to extract the number of elas t i c a l l y scattered events from the raw data has been discussed in section 3 . 3 . 3 , and the relevant error analysis in appendix VII. The expression for the absolute cross section, then, is the same as that of equation 3.3 where the symbols used have the meanings in table 3 .15. A number of corrections are referred to in that table; we now discuss their implementation. 3.3.4.1 BM1«BM2 Rate Loss This loss was due to a number of effects. The f i r s t is the effect of a beamspdt exceeding the target in extent. In this case, the flux rate of pions through the target which may cause scattering events is smaller than the BM1*BM2 monitor indicates. This correction i s dependent upon angle, as Figure 3.19 Peak f i t t i n g r e s u l t s from T T + 1 8 0 spectrum at 62.8 MeV, 110°. [RUN1109] Table 3.13 Ratio of Inelastic (4.44 MeV) Cross Sections to Elastic for v+ on 12C at 48.3 MeV d a ^ V d f i 6 (%) ±(%) (deg cm) da/dn 50.2 5.06 0.56 60.4 5.88 0.43 70.4 4.02 0.47 81.0 6.79 0.49 91.1 7.59 0.75 100.7 9.24 0.54 110.7 12.26 0.76 120.3 15.87 0.98 130.2 24.8 1.3 Table 3.14 Ratio of Inelastic (4.44 MeV) Cross Sections to Elastic for ir + on 1 2C at 62.8 MeV da^Vcm e (%) ±(%) (deg cm) da/dn 50.0 2.58 0.67 51.1 2.97 0.96 60.1 4.44 0.81 71.6 3.64 0.63 81.6 5.70 0.74 91.5 11.6 1.4 101.5 24.0 1.6 111.1 42.3 1.8 120.9 78.5 2.9 Table 3.15 Meaning of Symbols Used In the Calculation of Elastic Differential Scattering Cross Sections SYMBOL MEANING N ( ) Counts i n () e l a s t i c peak l ( ) BM1S-BM2 counts f o r () run 4> Angle of target normal to beam d i r e c t i o n J ( ) ( 6 ) Jacobian d^/d^* for () I BM1«BM2 Rate loss MWPC e f f i c i e n c y product Fr a c t i o n of pions s a t i s f y i n g cuts 4> F r a c t i o n of scattered pions surviving to Dipole «o S o l i d angle presented by spectrometer Center of Mass (CM) d i f f e r e n t i a l s o l i d angle dn* LABoratory (LAB) d i f f e r e n t i a l s o l i d angle 9 E f f e c t i v e CM sca t t e r i n g angle of Spectrometer 0(8) CM d i f f e r e n t i a l s c a t t e r i n g cross section T ( ) Target thickness i n g/cm2 f o r () A ( ) Atomic weight f o r () No Avogadro's number ( ) One of ( 1 2C, 1 4N, 1 6 0 , 1 8Q or eMpTy) the s i z e of target that the incident beam sees i s dependent upon angle. The c o r r e c t i o n was made by noting the average beamspot shape over a number of runs. A simple Monte Carlo simulation then gave a c o r r e c t i o n f a c t o r as a function of angle. The siz e of t h i s c o r r e c t i o n was found to be about 10% and 30% f o r the 48.3 and 62.8 MeV data sets r e s p e c t i v e l y . The next e f f e c t to consider i s that of multiple pions a r r i v i n g at the s c a t t e r i n g target per beam burst at the production target. The cyclotron time structure i s such that (at 100% duty f a c t o r ) , a burst of protons reaches the production target every 43 ns. As the proton beam f l u x increases, the p r o b a b i l i t y of having more than one pion produced at the same time (the FWHM of the proton beam burst i s 2.5 ns) and hence a r r i v e at the beam counters (which, with only lower energy l e v e l d i s c r i m i n a t i o n i n t e r p r e t s a double pion event to be a sin g l e more energetic p a r t i c l e ) increases. u 0 i s the average number of BM1«BM2 pion events observed per 43 ns time i n t e r v a l . p 0 * s proportional to I P ( i ) , where P(i ) i s the p r o b a b i l i t y of observing exactly i pions. u, the Poisson average number of events per time i n t e r v a l i s then u = - l o g e ( l - u 0 ) , so that the m u l t i p l i c a t i v e c o r r e c t i o n f a c t o r i s l o g e ( l - u Q ) - 1 / u Q . U 0 was taken to be the number of BM1»BM2 events per run divided by the number of beam buckets which a r r i v e d at the production target (1AT1) during the run's duration ( l i v e time only). Figure 3.20 shows u 0 / l o g e ( l - u 0 ) - 1 for various values of u 0 , along with the range of lip's observed i n these experiments. The c o r r e c t i o n was t y p i c a l l y less than 10%. The f l u x loss due to beam decay muons with pion TOF incident upon the s c a t t e r i n g target i s calculated from equations 3.5 and 3.6. This e f f e c t Rate c o r r e c t i o n due to multiple pions per beam burst. The range of the c o r r e c t i o n found i n these experiments i s shown as a shaded area. a l t e r s the e f f e c t i v e beam pion f r a c t i o n at the target, a f r a c t i o n which creates an apparent BM1»BM2 rate increase at the target. 3.3.4.2 Spectrometer Offset Angle The spectrometer o f f s e t was estimated by performing separate cross se c t i o n measurements at 50° f o r 1 2 C and 1 6 0 targets with the spectrometer on opposite sides of the Mil beam. As the cross section drops r a p i d l y with increasing s c a t t e r i n g angle i n t h i s angular region, the d i f f e r e n c e i n the cross sections between measured values on the d i f f e r e n t sides i s a measure of the angular o f f s e t of the beam with respect to the spectrometer. The cross sections were indeed observed to be d i f f e r e n t , but consistent with the d i f f e r e n c e i n the average s c a t t e r i n g angles caused by asymmetries i n the spectrometer acceptance. The measurement of the average s c a t t e r i n g angles i s discussed i n the next section. 3.3.4.3 Average Scattering Angles As the spectrometer target traceback provides t r a j e c t o r y information i n terms of the angles 6Q and <j>0, i t i s possible to average the s c a t t e r i n g angles on an event by event basis. Writing the polar angle of the p a r t i c l e ' s t r a j e c t o r y with respect to the spectrometer axis as t a n 2 ¥ = t a n 2 9 0 + t a n 2 <J»0. (3.29) The s c a t t e r i n g angle to f o r t h i s event i s given by: cos 9 - i (1 - cos 2 6 ) 1 / 2 tan 6 0 cos to = _ _ _ _ _ — _ — (3.30) ( t a n 2 T + 1 ) 1 / 2 104 where 6 i s the spectrometer angle and i ^ = +1 (-1) f o r the spectrometer placed on the l e f t ( r i g h t ) of the beam. For N events, where the index j ranges from 1 to N. Now 50 = 9 - S i s the o f f s e t generated by t h i s angle averaging process. The s c a t t e r i n g angles given f o r the measured absolute cross sections have been transformed to the CM frame by a function F of the masses and energies involved. Properly, t h i s should be: 9 c m = F(iij). In pr a c t i c e , the approximation 0 c m = F(0) - 59 has been used. 3.3.4.4 Solid Angle Determination Transport c a l c u l a t i o n s [S0B84, SOB84a] have shown the spectrometer acceptance to be around 17 msr. This number i s dependent upon the exact geometry of the spectrometer. A more precise determination of s o l i d angle could be made with extensive Monte Carlo simulations [ALT85], but here we have used the experimental data set to a r r i v e at a value. The polar angle ¥ of an event ray with respect to the spectrometer axis i s defined by tan 2Y of equation 3.29. Use of the r e l a t i o n : then y i e l d s a value of ¥ i n the range 0 < ¥ < TT. For a series of rays o r i g i n a t i n g at target coordinates (X Q,Y 0) = (0,0) the s o l i d angle subtended by the spectrometer i s : = (E arccos(o) .))/N, (3.31) ¥ = a r c s i n ( ( l + t a n " 2 ¥ r 1 / 2 ) . (3.32) An = (Total Number of events) x 2TT (1-cos YQ ) (3.33) (Number of events with * < ¥ 0 ) 1 0 5 where * 0 * s chosen such that the spectrometer s o l i d angle presented to rays with V < "P0 i s s t i l l geometrical and a s u f f i c i e n t number of events occurs i n t h i s region. For a point o f f - a x i s , ( X Q , Y 0 ) # ( 0 , 0 ) , the subtended s o l i d angle i s expected to be geometrical i n some d i r e c t i o n other than that of the spectrometer a x i s . A ray connecting ( X Q , Y Q ) with the geometrical center of the bending dipole i s defined by: 6 0 = -tan-^Xg/dg) ( 3 . 3 4 ) 4>0 = - t a n " 1 ( Y 0 / d 0 ) , The distance from the target center to the center of the spectrometer dipole being d Q = 1 3 8 0 mm. This ray i s assumed to define the d i r e c t i o n of maximum acceptance. Aft i s then calculated from equation 3.33 with: tan 2Y = t a n 2 ( 9 - 9 0 ) + t a n 2 C<p—*0 ) ( 3 . 3 5 ) Figure 3.21 shows a pl o t of Aft vs for TT + scattered from the 1 6 0 target at 6 2 . 8 MeV. Note that the curve r i s e s sharply and s t a b i l i s e s around •PQ = 4 0 mrad, before r i s i n g as the number of events with ¥ < ¥ Q becomes l e s s than would be the case i f the s o l i d angle were s t i l l geometrical, ( i e . a loss i n s o l i d angle occurs at the "edges" of the spectrometer entrance port.) The values of Aft determined were averaged over 1 2 C and 1 6 0 runs at each energy. The value of Aft for the 62.8 MeV experiment was found to be 1 6 . 8 ± 0 . 8 and at 4 8 . 3 MeV, 1 8 . 0 ± 0 . 4 . The decrease observed at 62.8 MeV i s due to the fact that, i n that experiment, the data was c o l l e c t e d with the 106 Figure 3.21 T y p i c a l observed s o l i d angle of spectrometer as a function of WQ. [RUN1228 , 62.8 MeV; 1 6 0 , 50°] 107 h o r i z o n t a l l y focussing element QT1 set to zero. The decrease of about 10% i s i n agreement with REVMOC c a l c u l a t i o n s [S0B84b]. 3.3.5 Ratio Evaluations and Corrections The form of the r a t i o c a l c u l a t i o n i s the same as that given i n section 3.2.3, equation 3.14. The error evaluation i s discussed i n appendix VII, where some e f f o r t i s applied to allowing for the e f f e c t s of u n c e r t a i n t i e s i n the peak shape which was used. The r a t i o analyses used the f i t t e d peak areas i n a l l cases (except for MT target subtractions). The i s o t o p i c c o r r e c t i o n f o r 1 6 0 , 1 2 C cross sections and r a t i o s i n v o l v i n g 1 6 0 , 1 1 +N and 1 2 C were c a r r i e d out as i n equation 3.13; that i s with simple adjustments to the number of target n u c l e i due to i s o t o p i c v a r i a t i o n s i n atomic number. In the case of the oxygen isotope r a t i o , the c o r r e c t i o n can be made more p r e c i s e l y because a major contaminant i n the H 2 1 8 0 i s H 2 1 6 0 . If 8-exp i s the experimental r a t i o of cross sections i n the lab of target X to target Y, and the experimental target X contains a f r a c t i o n FR(X) of X and (l-FR(X)) of Y, by weight, then o(X)FR(X)J(X) q(Y)(l-FR(X))J(Y) R A(X) MY) exp g(Y)T(X)J(Y) A(Y)T(Y) where the a's are the CM d i f f e r e n t i a l cross sections, A's the atomic weights, J's the Jacobian r a t i o s d ^ c m / d ^ a b , and T's the thicknesses i n g/cm2 of the targets. R e c a l l that H 20 i s 99.8% H 2 1 60 so that FR(Y) = 1, c l o s e l y . If R 0 i s the CM r a t i o of cross sections f o r pure targets, then equation 3.37 gives RQ from R e X p « 108 FR(X)J(X)A(Y) J(Y)A(X) +(1-FR(X))) T(Y) T(X) (3.37) 3.3.6 Results The d i f f e r e n t i a l e l a s t i c cross sections for p o s i t i v e pions on 1 2 C , 1 4N, 1 6 0 and 1 8 0 are given i n tables 3.16 through 3.23 . The errors are s t a t i s t i c a l . The 1 2 C and 1 6 0 data i s shown i n figures 3.22 and 3.23. Best f i t and "Global parameter" c a l c u l a t i o n s are shown for each. The Global parameters i n the 62.8 MeV c a l c u l a t i o n s do not reproduce the back angle data. The 1 2 C cross sections agree with those of other experimenters [DYT79, PRE81, BLE83, TAC84, SOB84a] as does the 48.3 MeV 1 G 0 data [PRE81]. No previous e l a s t i c cross section measurement for tr + on 1 6 0 at 62.8 MeV (or within a few MeV) e x i s t s . The absolute normalisation i s correct to about 10%. This uncertainty i s l a r g e l y due to the uncertainty i n the spectrometer s o l i d angle and muon decay c o r r e c t i o n s . The normalisation i s consistent with that obtained by comparing the ir +p cross section at 50° (LAB) with that of SAID [ARN82]. Tables 3.24 through 3.31 give r a t i o s of cross sections 1 1 + a / 1 2 o , 1 6 o / 1 2 a , 1 6o/ l l*a and 1 8 o / 1 6 o for TF+ s c a t t e r i n g on 1 2 C , 1J*N, 1 6 0 , and 1 8 0 . The r e s u l t s are displayed i n figures 3.24 through 3.29 along with f i t t e d c a l c u l a t i o n s . These c a l c u l a t i o n s are discussed i n chapter IV. 3.4 Summary In chapter I I I , the extraction of the e l a s t i c cross sections and t h e i r 109 r a t i o s from the raw data has been described. A number of techniques common to nuclear physics have been discussed, such as the use of PID ( p a r t i c l e i d e n t i f i c a t i o n ) cuts, peak f i t t i n g , i s o t o p i c c o r r e c t i o n to the cross sections, and compensation for p a r t i c l e decay. Some techniques more s p e c i f i c to the p a r t i c u l a r apparatus of t h i s experiment have also been considered, such as the QQD s o l i d angle and average angle evaluations. Results have been presented f o r the e l a s t i c d i f f e r e n t i a l cross sections and t h e i r r a t i o s f o r 1 2 C , n B at 38.6 and 47.7 MeV and 1 2C, 1'*N, 1 60 / 1 8 0 , 1 6 0 at 48.3 and 62.8 MeV. The experimental cross section r a t i o s are free from many of the systematic e f f e c t s which need to be allowed f o r i n a r r i v i n g at the absolute cross sections. The experimental r a t i o s are thus the strength of t h i s work. In chapter IV, an analysis w i l l be made to extract from them proton matter differences f or the corresponding n u c l e i . Table 3.16 Differential Cross Sections for Elastic Scattering of ir+ on 12C at 48.3 MeV 6 (deg cm) do/dQ. (mb/sr) + (mb/sr) 50.2 3.67 0.23 60.4 2.52 0.09 70.4 2.44 0.18 81.0 3.52 0.16 91.1 4.28 0.29 100.7 5.64 0.25 110.7 6.44 0.38 120.3 6.49 0.42 130.2 6.32 0.41 Table 3.17 Differential Cross Sections for Elastic Scattering of i r + on 1<fN at 48.3 MeV e (deg cm) dcr/dft (mb/sr) + (mb/sr) 50.1 4.76 0.31 60.3 3.47 0.23 70.4 3.54 0.26 80.8 4.95 0.41 90.8 5.75 0.45 101.1 6.97 0.39 110.9 8.09 0.53 120.2 7.12 0.61 130.0 6.83 0.68 Table 3.18 Differential Cross Sections for Elastic Scattering of * + on 1 60 at 48.3 MeV 6 (deg cm) da/dft (mb/sr) + (mb/sr) 50.0 5.24 0.26 60.2 3.90 0.19 70.3 4.24 0.23 80.7 5.86 0.40 90.7 6.42 0.42 101.0 7.34 0.30 110.8 7.94 0.47 120.1 6.95 0.50 129.9 5.99 0.47 Table 3.19 Differential Cross Sections for Elastic Scattering of n + on *80 at 48.3 MeV 0 (deg cm) do/dtt (mb/sr) + (mb/sr) 49.9 6.39 0.36 60.1 3.78 0.23 70.2 3.63 0.22 80.6 4.99 0.43 90.6 5.35 0.40 100.9 6.03 0.28 110.7 6.72 0.47 120.0 5.23 0.44 129.8 4.19 0.45 Table 3.20 Differential Cross Sections for Elastic Scattering of * + on 1 2C at 62.8 MeV 0 (deg cm) do/dti (mb/sr) + (mb/sr) 50.0 6.72 0.48 51.1 5.83 0.51 60.1 2.53 0.16 71.6 2.26 0.13 81.6 2.90 0.16 91.5 3.49 0.21 101.5 3.96 0.19 111.1 3.23 0.16 120.9 2.72 0.11 Table 3.21 Differential Cross Sections for Elastic Scattering of n+ on 1<fN at 62.8 MeV 6 (deg cm) do/dQ. (mb/sr) + (mb/sr) 61.4 3.33 0.27 71.5 2.74 0.19 81.3 3.79 0.21 91.1 4.73 0.47 101.3 4.93 0.35 111.0 3.54 0.18 120.6 3.09 0.18 Table 3.22 Differential Cross Sections for Elastic Scattering of if+ on 160 at 62.8 MeV 9 (deg cm) do7d°, (mb/sr) + (mb/sr) 49.7 8.77 0.77 51.0 7.06 0.62 61.3 3.61 0.22 71.4 3.16 0.16 81.2 4.09 0.18 91.0 4.74 0.32 101.2 4.18 0.24 110.9 2.81 0.12 120.5 1.89 0.09 Table 3.23 Differential Cross Sections for Elastic Scattering of i r + on 1 80 at 62.8 MeV e (deg cm) da/dft (mb/sr) + (mb/sr) 61.2 3.14 0.32 71.3 2.56 0.18 81.1 3.44 0.21 90.9 3.59 0.30 101.1 3.13 0.24 110.8 1.96 0.11 120.4 1.20 0.10 118 10 CO _Q E 10 7 "O b TD 10' 30 50 70 90 110 130 150 6 (degrees, CM) Figure 3.22 E l a s t i c d i f f e r e n t i a l cross sections of TT + incident on 1 2 C and 1 6 0 at 48.3 MeV. Calculations use densities derived from electron s c a t t e r i n g (FL) and the SMC81 p o t e n t i a l . S o l i d , Set E50; broken, Set Ef50f. Figure 3.23 E l a s t i c d i f f e r e n t i a l cross sections of I T + incident on 1 2 C and 1 6 0 at 62.8 MeV. Calculations use densities derived from electron s c a t t e r i n g (FL) and the SMC81 p o t e n t i a l . S o l i d , Set EBLE65f; long dashes, Set EC65f (Set E065f for 1 6 0 c a l c u l a t i o n ) ; short dashes, Set E65; and dots Set EBLE65co. Table 3.24 Differential Cross Section Ratios of Elastic Scattering of n+ on 1HN and 12C at 48.3 MeV 9 (deg cm) l^o/ l Z a + 50.1 1.300 0.043 60.3 1.374 0.040 70.4 1.440 0.063 80.8 1.419 0.062 91.0 1.349 0.065 100.8 1.241 0.039 110.8 1.260 0.041 120.2 1.101 0.046 130.0 1.082 0.047 Table 3.25 Differential Cross Section Ratios of Elastic Scattering of * + on 1 6 0 and 12C at 48.3 MeV e (deg cm) 16 a/12 a + 50.1 1.432 0.045 60.3 1.564 0.045 70.4 1.739 0.080 80.8 1.678 0.069 90.9 1.506 0.069 100.8 1.311 0.039 110.8 1.236 0.036 120.2 1.075 0.042 130.0 1.950 0.038 Table 3.26 Differential Cross Section Ratios of Elastic Scattering of n+ on 1 60 and lkH at 48.3 MeV 6 (deg cm) + 50.0 1.101 0.035 60.2 1.124 0.035 70.3 1.197 0.045 80.7 1.183 0.058 90.8 1.116 0.050 100.7 1.053 0.033 110.7 0.981 0.030 120.1 0.976 0.043 129.9 0.877 0.044 Table 3.27 Differential Cross Section Ratios of Elastic Scattering of w+ on 180 and 160 at 48.3 MeV 9 (deg cm) 18 0/16 a + 50.0 1.220 0.029 60.2 0.970 0.024 70.2 0.855 0.021 80.7 0.851 0.045 90.7 0.833 0.035 101.0 0.821 0.018 110.8 0.846 0.029 120.0 0.752 0.032 129.9 0.699 0.040 Table 3.28 Differential Cross Section Ratios of Elastic Scattering of * + on 1 WN and 1 2C at 62.8 MeV 0 (deg cm) + 60.8 1.318 0.092 71.6 1.214 0.077 81.4 1.309 0.067 91.3 1.343 0.075 101.5 1.247 0.066 111.1 1.228 0.050 120.8 1.139 0.050 Table 3.29 Differential Cross Section Ratios of Elastic Scattering of v+ on 160 and 12C at 62.8 MeV 9 (deg cm) 16 a/12 a + 49.8 1.306 0.108 51.0 1.211 0.120 60.7 1.428 0.099 71.5 1.400 0.084 81.4 1.412 0.069 91.2 1.358 0.072 101.4 1.057 0.054 111.0 0.872 0.039 120.7 0.695 0.034 Table 3.30 Differential Cross Section Ratios of Elastic Scattering of w+ on 1 60 and llfN at 62.8 MeV 6 (deg cm) + 60.6 1.084 0.078 71.4 1.153 0.075 81.3 1.079 0.050 91.1 1.002 0.062 101.3 0.848 0.045 110.9 0.710 0.032 120.6 0.611 0.031 Table 3.31 Differential Cross Section Ratios of Elastic Scattering of i r + on 1 80 and 1 60 at 62.8 MeV e (deg cm) 18 a/16 C T + 61.2 0.870 0.081 71.3 0.811 0.053 81.2 0.841 0.043 90.9 0.757 0.036 101.1 0.748 0.045 110.9 0.779 0.039 120.4 0.633 0.051 0.9 I • 1 • 1 • 1 • 1 • 1— 30 50 70 90 110 130 6 (degrees, CM) Figure 3.24 Ratios of e l a s t i c cross sections for T T + at 48.3 MeV:  lho/12o. C a l c u l a t i o n uses Set Ef50f i n an SMC81 p o t e n t i a l and a FL parameterisation of a model independent electron s c a t t e r i n g density f o r 1 2 C . The 1 I +N proton density i s a best f i t FL form ( c f . figure 4.15). 3 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 T 1 1 r i 1 r - 48.3 MeV TT + -i I . L 30 50 70 90 110 130 6 (degrees, CM) Figure 3.25 Ratios of e l a s t i c cross sections f or TT + at 48.3 MeV: 1 6 a / l l + o . C a l c u l a t i o n uses Set Ef50f i n an SMC81 p o t e n t i a l and FL parameterisations of the model independent electron s c a t t e r i n g d e n s i t i e s of 1 6 0 and 1 2 C . The l i +N proton density i s a best f i t FL form ( c f . fig u r e 4.16). Figure 3.26 Ratios of e l a s t i c cross sections f o r TT + at 48.3 MeV: 1 8 o 7 1 6 a . C a l c u l a t i o n uses Set EIM50 i n an SMC81 p o t e n t i a l and a MG form for the 1 6 0 matter d e n s i t i e s . The 1 8 0 proton density i s a best f i t FL form ( c f . f i g u r e 4.13). Figure 3.27 Ratios of e l a s t i c cross sections f or i r + at 62.8 MeV: 1 1 + o 7 1 2 a . C a l c u l a t i o n uses Set EC65f i n an SMC81 p o t e n t i a l and a FL parameterisation of a model independent electron s c a t t e r i n g density f o r 1 2 C . The ll*N proton density i s a best f i t FL form ( c f . fi g u r e 4.17). Figure 3.28 Ratios of e l a s t i c cross sections f o r T T + at 62.8 MeV: 1 6a/ 1'*o. C a l c u l a t i o n uses Set E065f i n an SMC81 p o t e n t i a l and FL parameterisations of the model independent electron s c a t t e r i n g d e n s i t i e s of 1 6 0 and 1 2 C . The ll*N proton density i s a best f i t FL form ( c f . fig u r e 4.18). o O J O _ ci O CP CO - g O _| CO O o o (0 9 l) UP/-op / (0 f t l) U P/-°P 8 1 Figure 3.29 Ratios of e l a s t i c cross sections f o r i r + at 62.8 MeV: 1 8 o 7 1 6 a . C a l c u l a t i o n uses Set EIM65 i n an SMC81 p o t e n t i a l and a MG form f o r the 1 6 0 matter d e n s i t i e s . The 1 8 0 proton density i s a best f i t FL form ( c f . figure 4.14). Chapter IV Interpretations 4.1 Introduction To t h i s point, we have discussed aspects of the n + r a t i o experiments pertaining to data a c q u i s i t i o n and the extraction and c o r r e c t i o n of cross section and r a t i o r e s u l t s . The cross section r a t i o s , i t w i l l be r e c a l l e d , are free from many of the systematic e f f e c t s which were considered i n extracting absolute cross sections. We have previously indicated that o p t i c a l model c a l c u l a t i o n s of these r a t i o s are s e n s i t i v e to proton matter differences between the n u c l e i , without strong dependence upon the d e t a i l s of the p o t e n t i a l i t s e l f . In the present chapter, we perform such o p t i c a l model c a l c u l a t i o n s and extract proton matter d i s t r i b u t i o n d i f f e r e n c e information from the r a t i o s . There s t i l l e x i s t s some p o s s i b i l i t y of ambiguity i n t h i s e x t r a c t i o n (see section 1.5), because the nuclear physics of the ir-nucleus i n t e r a c t i o n and the nuclear structure (matter d i s t r i b u t i o n s ) of the nucleus are folded together i n the o p t i c a l p o t e n t i a l . Our o p t i c a l p o t e n t i a l i s pararaeterised from analyses of many independent data sets [STR79,STR80,CAR82] and t a i l o r e d to the current analyses by f i t t i n g to reference cross sections to determine a s e l e c t i o n of the o p t i c a l parameters. The nuclear structure parameters are then determined by f i t t i n g to the cross section r a t i o s . We separate the analysis here i n t o two parts. The f i r s t deals with analyses of the CB experiment i n terms of simple RMS r a d i i of proton matter d i s t r i b u t i o n s . In t h i s experiment the unknown nucleus i s an isotone of the reference. The second part deals with analysis of the CNO2 experiments with 135 the use of Fourier expansions of the density d i f f e r e n c e s . The 1 8 0 / 1 6 0 measurement Is considered as a p r e c i s i o n t e s t , since electron scatterers have measured A(p(r)) between the isotopes over the extent of the nuclear volume. The 1 2C, l l +N, 1 60 (CNO) data set i s a test of the TT + r a t i o technique i n measuring A(p +(r))=A(p_(r)) i n n u c l e i with proton number equal to neutron number, but where the nucleon content varies s i g n i f i c a n t l y with nucleus. 4.2 Optical Potential Analysis In the opening chapter, i t was shown that the r a t i o of TT + e l a s t i c d i f f e r e n t i a l s c a t t e r i n g cross sections i s extremely s e n s i t i v e to proton matter differences between n u c l e i . To perform a quantitative i n t e r p r e t a t i o n of any experimental data, one must use a more sophisticated model. This model should include information about nuclear structure, as well as kinematic, c o r r e l a t i v e , and absorptive processes. In the low energy regime (E«A, where A i s the energy at which the (3,3) resonance peaks: T^ISO MeV), some of the most promising models describing pion s c a t t e r i n g processes are the o p t i c a l models. So named because of t h e i r reminiscence of c l a s s i c a l photon s c a t t e r i n g , these models develop i n t e r a c t i o n p o t e n t i a l s ( o p t i c a l p o t entials) which do not depend upon i n d i v i d u a l nucleon coordinates (equation 1.33). Several excellent o p t i c a l p o tentials have been developed to explain the systematic e f f e c t s observed i n pion-nucleus s c a t t e r i n g . The Landau-Thomas theory (LT) [LAN78, LAN82] extends the formalism of Kerman et a l . (Kerman, McManus and Thaler or KMT) [KER59] to pion-nucleus s c a t t e r i n g , performing c a l c u l a t i o n s i n a momentum space framework. It uses a semi-factored approximation i n i t s f i r s t order o p t i c a l p o t e n t i a l (thereby neglecting any Ericson-Ericson Lorentz-Lorenz ( E 2 L 2 ) e f f e c t s ) and chooses a separable pion-nucleon t-matrix which i s simply related to observed phase s h i f t s . It includes angle transformation and Pau l i corrections, as well as true pion absorption i n the nucleus. Other p o t e n t i a l s , such as the Colorado [DIG77] po t e n t i a l and that of St r i e k e r , McManus and £arr (SMC) [STR79, STR80, CAR82], have t h e i r o r i g i n s i n the E r i c s o n - K i s s l i n g e r p o t e n t i a l . This Ericson [ERI66] form takes the scat t e r i n g amplitude of equation 1.1, writes the appropriate multiple s c a t t e r i n g equations, and a r r i v e s at the v e l o c i t y dependent K i s s l i n g e r p o t e n t i a l [KIS55] (equation 1.33). The v e l o c i t y dependence arises from the p-wave (k#_k') term i n the scattering amplitude and i s important i n charact e r i s i n g the i n t e r a c t i o n . Short range c o r r e l a t i o n s between nucleons i n nuclear matter r e s u l t i n an e f f e c t s i m i l a r to that induced by a medium on the s c a t t e r i n g of dipole ( o p t i c a l ) r a d i a t i o n : the celebrated E 2 L 2 e f f e c t weakens the p-wave a t t r a c t i o n of pions to a nucleus. Generalisations allow for P a u l i (increases s-wave repulsion) and kinematic e f f e c t s . Absorptive e f f e c t s (two-nucleon) are included as complex 'rho-squared terms' to be added to those of the otherwise f i r s t order o p t i c a l p o t e n t i a l . The SMC theory a l t e r s the Ericson p o t e n t i a l by removing the higher order corrections to the p-wave absorption [STR79] and including e f f e c t s due to the transformation between pion-nucleus and pion-nucleon center of momentum coordinate systems (angle transformations) (V 2p,V 2p 2)[THI76]. The Colorado p o t e n t i a l i s also of the K i s s l i n g e r form, but allows for absorptive e f f e c t s by the i n c l u s i o n of a semi-phenomenological s-wave term. S i c i l i a n o and Johnson [J0H80, J0H83, JOH83a, JOH83b] have spent some e f f o r t i n describing the i s o s p i n dependence of the o p t i c a l p o t e n t i a l . The general form f o r a TT2N type of i n t e r a c t i o n generates isotensor terms i n the ir-nucleus i n t e r a c t i o n i n much the same way that the TTN i n t e r a c t i o n generates i s o s c a l a r and isovector terms. (The isotensor terms are characterised by terms l i k e (T«n) 2 i n the i s o s p i n v a r i a b l e s . ) Such isotensor terms are not considered by the SMC p o t e n t i a l s . S i c i l i a n o and Johnson, however, do include them. Further work by S i c i l i a n o , i n p a r t i c u l a r , has extended the work to low energy, where he i s currently examining the e f f e c t s of p-wave cor r e l a t i o n s ( E 2 L 2 , p exchange, and exchange graphs) on charge exchange reactions [J0H83b, IR083, C0084, ALT85a]. The r e l a t i v e importance of isovector and isotensor terms i n the o p t i c a l p o t e n t i a l w i l l be most s e n s i t i v e l y probed by the current generation of single and double charge exchange reactions. 4.2.1 SMC P o t e n t i a l s To demonstrate that matter r a d i i obtained through analyses of pion cross section r a t i o data are independent of p o t e n t i a l model, some sampling of a v a i l a b l e o p t i c a l model approaches should be made. This has i n fact been the p r a c t i c e of the 'PISCATtering' group at TRIUMF [JOH79, BAR80]. The r e s u l t s of such analyses w i l l be discussed l a t e r . For the purposes of t h i s a n a l y s i s , however, we concern ourselves primarily with the SMC p o t e n t i a l . 4.2.1.1 The P o t e n t i a l The SMC p o t e n t i a l , U, i s written as: 2_U = -4TT [b(r)+B(r)+ V 2c(r)+ V 2 C ( r ) l + with: 2 2 C 0 4TT [v«L(r)c(r)V + — V»p 2(r)V + 2wV c(r)] (4.1) p2 b(r)-P 1[b 0p(r)-e i rb 16p(r)] (4.2) b o - b o - S C b o ^ b ! 2 ) ^ ^ (4.3) c ( r ) = [ c 0 p ( r ) - e 7 r c 1 6 p ( r ) ] / P 1 (4.4) B(r)= P 2 B 0 p 2 ( r ) (4.5) C( r ) = C 0 p 2 ( r ) / P 2 (4.6) 6p(r)=p-(r)-p +(r) (4.7) L(r) = [ l + - ^ i A l _ _ c ( r ) ] - l (4.8) pi-rrfe7A> ( 4- 9 ) P2=rTT e7A) (*-10> 5 = 1 +"(e/A) ( 4 * U ) Here k^ ab an<* w l a b a r e t n e momentum and t o t a l energy i n the lab and co i s the t o t a l energy of the pion i n the ACM system, i s the pion charge, A Q the nuclear mass per nucleon (=931 MeV) , kf the fermi momentum (=1.4 fm - 1) and e i s IO/AQ. The above, so c a l l e d SMC79 p o t e n t i a l , was the form o r i g i n a l l y introduced [STR79]. Isovector absorption terms may be included: B ( r ) = P 2 [ B 0 p 2 ( r ) - e i r B l P ( r ) 6 p ( r ) ] (4.12) C ( r ) = [ c 0 p 2 ( r ) - e i r C l P ( r ) 6 p ( r ) ] / P 2 , (4.13) but the e f f e c t s of Bj and Cj are small i n e l a s t i c s c a t t e r i n g . The exact treatment of the E 2 L 2 e f f e c t i s not w e l l defined, but i n l a t e r papers the absorption i s included i n L ( r ) [STR80, CAR82] so that the term: V»L(r)c(r)V + V«C(r)V = 7»[L(r)c(r)+C(r)]v (4.14) becomes: V«[L(r)[c(r)+C(r)]]v, (4.15) where: L( r ) = [ l + 4 T r X ( c ( r ) + C ( r ) ) / 3 ] _ 1 (4.16) The f a c t o r (A-l)/A previously accompanying X, due to Thies [THI76], has been dropped. This form of the SMC p o t e n t i a l , with B^CjMD, we re f e r to here as the SMC81 p o t e n t i a l . 4.2.1.2 The C a l c u l a t i o n The e l a s t i c s c a t t e r i n g cross sections are calculated i n center of momentum coordinates with use of the Krell-Thomas code [KRE68, TH081]. This code matches the Coulomb wave functions of p a r t i c l e s approaching a nucleus with solutions to the wave equation: (V 2+k 2-2uU) ¥(r) = 0 (4.17) at about 15 fm from the nuclear center. Partial-wave phase-shifts are thereby obtained. The o p t i c a l parameters are somewhat el u s i v e , but may be obtained [KRE69, MCM79, STR80] from f i t t i n g to pionic atom l e v e l s h i f t s and l i n e widths, along with data obtained i n pion-nucleon s c a t t e r i n g experiments ( s c a t t e r i n g lengths). The E 2 L 2 parameter, A, i s often assumed to have the long wavelength l i m i t i n g value of 1, even though there i s some evidence that 1.2 i s a better value [THI76]. A n u l l value i n t h i s parameter would in d i c a t e an absence of the e f f e c t . Values as high as 2 have been suggested [JEN83]. Proton and neutron matter d e n s i t i e s i n l i g h t n u c l e i are assumed to have a Modified Gaussian (MG) form: p (r) = T P f J 0 (1 + T ( r / a ) 2 ) e " ( r / a ) 2 (4.18) The quantity T can be related to the number of nucleons, T, i n a given i s o s p i n state ( i . e . : N or Z) and to the relevant RMS matter radius through the s h e l l model [ELT61]. 4.3 CB Experiment 4.3.1 Cross Section Calculations with the SMC Potential If the o p t i c a l parameters are treated as free (or constrained) parameters, the o p t i c a l codes may be u t i l i s e d i n the f i t t i n g of experimental nuclear cross section data. This can admittedly be a somewhat r i s k y p r a c t i c e , as the parameters are numerous and i n some cases strongly co r r e l a t e d . Certain semi-ad hoc assumptions [KRE69] prove u s e f u l i n providing con s t r a i n t s . For example, the absorption parameters may be assumed to be of the form i c ( l - i ) where K has a r e a l value. 141 In order to avoid misleading convergence, an i n t e r a c t i v e f i t t i n g routine VIEWFIT was written. This routine allows the generation of r e l i a b l e f i t s by presenting x 2 and a v i s u a l display of c a l c u l a t i o n and experiment for user al t e r e d parameter sets. F i t s generated i n t h i s manner are not i n the s t r i c t e s t sense 'best f i t s ' , but are hereafter referred to as such. (In f a c t , i f the trend of the data i s of more importance than, say, the normalisation, t h i s method may well y i e l d the better ' f i t ' . ) Appendix V discusses an a l t e r n a t i v e normalisation-independent f i t t i n g approach. Table 4.1 shows several sets of parameters referred to i n the analysis of t h i s experiment. Set 0 i s the [STR79] Global set 1. The imaginary parts of the scattering lengths and volumes are those suggested by Strieker et a l . for 50 MeV pion s c a t t e r i n g . Set 1 i s i d e n t i c a l to set 0, but uses the parameters suggested f o r pions of energy 30 MeV. These two sets d i f f e r l i t t l e i n t h e i r predicted cross sections. Sets l a and lb use set 1 as the i n i t i a l coordinates i n a true X 2 minimisation on the 1 2 C cross section data of t h i s experiment. In t h i s f i t t i n g an assumption for the true absorption terms was made: BQ=K 1(1-I) and CQ=K 2(1-1). The free parameters were <1, K 2, and X (The E 2 L 2 parameter, equation 4.16). The f i t t i n g was c a r r i e d out with a modified version of the Krell-Thomas code, u t i l i s i n g a FORTRAN routine FLETCH [PAT78] i n the parameter search. Figures 3.8 and 3.9 show the r e s u l t s of the c a l c u l a t i o n s : reproducing the scattering requires a good knowledge of the higher order s c a t t e r i n g processes (absorption, E 2 L 2 ) . Parameter set 2 i s a 'best f i t ' ( i e : VIEWFIT) set obtained for the 40 MeV data of Blecher et a l . [BLE79]. This data and the f i t t e d curve are Table 4.1 Various Optical Parameter Sets C,11B Experiment: SMC79 Potential Para-meter Parameter Set (Real and Imaginary Parts) 0 1 l a l b 2 X 1.000 1.000 0.765 1.420 0.875 b0 -.028 .002 -.028 .002 -.028 .002 -.028 .002 -.026 .001 -.080 .000 -.078-.001 -.078-.001 -.078-.001 -.080 .000 Bo -.043 .043 -.043 .043 -.032 .032 -.047 .047 -.026 .038 .270 .002 .265 .0025 .265 .0025 .265 .0025 .267 .0025 c l .220 .001 .219 .001 .219 .001 .219 .001 .220 .001 co -.100 .100 -.099 .099 -.056 .056 -.009 .009 -.065 .065 143 shown i n f i g u r e 4.1. Figure 4.2 demonstrates that set 2 a l s o gives a reasonable f i t to the 50 MeV data obtained by the same group [MOI78]. These data w i l l be r e f e r r e d to h e r e a f t e r as the 'Los Alamos' data. I t must be emphasised at t h i s p o i n t that d i s c r e p a n c i e s between the Los Alamos data set and that of the present experiment are not worrisome. The 40 MeV data sets agree reasonably, the s l i g h t l y higher back angle data of t h i s experiment Is p a r t i a l l y the r e s u l t of the d i f f e r e n c e between 40 MeV s c a t t e r i n g and that at 38.6 MeV. There are no d i s c r e p a n c i e s i f one assumes a n o r m a l i s a t i o n d i f f e r e n c e of the order of 8% between the data s e t s . (The Los Alamos data claims 4% to 7% u n c e r t a i n t y i n t h i s q u a n t i t y . ) The 47.7 MeV cross s e c t i o n data disagrees at back angles by an amount greater than can be a t t r i b u t e d to the energy d i f f e r e n c e between 49.9 and 47.7 MeV. The data at forward angles seems to slump w i t h respect to other measurements, as discussed i n s e c t i o n 2.2.2.1. In any case, i f t h i s cross s e c t i o n data s u f f e r s from uncorrected systematic e f f e c t s such as might accompany the angle dependent gain i l l u s t r a t e d i n f i g u r e 2.9, no e f f e c t upon the r a t i o data should be present. 4.3.2 Generic Low Energy Pion Scattering The cross s e c t i o n s i n f i g u r e s 3.8, 3.9, 4.1, and 4.2 a l l e x h i b i t f e a t u r e s c h a r a c t e r i s t i c of low energy pion s c a t t e r i n g ' on l i g h t n u c l e i . A dip at 60° I s due to an i n t e r f e r e n c e between the r e p u l s i v e s-wave and a t t r a c t i v e p-wave p a r t s of the p o t e n t i a l . I t i s f i l l e d i n by the e f f e c t s of imaginary terms i n the s c a t t e r i n g lengths and volumes. This d i p i s seen f o r pions of both p o s i t i v e and negative charge. For p o s i t i v e pions a d e s t r u c t i v e i n t e r f e r e n c e a l s o occurs between the pion-nuclear p-wave and Figure 4.1 The 40 MeV Los Alamos T T + , 1 2 C e l a s t i c cross section data [BLE79]. O p t i c a l c a l c u l a t i o n s use parameter set 1 (dashed curve) and 2 ( s o l i d curve). o 00 O ^ 10 to CJ ( J S / quu ) {e)**zi Figure 4.2 The 49.9 MeV Los Alamos T T + , 1 2 C e l a s t i c cross section data [MOI78]. O p t i c a l c a l c u l a t i o n s use parameter set 1 (dashed curve) and 2 ( s o l i d curve). Coulomb amplitudes. For negative p i o n s , t h i s i n t e r f e r e n c e i s c o n s t r u c t i v e . The trend at l a r g e angles i s a dip toward the f i r s t d i f f r a c t i v e minimum. This trend becomes more apparent as i n c i d e n t pion energy i s increased (compare f i g u r e s 4.1 and 4.2). Figure 4.3 c o n t r a s t s the e l a s t i c s c a t t e r i n g of p o s i t i v e and negative pions at 43.1 MeV on 1 2C. The form of the d i f f e r e n t i a l e l a s t i c s c a t t e r i n g cross s e c t i o n s i n t u r n p r e d i c t s the form of t h e i r r a t i o s . For small angle s c a t t e r i n g the r a t i o w i l l go roughly as the r a t i o of the f o u r t h power of the charges of the r e s p e c t i v e n u c l e i . At l a r g e r angles (9>30°) the nuclear s c a t t e r i n g begins to predominate. As one chooses the convention of always using the heavier nucleus' cross s e c t i o n as the numerator i n the r a t i o , the r a t i o w i l l r i s e above u n i t y . The p r e c i s e manner i n which t h i s occurs i s dependent upon the r e l a t i v e s i z e s of the s and p partial-wave amplitudes i n the n u c l e i . A f t e r reaching a maximum, the r a t i o again drops as the cross s e c t i o n s dip toward t h e i r d i f f r a c t i v e minima. Figure 4.4 shows c a l c u l a t e d cross s e c t i o n s f o r s c a t t e r i n g from the isotones 1 2C and 1 1 B at 43.1 MeV. The r a t i o of these i s shown i n Figure 4.5. 4.3.3 Ratio Calculations We have seen that o p t i c a l p o t e n t i a l models use a la r g e number of parameters i n t h e i r r e p r e s e n t a t i o n of the physics of p r o j e c t i l e - n u c l e u s i n t e r a c t i o n s . I f such models are reasonable, one expects the parameters to be independent of nucleus or, at worst, slowly v a r y i n g . The e f f e c t on a r a t i o of such v a r i a t i o n s would be s m a l l . Furthermore, a r a t i o w i l l tend to Figure 4.3 Optical model calculations (parameter set 2) contrast the natures of charged pion scattering from 1 2C at 43.1 MeV. Character of Coulomb-nuclear interference distinguishes pion charge states. Figure 4.4 O p t i c a l model c a l c u l a t i o n s (parameter set 2) of s c a t t e r i n g of p o s i t i v e pions on 1 2 C (dashed curve) and n B at 43.1 MeV. Notice the r e l a t i v e magnitudes and the s h i f t i n the minimum. 20 40 60 80 100 120 140 160 6 (CM.) Figure 4.5 Ratio of cross sections i n figure 4.4. 150 eliminate the e f f e c t s of unce r t a i n t i e s i n the precise values of the o p t i c a l parameters. If we assume f or the moment that the parameters themselves are well known, then the differe n c e between n u c l e i i s characterised by the matter d i s t r i b u t i o n and the is o s p i n of nucleons i n the nucleus. We parameterise the proton (neutron) matter density according to equation 4.18, where: POO = [ ^ - { l + ^ } a 3 ] - l . (4.19) For n u c l e i with T protons (neutrons): T = (T-T 0)/3 (4.20) In the s h e l l model, T Q for p - s h e l l n u c l e i (T = {2,3 8}) i s given the value 2; t h i s value Is used here unless s p e c i f i c a l l y stated to the contrary. The (2m) t n moment i s : <r 2 m> = *3/2 P q o i L ^ l (2m+l)!! { l + T ( 2 m + 3 > } (4.21) so that: The RMS r a d i i , r( ) (where ( ) e {+,protons; -,neutrons}), characterise the matter d i s t r i b u t i o n s and the r a t i o . For protons, r+ i s related to the RMS charge radius (appendix X), r 0 , by: r + 2 = r 2 - r 2 (4.23) T 0 pp where r p p i s the RMS charge radius of a si n g l e proton (0.8 fm [COL67, AND77]). 151 The experimental r a t i o data were f i t t e d to the r a t i o s predicted by the SMC79 p o t e n t i a l using parameter set 2 with n r + a free parameter. The value f o r 1 2 r + (r+ f o r 1 2C) was assumed to be equal to 1 2 r _ and was obtained from the experimentally determined charge radius, 1 2 r Q , of 2.44±0.02 fm [KLI73] f o r that nucleus. The value of 1 1 r _ was also taken to be equal to 1 2 r _ . The value f o r 1 0 r 0 of 2.43±0.06 fm has been measured [BEE79, 0LI81] and y i e l d s the assumed value f o r 1 0 r + . It was assumed that 1 0 r + = 1 0 r _ . F i t t i n g was performed i n the usual manner, searching for a minimum i n : X 2< l lr+> = E [ ] 2 (4.24) i 6 i Rj[ and R Q i are the experimental (corrected) and calculated r a t i o s r e s p e c t i v e l y . The 6^ are the (corrected) experimental e r r o r s . The quantity h(6) (see table 3.5 and equations 3.12 through 3.14) was allowed to vary with 1 J r + . x 2 was evaluated at i t s approximate minimum i n ^ r ^ . and at 6 other points located symmetrically about that point. These values of *lv+ were separated by about 1/3 of the s t a t i s t i c a l f i t t i n g e r r o r . A parabola of the form: X2 ( n r + ) = a + b ( n r + ) + c ( n r + ) 2 (4.25) was f i t t e d to t h i s d i s t r i b u t i o n i n order to determine the precise l o c a t i o n of the minimum at f , defined by: r = (-b/2c) ± l / / c (4.26) The uncertainty i s the average displacement i n **r+ required to increase the x 2 by unity from i t s value at the minimum [CLI70]. A t y p i c a l X 2 p l o t f o r each data set i s shown i n fi g u r e 4.6, along with the r e s u l t s of 21 Figure 4.6 X 2 values generated from equation 4.24. S o l i d curves are f i t s to equation 4.25. Dashed l i n e s indicate unity increases i n x 2 from values at the respective minima. 153 f i t s to equation 4.25. The goodness of the p a r a b o l i c f i t s i n d i c a t e that the r a t i o s are l i n e a r i n 1 1 r + i n the regions considered. The x2's f o r the 38.6 MeV r a t i o are smaller than f o r the 47.7 MeV data; t h i s suggests that the p o t e n t i a l parameter set b e t t e r describes the former. The r e s u l t a n t values of f are given i n t a b l e 4.2 along w i t h the r e s u l t s of s i m i l a r c a l c u l a t i o n s u sing o p t i c a l parameter set 0. 4.3.4 Sensitivities to Higher Moments As was seen i n chapter I , low energy s c a t t e r i n g i s expected to be s e n s i t i v e only to the sma l l e s t even moment of the nuclear matter d i s t r i b u t i o n . To the extent that t h i s i s t r u e , a l l reasonable matter d i s t r i b u t i o n s should y i e l d i d e n t i c a l r e s u l t s i n our r a t i o a n a l y s i s . To t e s t t h i s , four d i f f e r e n t d e n s i t y d i s t r i b u t i o n s were used: 1) M o d i f i e d Gaussian (MG) 2) M o d i f i e d £aussian Core + Valence Nucleons (MGCV) 3) Constrained Fermi (CF) 4) Constrained Fermi Core + Valence Nucleons (CFCV) Each d i s t r i b u t i o n c o n s i s t s of a core d i s t r i b u t i o n ( 1 2 C ) to which the d i s t r i b u t i o n of s i n g l e nucleons i n a harmonic p o t e n t i a l i s added. The constrained Fermi d i s t r i b u t i o n i s an ord i n a r y Fermi d i s t r i b u t i o n w i t h quadrupole matter moment equal to that of the MG d i s t r i b u t i o n w i t h the same r+. These d i s t r i b u t i o n s are discussed i n more d e t a i l i n Appendix IV and summarised i n t a b l e A4.1. Table 4.2 RMS Proton Matter Radius Differences: H r + _ 1 2 r + Derived from Optical Model Calculations Parameter Set n r + - 1 2 r + (fm) 38.6 MeV 47.7 MeV 0 2 -.022 (29) -.031 (23) -.078 (15) -.071 (11) 4.3.4.1 Modified Density Results Table 4.3 shows the best values of f for the various forms of density. A l l of these c a l c u l a t i o n s u t i l i s e parameter set 2 i n the o p t i c a l p o t e n t i a l . (Figure A4.1 i l l u s t r a t e s the density d i s t r i b u t i o n s f o r 1 1r+=2.25 fm.) The core + valence d i s t r i b u t i o n s use a 1 2 C core f o r the c a l c u l a t i o n of U B and 1 0 B matter d i s t r i b u t i o n s (1 and 2 valence holes r e s p e c t i v e l y ) . The 1 2 C and 1 0 B RMS r a d i i are assumed to be those previously quoted. There i s some s e n s i t i v i t y to the exact form of the d i s t r i b u t i o n used, although each f i s l e s s than i t s f i t t i n g error from the mean. It may w e l l be that the Fermi d i s t r i b u t i o n i s not a reasonable form to use; the MG form i s c l o s e r to r e a l i t y f o r l i g h t n u c l e i . In any case, the r e s u l t s are not t o t a l l y independent of the higher r a d i a l moments of the assumed mass d i s t r i b u t i o n s ( i n c l u d i n g those higher than <r**>). The s e n s i t i v i t y to the precise form of the mass d i s t r i b u t i o n can be made more quantitative by the preparation of x 2 contour p l o t s . These plot s are generated by i n t e r p o l a t i n g x 2 values evaluated on a 7 by 7 g r i d centered near ?. Figures 4.7 and 4.8 show some r e s u l t s f o r the MGCV density; RCOP i s the core radius f o r the proton d i s t r i b u t i o n and RAD i s ^ r ^ . . Assuming r ( c o r e ) to be the independently known quantity, the v a r i a t i o n i n ? with r ( c o r e ) can be found. The r e s u l t s of that c a l c u l a t i o n are given i n table 4.4. Table 4.3 Results of Tests for Sensitivity of Derived Proton Matter Density Difference to Details of the Matter Distribution 12C, nB Experiment D i s t r i b u t i o n ( l l r + _ 1 2 r + ) ( f m ) 38.6 MeV 47.7 MeV MG -.031 (23) -.071 ( I D MGCV -.042 (20) -.077 (13) CF -.050 (21) -.083 (9) CFCV -.039 (22) -.088 (14) MEAN -.041 (11) -.080 (6) 157 Figure 4.7 Contour pl o t for MGCV density at 38.6 MeV. RAD Is n r + and RCOP i s the core radius. Figure 4 . 8 C o « o „ r p l o t f o r M G C V d e n s U y > t M e y _ ^ i s ^ ^ RCOP i s the core radius. Table 4.4 Dependence of Measured 11t on the Assumed Core Radius (Units are am per percent change) D i s t r i b u t i o n Data Set 38.6 MeV 47.7 MeV MGCV 2.9 -3.8 160 4.3.5 Sensitivities to Optical Parameters For the i s o r a t i o method of determining the matter radius of a nucleus to be considered v i a b l e , i t should be demonstrated that the r e s u l t s are not strongly affected by uncertainties i n the o p t i c a l parameters. This has been accomplished here through the use of the contour p l o t t i n g routine previously mentioned. Table 4.5 shows the r e s u l t s of these c a l c u l a t i o n s . Each parameter i s a l t e r e d from i t s value In set 2 for both n u c l e i , then a redetermination of the best f i t radius i s made. Figures 4.9 and 4.10 show the accompanying contour plots for R e ( c Q ) , which demonstrates by f a r the strongest e f f e c t upon the RMS radius. Notice, though, that parameter sets 0 and 2 d i f f e r by only a percent i n t h i s parameter. An o v e r a l l uncertainty of 5 % i n our knowledge of the o p t i c a l parameters would contribute uncertainties i n f of somewhat less than the s t a t i s t i c a l f i t t i n g e r r o r s . 4.3.6 Miscellaneous Sensitivities Table 4.6 demonstrates the s e n s i t i v i t y of the determinations of f to various miscellaneous parameters. The experimental uncertainty i n 1 0 r + (~2.5%) y i e l d s a 13 am uncertainty i n f . The boron target thickness uncertainty, 3 mg/cm2, r e s u l t s i n an ~8 am uncertainty i n ?. The uncertainty i n the mean energy at the target center i s of the order of 1/2 %, giving an uncertainty i n r of ~1 am. The shown 1 0 B model dependency r e s u l t s i f the Colorado p o t e n t i a l instead of SMC79 i s used i n the c a l c u l a t i o n of the correction factor k(9) discussed i n chapter III (table 3.5 and equations 3.12 through 3.14). Using Table 4.5 Sensitivities of 1 1 r to Optical Parameters. (Units are am per percent change) Parameter 38.6 MeV 47.7 MeV Re(b Q) 0.12 -0.12 Im(b 0) 0.054 0.028 Re(b 1) -0.68 -0.15 I n K b ^ t -0.011 0.002 Re(B Q) 0.013 -0.056 Im(B 0) 0.47 0.26 Re(c Q) 3.48 2.02 Im(c Q) 0.015 0.002 Re(c x) -1.08 -0.73 Im(c 1) 0.18 -0.20 Re(C 0) -0.29 -0.11 Im(C 0) 0.21 0.13 X -0.22 -0.34 Quadrature 3.77 2.19 t c e n t r a l value of 0.001 assumed when set 2 value i s 0 Figure 4.9 Contour plot for Re(c Q) at 38.6 MeV. RAD i s 1 1 r + and RECS i s R e ( c 0 ) . Figure 4.10 Contour plot f o r Re(c Q) at 47.7 MeV. RAD i s l x r + and RECS i s R e ( c 0 ) . Table 4.6 Miscellaneous Parameter Dependencies 1 2C, n B Experiment Quantity Dependence (38.6 MeV) (47.7 MeV) Units 10 r + = 1 o r _ -0.23 -0.22 am / (am change) 1 0 B co r r e c t i o n model dependency -18 -10 am Ratio Normalisation 11.8 8.9 am / (% change) Energy 1.2 2.5 am / (% change) 1 2 R + - 1 2 r _ « l l r _ 0.97 1.20 am / (am change) t h i s k(6) i n the SMC79 r a t i o c a l c u l a t i o n r e s u l t s i n the indicated change i n f . The e f f e c t of varying the assumed value of 1 2 r + i s close to 1 am per am change i n ^ r + i n d i c a t i n g that the r a t i o technique a c t u a l l y measures differences i n the r a d i i of n u c l e i and not the absolute r a d i i . Excluding t h i s dependence, then, table 4.6 implies uncertainties of about 23 am at 38.6 MeV and 18 am at 47.7 MeV. 4.3.7 Optical Model Dependence of Results It has been noted that there e x i s t , at present, several o p t i c a l p o t e n t i a l s which s u c c e s s f u l l y describe low energy pion-nucleus s c a t t e r i n g . The analysis here has used the SMC79 approach almost e x c l u s i v e l y . In the analysis of [BAR80], i t has i n f a c t been shown that the dependence upon the choice of o p t i c a l model i s not strong. Table 4.7 summarises the r e s u l t s . 4.4 CNO2 Experiments 4.4.1 Cross Section Fitting In the sections that follow, the SMC81 form of the p o t e n t i a l i s used. The protocol observed i n cross section f i t t i n g s on the CNO2 data sets also d i f f e r e d from that described i n section 4.3.1. In general, one might expect a global set of parameters to represent a p a r t i c u l a r cross section well, with the need for f i n e tuning of some subset of the parameters. The o p t i c a l parameters are heavily i n t e r c o r r e l a t e d , though, so that trends i n a cross section can be f i t t e d by changes i n a number of d i f f e r e n t parameters. Given a cross section and a r a t i o , however, one may f i n d that parameters whose v a r i a t i o n s produce equally good f i t s to the cross section Table 4.7 Deviation of  1 1 f from the Potential Averaged Means, in Colorado, LT and SMC79 Analyses.  1 1 1 i s the Average over Optical Parameter Sets 0 and 2. (Units are am) P o t e n t i a l Data Set 38.6 MeV 47.7 MeV SMC 0. 4. Colorado 5. -2. LT -5. -2. may, when used i n a r a t i o c a l c u l a t i o n , have markedly d i f f e r e n t e f f e c t s upon how well the r a t i o c a l c u l a t i o n f i t s the r a t i o data. It was found by studying the i r + 0 1 8 / 0 1 6 data at 48.3 MeV that changes i n the imaginary parts of b Q, B Q and C Q with nucleus affected the form of the r a t i o markedly, e s p e c i a l l y at the three forward angle points. Hence the cross section f i t t i n g s were performed as regular x 2 minimisations searching only upon ImbQ, ImBQ and ImCQ (Sets EIM50, EIM65; f i g u r e 4.11). Table 4.8 l i s t s these o p t i c a l parameter sets. Sets E50 and E65 are global Set E of reference SCM80 scaled q u a d r a t i c a l l y to the energies of these experiments. In the analysis of the CNO data, i t was intended to f i t cross sections to 1 2 C and 1 6 0 simultaneously to a r r i v e at an o p t i c a l p o t e n t i a l v a l i d throughout the range of atomic number encountered. For the 48.3 MeV data, t h i s was accomplished by varying a large number of the o p t i c a l parameters to a r r i v e at a s u i t a b l e p o t e n t i a l (Set E f 5 0 f ) . In the case of the 62.8 MeV data, i t was found to be very d i f f i c u l t to a r r i v e at such a parameter set. This was exacerbated by the lack of f a r forward angle data i n the cross section measurement; such data appear to be very important i n determining a number of the parameters ( e s p e c i a l l y the E 2 L 2 parameter, X) which contribute to the nuclear amplitude's interference with the Coulomb. The exact systematic change i n t h i s interference with nucleus does not appear to be c o r r e c t l y reproduced by the current form of the o p t i c a l p o t e n t i a l . The e f f e c t s of a lack of spin saturation i n the 1 2 C nucleus were considered [BR078] but found to produce e f f e c t s of at l e a s t an order of magnitude too small to reproduce the observed differences between c a l c u l a t i o n and experiment. (/> E <3 b X J 8 2-8 0 62.8 MeV 40 60 80 100 120 ANGLE (deg. cm.) 140 Figure 4.11 E l a s t i c d i f f e r e n t i a l cross sections of i r + on 1 6 0 at 62.8 MeV and 48.3 MeV showing SMC81 p o t e n t i a l c a l c u l a t i o n s with parameter sets E (E50 and E65; broken curves) and the f i t t e d sets EIM (EIM50 and EIM65). Matter densities are of the MG form. Table 4.8 Optical Parameter Sets (SMC81) 1 80, 1 60 Experiments O p t i c a l Parametert E50 EIM50 E65 EIM65 X 1.4 1.4 1.4 1.4 Reb 0 -.0427 -.0463 Imbg .0040 .0829 .0062 .0490 Rebj -.0920 -.0920 Imbj -.0013 -.0018 ReB 0 -.0048 -.0069 ImB0 .0285 -.1821 .0192 -.0930 Rec 0 .2472 .2513 Imc0 .0092 C .0156 Recj .1622 .1678 Imc^ .0043 .0069 ReC 0 .0447 .0481 ImC0 .0694 -.2230 .0543 -.0058 t expressed i n units of m^/CRc) . Entries l e f t blank are unchanged from the preceding column. The d i f f i c u l t i e s i n obtaining simultaneous f i t s to the 1 2 C and 1 6 0 62.8 MeV data are doubtless complicated by the increased importance of absorption and of the increasingly d i f f r a c t i v e shape of the e l a s t i c cross sections at the higher energies. The r a t i o analysis i n the 62.8 MeV CNO data w i l l r e f e r to four o p t i c a l parameter sets, which are summarised i n table 4.9: i ) Set EC65f: r e s u l t i n g from a f i t to the 62.8 MeV 1 2 C cross section i n varying ImbQ, B Q and C Q. i i ) Set E065f: as above but for the 62.8 MeV 1 6 0 data. i i i ) Set EBLE65f: r e s u l t i n g from a f i t to the data of Blecher et a l . [BLE83] at 65 MeV on 1 2 C (including points as f a r forward as 25° i n the angular d i s t r i b u t i o n ) . i v ) Set EBLE65co: a simultaneous f i t to the 1 2 C and 1 6 0 data of t h i s experiment, s t a r t i n g at set EBLE65f and thence varying ImbQ, Imc 0, B Q and C Q. Figures 3.22 and 3.23 showed the 1 2 C and 1 6 0 cross sections along with c a l c u l a t i o n s using the above parameter sets. Figure 4.12 i l l u s t r a t e s the data of Blecher et a l . , along with s i m i l a r c a l c u l a t i o n s . 4.4.2 Density Distribution Difference Analysis E a r l y nuclear s i z e studies with charged p a r t i c l e s a l l used simple a n a l y t i c forms to describe the r a d i a l density of the nucleus [BAR81]. The most common of these were the MG and the Fermi d i s t r i b u t i o n s , discussed i n sect i o n 4.3.3 and appendix IV. These forms have been used i n our studies Table 4.9 Optical Parameter Sets (SMC81) 1 2C, 1 , fN, 1 60 Experiments t E50 EF50f E65 EC65f E065f EBLE65f EBLE65co X 1.4 1.072 1.4 1.229 Reb 0 -.0427 -.0231 -.0463 -.0553 Imbg .0040 -.0272 .0062 .0267 .0323 .0466 -.0544 Reb x -.0920 -.0920 Imbj -.0013 -.0018 ReB Q -.0048 -.0219 -.0069 .0405 .0197 .0592 .0982 ImBg .0285 .1228 .0192 -.0324 -.0556 -.1192 .1896 Recg .2472 .1682 .2513 .2860 Imc 0 .0092 -.0923 .0156 -.0064 .0680 Rec x .1622 .1678 Imcj .0043 .0069 ReCg .0447 .1878 .0481 -.0366 -.0175 -.1022 -.1673 ImCg .0694 .3621 .0543 .0728 .0930 .1942 .1388 t o p t i c a l parameters are expressed i n units of m^ /Cnc) . E n t r i e s l e f t blank are unchanged from the preceding column. 1 0 ° I — I — I — I — I — I — I 10 30 50 70 90 110 130 9 (degrees, CM) Figure 4.12 E l a s t i c d i f f e r e n t i a l cross sections at 65 MeV for i r + on 1 2 C as measured by Blecher et a l . Calculations with various parameter sets are shown: S o l i d , Set EBLE65F; Long dashed, Set EC65F; Short dashed, Set E65; Dotted, Set EBLE65co. Matter de n s i t i e s are of the FL form. of the proton d i s t r i b u t i o n i n r e l a t i v e to 1 2 C . The use of these a n a l y t i c forms l i m i t s the analysis to the gross features of a r a d i a l d i s t r i b u t i o n ( f o r example the RMS radius) and ignores a probe's p o t e n t i a l f o r more d e t a i l e d r a d i a l study. More s e r i o u s l y , i t can r e s u l t i n r e s u l t s dependent upon the model used f o r the density, and i n underestimates i n the uncertainties i n the derived moments. In recent years, electron s c a t t e r i n g data have been analysed 'model independently' using nearly complete sets of orthogonal functions to represent the density d i s t r i b u t i o n s of nucleons within n u c l e i [FRI75, FRI78]. Fourier Bessel (FB) expansion i s e s p e c i a l l y well suited to e x t r a c t i o n of charge d i s t r i b u t i o n s from e l a s t i c e lectron s c a t t e r i n g data because the Coulomb charge form f a c t o r i s e s s e n t i a l l y the 3-dimensional Fourier transform of the charge d i s t r i b u t i o n . The expansion i s written: p(r) = P l ( r ) + I a n sin[ § 1 ] r " l r < R c c , \ r > R = P l ( r ) c where R c i s a cutoff radius. The orthogonality of the expansion allows the e x t r a c t i o n of the greatest amount of information per added degree of freedom. The s t a r t i n g density, p j , i s often taken to be a Fermi of MG form. The FB and also a Fourler-Laguerre (FL) expansion: P(r) = P l ( r ) + Z a n L l / 2 ( 2 ( r / a ) 2 ) e _ ^ r / a ) 2 (4.28) were used i n these experiments. The parameter a may be related to the strength of the nuclear p o t e n t i a l , and L i s the Laguerre polynomial. The mathematical properties of and s t a t i s t i c a l analysis techniques appropriate to these Fourier expansions are discussed i n appendix VIII. In 174 implementing them i n the o p t i c a l model a n a l y s i s , the normalisations were chosen such that: / P l ( r ) r 2 d r = T ; / p F ( r ) r 2 d r = 0 and / Ll/2 ( 2 ( r / a ) 2 ) L ^ 2 ( 2 ( r / a ) 2 ) e - 2 ( r / a ) 2 r 2 d r = fi a"(nx) ( 4 ' 2 9 ) m n mn where T i s the number of neutrons or protons and Pp i s the Fourier component of the density. The value of a Q was chosen to implement the second i n t e g r a l condition, and hence was never a free parameter. In addition, a constraint requiring p o s i t i v e d e n s i t i e s everywhere was imposed on the c a l c u l a t i o n s . The scale lengths a=1.5 fm and Rc=6 fm, of equations 4.27 and 4.28, were treated as free parameters. The 'reference' and 'unknown' n u c l e i had t h e i r d e n s i t i e s parameterised i n Fourier forms. The reference n u c l e i ( 1 2 C , 1 6 0 ) however are reasonably well described by the MG forms; i n the 1 8 0 / 1 6 0 analysis the Fourier c o e f f i c i e n t s a j of the reference were set to zero. The CNO analysis used FL parameterisations for the reference n u c l e i unfolded from the model independent charge density d i s t r i b u t i o n s [CAR80, REU82, NOR82]. These are tabulated i n table 4.10. The actual r a t i o f i t t i n g s were accomplished with the SMC81 o p t i c a l p o t e n t i a l accessed through the program PIFIT [GYL84]. PIFIT was modified to allow Fourier representations of both proton and neutron matter d i s t r i b u t i o n s simultaneously and to deal properly with charged nucleons. P a r a l l e l code was written to implement the FL density option. The x 2 of the f i t (equation 4.24) was again used as the goodness of f i t c r i t e r i o n . Table 4.10 Fourier Laguerre Parameterisations of Reference Densities Parameter 1 2 C 16 Q Proton MG RMS (fm) 2.3098 2.3985 2.5735 Neutron MG RMS (fm) 2.2725 2.3542 2.5203 FL parameters ( a - 3 7 " 2 normalisation) a p (fm) 1.48781 1.54542 a n (fm) 1.46380 1.51348 a 0 t a l -.11254 - 1 -.96695 - 2 a 2 -.69905 - 2 -.16652"2 a3 .15982 - 2 .92090 - 2 a5 .19844"2 •23373"2 o 6 t -.27029 - 1 2 -.22989~ 2 t a Q and a g are determined here by constraints on volume and RMS in t e g r a l s of the Fourier component. 4.4.3 1 80/ I 60 Ratio Experiments 4.4.3.1 Matter Distribution Determinations The density of the proton matter d i s t r i b u t i o n of 1 8 0 was taken to have a Fourier form (equation 4.27) where P l i s MG with r+=2.67 fm. The neutron d i s t r i b u t i o n of 1 8 0 was assumed, also, to be MG with radius obtained from the measured 1 8 0 - 1 6 0 neutron radius d i f f e r e n c e , .21 fm [JOH79]. The proton d i s t r i b u t i o n of 1 6 0 was assumed MG with r+=2.59 fm. The neutron d i s t r i b u t i o n of 1 6 0 was taken equal to i t s proton d i s t r i b u t i o n . The r a t i o s were f i t t e d by allowing a n (1 < n < nmax) and R c f o r the heavier nucleus to vary, subject to constraints which required that the Fourier sum's volume i n t e g r a l was zero and a l l de n s i t i e s uniformly p o s i t i v e . Error bands were plotted by c a l c u l a t i o n of the propogation of errors f or the density function and include the e f f e c t s of the o f f diagonal terms of the error matrix. To t h i s error was added a completeness error [FRI75], r e s u l t i n g from the truncation of the Fourier series at nmax terms and r e f l e c t i n g the i n a b i l i t y of the density expansion to model a complete set of d e n s i t i e s . The completeness error was determined by f i t t i n g d i r e c t l y to a r e a l i s t i c density ( i n t h i s case the proton matter density d i s t r i b u t i o n f o r 1 8 0 from e l e c t r o n s c a t t e r i n g [MIS79, NOR82]). The diffe r e n c e between the f i t and the r e a l i s t i c density was taken to be the completeness error . The r e s u l t s are presented g r a p h i c a l l y i n figures 4.13b and 4.14b f o r the two energies. Table 4.11 shows the RMS radius difference as derived under various assumptions about the density d i f f e r e n c e s . In chapter V i t w i l l be seen that these r a d i i agree well with those obtained with other probes. I O O O 5.0 2.5 0 -2.5 -5.0 - 7 . 5 h -10.0 I 5.0 2.5 0 1 i 1 1 (a) • IB ii W^/ 48.3 MeV • i i * -2.5 -5.0 -7.5 10.0 1 1 1 1 1 (b) i /IF • i 7 | | ir* Ratio * 3HLDQE ••Scatl.ring -i i i 2 3 4 5 RADIUS (fm) Figure 4.13 Proton matter density differences ( 1 8 P p ( r ) - 1 6 p p ( r ) ) derived from 48.3 MeV ir+ r a t i o s ( c f . fig u r e 3.26). (a) V a r i a t i o n i n derived Ap(r) with +10% v a r i a t i o n s (except as indicated) i n o p t i c a l parameters from Set EIM50 values: Broken curve (+5%), ImbQ; Dotted curve, ImBo; i ) , Rebo; i i i ) , Imco; i v ) , ImCg; i i ) , those remaining, (b) Best f i t FL density with Set EIM50 o p t i c a l parameters. Err o r envelope includes completeness error. E l e c t r o n s c a t t e r i n g matter den s i t i e s are shown f o r comparison. Figure 4.14 Same as fi g u r e 4.13 at 62.8 MeV ( c f . figure 3.29). (a) V a r i a t i o n i n derived Ap(r) with +10% va r i a t i o n s i n o p t i c a l parameters from Set E65 values: i ) , ImB0; i i ) , Imc 0; i v ) , X; i i i ) , those remaining, (b) Best f i t FB density with Set E65 o p t i c a l parameters. Error envelope includes completeness error. E l e c t r o n s c a t t e r i n g matter den s i t i e s are shown f o r comparison. Table 4.11 Proton Matter Distribution Differences: 1 80- 1 60 Energy O p t i c a l Parameter Set Density nmaxt, nx 6 < r 2 > l / 2 t t 62.8 E65 FB 4 0 .091(26) EIM65 FL 3 0 .079(12) E65 F L t t t 3 0 .100(37) E65 FL 3 0 .097(28) 48.3 EIM50 FL 3 0 .084(11) EIM50 FB 3 0 .087(15) t T y p i c a l values: Rc=6.0 fm (FB); a=1.5 fm (FL) tt Errors shown i n parenthesis are s t a t i s t i c a l , t t t In t h i s case only, the 1 8 0 neutron matter RMS radius was set equal to i t s proton matter RMS radius. 180 Analysis of e l a s t i c d i f f e r e n t i a l s c a t t e r i n g cross section r a t i o s of electrons on 1 8 0 and 1 6 0 [MIS79], has provided a p r e c i s i o n FB expansion f o r the charge d i s t r i b u t i o n d i f f e r e n c e between isotopes [N0R82]. To a r r i v e at a proton matter d i s t r i b u t i o n d i f f e r e n c e d i s t r i b u t i o n , the e f f e c t s of the proton charge form f a c t o r have been unfolded from the charge density d i f f e r e n c e by assuming a 3 Gaussian parameterisation of that f a c t o r , a f t e r Chandra and Sauer [CHA76]. The unfolding i s discussed at more length i n appendices IX and X. In figures 4.13b and 4.14b, the electron s c a t t e r i n g matter d i s t r i b u t i o n s are seen to be i n agreement with those derived from the pion r a t i o data. 4.4.3.2 O p t i c a l Parameter S e n s i t i v i t i e s The gradient p o t e n t i a l c h a r a c t e r i s i n g pion s c a t t e r i n g from the nucleus obscures understanding of the c o r r e l a t i o n s between the o p t i c a l p o t e n t i a l parameters and the matter density d i f f e r e n c e ; that i s , the manner i n which a change i n Ap(r) i s mimicked by changes i n the o p t i c a l p o t e n t i a l parameters. Figures 4.13a and 4.14a show the e f f e c t s of +10% changes i n the various o p t i c a l parameters upon the determination of the matter d i s t r i b u t i o n d i f f e r e n c e s . At 48.3 MeV ( f i g u r e 4.13a), the changes are from Set EIM50 parameters, where the imaginary terms ImbQ, ImB0, and ImC0 produce the most dramatic changes i n _ p ( r ) . In f a c t , Imb0 i s so i n f l u e n t i a l (a +5% change i s shown for t h i s v a r i a b l e at 48.3 MeV) that the Set E parameters w i l l not allow a reasonable f i t to Ap(r) for the 48.3 MeV data set. The 62.8 MeV Ap(r) ( f i g u r e 4.14a) i s s e n s i t i v e to the E 2 L 2 parameter, X, i n addition to the imaginary terms referred to above. This f i g u r e shows v a r i a t i o n s from Set E65 parameters, parameters which represent the r a t i o s much better than they do the cross sections (see f i g u r e 4.11). The precise r a d i a l s e n s i t i v i t y of the probe [FRI82, FRI83] i s determined by the d e t a i l s of i t s i n t e r a c t i o n with the nucleus, i n p a r t i c u l a r the dependence of the i n t e r a c t i o n upon the nuclear de n s i t i e s and t h e i r d e r i v a t i v e s . 4.4.4 l 2C, l l |H l 1 60 (CNO) Experiments 4.4.4.1 Matter Distribution Difference Determinations In t h i s a n a l y s i s , the proton and neutron matter d i s t r i b u t i o n s were assumed to be strongly correlated. This i s expected to be the case i n ll*N, e s p e c i a l l y , where the nuclear i n t r i n s i c and i s o t o p i c spins (T=0, J=l) suggest the presence of a quasi-deuteron outside of a ( r e l a t i v e l y i n e r t ) 1 2 C core. The c o r r e l a t i o n was assumed to be of the form [LAW80]: 1 / 2 a 2 = a 2 (1-6 ) where: (4.30) p n p (Z-l)e 2M ( Z - l ) 6 = - .0229 (fm) (4.31) p R 3 n 2 a* Z 3 This r e l a t i o n was taken to apply to the RMS r a d i i of the s t a r t i n g d i s t r i b u t i o n s Pj and the FL scale parameters a for a l l d i s t r i b u t i o n s ; i t provides a simple s c a l i n g of proton matter siz e r e l a t i v e to the neutron matter s i z e to allow f o r Coulomb repulsion e f f e c t s i n the nucleus. The FL parameterisations of the reference n u c l e i d i s t r i b u t i o n s (and the s t a r t i n g density f o r l l fN) are l i s t e d i n table 4.10. The forms used f o r ,(r) were MG i n a l l cases, as was the case i n section 4.4.3. The r a t i o s were f i t t e d by allowing the and a f o r the X^N nuclear proton d i s t r i b u t i o n to vary, again subject to constraints on the FL component to enforce zero o v e r a l l mass contribution and uniformly p o s i t i v e d e n s i t i e s . Each of the 1 4N: 1 2C and 1 6 0 : 1 4 N r a t i o s was analysed, with respect to the appropriate reference nucleus. A l l density d i f f e r e n c e d i s t r i b u t i o n s , however, are shown wrt 1 2 C . This allows easy intercomparison of the r a t i o r e s u l t s , and adds n e g l i g i b l e uncertainty except at small values (<1 fm) of the radius, where the 1 2 C and 1 6 0 electron s c a t t e r i n g r e s u l t s are most uncertain (and the pion r a t i o has i n s u f f i c i e n t momentum transfer to probe i n any case). Figures 4.15 through 4.18 present the r e s u l t s f o r 1**p+(r)- 1 2p+(r). Above about 1.8 fm, the density differences are i n t e r n a l l y consistent. Also shown are the r e s u l t s of a c a l c u l a t ion with a Self Consistent Single P a r t i c l e P o t e n t i a l (SCSPP) optimised i n the l*°Ca region [HOD85]. The electron s c a t t e r i n g matter density assumes a simple form f or the 1 < +N charge density [SCH75] and a FB model independent form [CAR80] for the 1 2 C density. No model independent e l e c t r o n s c a t t e r i n g charge density measurements for 1**N, or of the 1 2C, l l fN charge density difference for that matter, i s presently a v a i l a b l e . The pion r a t i o c a l c u l a t i o n indicates a s h i f t of the proton matter d i s t r i b u t i o n d i f f e r e n c e r e l a t i v e to that indicated by the ele c t r o n s c a t t e r i n g experiment. The l o c a t i o n of the maximum at about 2 fm i s the same as suggested by the SCSPP c a l c u l a t i o n . There i s a serious deficiency i n the electron s c a t t e r i n g data i n the present case. The a v a i l a b l e electron s c a t t e r i n g analyses were not designed to minimise systematic problems i n the data, and unce r t a i n t i e s i n the electromagnetic form factors are present i n d i f f e r e n t ways for the two 40 20 o 0 CM a . 'tf-20 h . Q . - 4 0 — i 1 r 48.3 MeV 0 1 2 3 4 5 RADIUS (fm) Figure 4.15 Proton matter density differences ( l l f p p ( r ) - 1 2 p p ( r ) ) derived from 48.3 MeV i r + , 1 4 | N / 1 2 C r a t i o s ( c f . f i g u r e 3.24). Best f i t FL density with Set Ef50f o p t i c a l parameters. E r r o r envelope i s s t a t i s t i c a l . Dashed curve i s di f f e r e n c e of model independent electron s c a t t e r i n g derived proton matter density i n 1 2 C and the best a v a i l a b l e MG density for 1 1 +N. Also shown are SCSPP ca l c u l a t i o n s with a 'standard' nuclear p o t e n t i a l developed i n the 4 0 C a region [H0D85]. 184 40 ro 20 0 CN Q. Q . ' ^ -20 —40 T 1 1 r 48.3 MeV •» A A 0 1 2 3 4 5 RADIUS (fm) Figure 4.16 Same as f i g u r e 4.15 f o r ( l l t p p ( r ) - 1 2 P p ( r ) ) derived from 48.3 MeV ir+, 1 60/ 1 J tN r a t i o s with Set EF50f ( c f . figure 3.25). 185 40 to I 20 0 CM '_-20 Q . 40 T 1 1 1 r 62.8 MeV 7T + Ratio - - (e.e) ' SCSPP J L 0 1 2 3 4 5 RADIUS (fm) Figure 4.17 Same as f i g u r e 4.15 f o r ( 1 4 p p ( r ) - 1 2 p p ( r ) derived from 62.8 MeV ir+.^N/^C r a t i o s with Set EC65f ( c f . figure 3.27). 40 20 0 CM Q. L-20 Q . - 4 0 0 62.8 MeV 7T + Ratio - - (e,e) * SCSPP j i i i i 1 2 3 4 5 RADIUS (fm) Figure 4.18 Same as f i g u r e 4.15 f o r (1J*p ( r ) - 1 2 p D ( r ) ) derived from 48.3 MeV ir+, 1 60/ l l*N r a t i o s with Set E065f ( c f . fig u r e 3.28). n u c l e i . This alone w i l l produce large model er r o r s . Furthermore, use of the simple 2 parameter MG form i n the analysis of the electron s c a t t e r i n g 1J*N data may give misleading r e s u l t s , e s p e c i a l l y at small nuclear r a d i i , through lack of completeness of the density form. In short, though no uncertainties are shown i n the e l e c t r o n s c a t t e r i n g matter den s i t i e s (none are published), of figures 4.15 through 4.18, the r e l a t i v e errors are doubtless f a r larger than those of the electron s c a t t e r i n g matter d e n s i t i e s shown i n figures 4.13 and 4.14 for the oxygen isotopes, where a p r e c i s i o n analysis was undertaken. Table 4.12 shows derived RMS r a d i i f o r various parameter sets and r a t i o s . These r a d i i are s e l f consistent; i n chapter V i t w i l l be seen that they are also consistent with the r e s u l t s from other probes. No c a l c u l a t i o n has been shown with set EBLE65co; those c a l c u l a t i o n s would not converge near the shape of the experimental r a t i o s , thereby suggesting an inappropriate o p t i c a l parameter set f o r 1 1 +N. Figure 4.19 compares the shape of the cross section r a t i o (Set EF50f) at 48.3 MeV with the (best f i t ) FL matter density and a simple MG density of the same RMS radius. It i s evident that a s i z e determination f o r t h i s nucleus with a simple MG form would y i e l d an incorrect value for the RMS radius. I n c i d e n t a l l y , the e f f e c t of the added Fourier terms (figures 4.15 through 4.18) i s to s h i f t the nucleon density towards the center of the nucleus, r e l a t i v e to what might otherwise have been expected. 4.5 Summary Chapter IV has seen discussion of the o p t i c a l model analysis of the cross section and r a t i o data presented i n chapter I I I . Some overview of the Table 4.12 Proton Matter Distribution Differences: 1 2C, l l fN, 1 60 Experiment Energy O p t i c a l Parameter Set Ratio Density nmaxt,nx 6 < r 2 > 1 / 2 t t 48.3 EF50f FL 5 -3 .106(11) 62.8 EC65f 3 -3 .002(94) 62.8 EBLE65f 3 -3 .017(75) 48.3 EF50f 160.14N 4 -3 .162(11) 62.8 E065f 4 -3 .198(51) 62.8 EBLE65f 3 -3 .157(12) T y p i c a l values: Rc=6.0 fm (FB); a=1.5 fm (FL) Errors shown i n parenthesis are s t a t i s t i c a l . Difference i s wrt the reference: RMS radius of heavier n u c l e i l e s s that of the l i g h t e r . 30 50 70 90 110 130 150 6 (degrees, CM) Figure 4.19 Ratio c a l c u l a t i o n (SMC81, Set Ef50f) with the best f i t density ( c f . figure 4.15) and a MG ( s o l i d ) density with the same RMS radius. In this case use of s o l e l y the MG form would lead to a serious overestimation of the 1 4 N RMS radius. 190 a v a i l a b l e o p t i c a l models has been made, with p a r t i c u l a r emphasis on the SMC coordinate space p o t e n t i a l . The 1 2 C , n B experiments used the SMC79 form of the p o t e n t i a l to extract differences i n proton matter r a d i i between those n u c l e i , assuming a modified Gaussian (MG) density form. The systematic uncertainties i n these extractions were examined: i n p a r t i c u l a r , s e n s i t i v i t i e s to density form (tables 4.3 and 4.4) and o p t i c a l p o t e n t i a l parameter values (table 4.5) were found to be of s i z e comparable to the s t a t i s t i c a l errors (=.020 fm). The analysis did not incorporate Fourier descriptions of the d e n s i t i e s : no proton matter d i f f e r e n c e d i s t r i b u t i o n analysis (as a function of nuclear radius) was attempted. This was deemed the wisest course as the 20% 1 0 B content i n the boron target would have made i n t e r p r e t a t i o n of such analyses d i f f i c u l t . The 1 8 0 , 1 6 0 r a t i o analyses used the SMC81 form of the o p t i c a l p o t e n t i a l , simple assumptions about the neutron matter density differences between the isotopes, and a Fourier Laguerre (FL) parameterisation of the proton matter density d i f f e r e n c e between the isotopes. The e f f e c t s of o p t i c a l parameter uncertainty was discussed and noted to be of the same order as the error bands ( s t a t i s t i c a l + completeness) f o r the density d i f f e r e n c e s . The 1 2C, 1 1 +N, 1 60 r a t i o s were analysed with the SMC81 p o t e n t i a l to derive the density d i s t r i b u t i o n differences between the n u c l e i under the assumption of correlated neutron-proton matter density d i s t r i b u t i o n d i f f e r e n c e s . The density differences were evaluated wrt the appropriate reference ( 1 2 C f o r 1Ha/12a and 1 6 0 f o r 16o/lka), but i l l u s t r a t e d wrt 1 2 C . No p r e c i s i o n electron s c a t t e r i n g measurement of the 1 I +N charge density d i s t r i b u t i o n , with which to compare, i s currently a v a i l a b l e . Hence comparison of the pion r a t i o derived proton matter density differences with those from e l e c t r o n s c a t t e r i n g i s d i f f i c u l t and perhaps misleading. The current r e s u l t s suggest that the proton matter i n 1J*N i s concentrated c l o s e r to the nuclear center than indicated by the a v a i l a b l e (non-precision) e l e c t r o n s c a t t e r i n g data. The concentration i s at about 2 fm: close to the radius, but to a greater extent than that, predicted by a SCSPP c a l c u l a t i o n . In each of the analyses, f i t t i n g of the o p t i c a l p o t e n t i a l parameters (but i n t h i s case not the density d i s t r i b u t i o n parameters) to the reference nucleus' d i s t r i b u t i o n of e l a s t i c d i f f e r e n t i a l cross sections was made. 192 Chapter V Discussion and Summary 5.1 Overview The r a t i o s of p o s i t i v e pion e l a s t i c s c attering d i f f e r e n t i a l cross sections have been used to extract proton matter density information on the n u c l e i n B , 1 I +N and 1 8 0 r e l a t i v e to the 'reference n u c l e i ' 1 2 C and 1 6 0 . These measurements have been performed at a v a r i e t y of energies between 35 and 65 MeV. Two types of apparatus were used. A I T + stopping telescope was used i n experiments on 1 2C, 1 1B' at 38.6 MeV and 47.7 MeV. Ratio and cross section data at 48.3 and 62.8 MeV were taken with the QQD magnetic spectrometer. The experimental techniques were designed to minimise systematic uncertainty i n the r a t i o data (systematic uncertainties which i n v a r i a b l y occur i n absolute cross section measurement). A l l measurements were performed i n the meson h a l l of the TRIUMF f a c i l i t y . In Chapter I, the expected s e n s i t i v i t y of i r + scattering to proton d i s t r i b u t i o n excess was outlined. I t was shown that, i n the low energy regime, the proton RMS matter radius, r+, a f f e c t s the normalisation of a cross section r a t i o c a l c u l a t i o n i n a simple way. This allows measurement of, at l e a s t , r+. The r a t i o s were analysed with a v a r i e t y of o p t i c a l p o t e n t i a l s to derive ( r e l a t i v e ) RMS r a d i i and, i n some cases, r a d i a l differences of the proton matter density d i s t r i b u t i o n s . The o p t i c a l p o t e n t i a l s were optimised by f i t t i n g to the e l a s t i c cross section data. In that which follows, the analysis which has been described i n some d e t a i l i n e a r l i e r chapters w i l l be summarised, and the r e s u l t s of the present experiments compared to those arr i v e d at v i a other nuclear density measurement techniques. 5.2 O p t i c a l P o t e n t i a l F i t t i n g I m p l i c i t i n the analysis discussed i n t h i s thesis has been the existence of an o p t i c a l p o t e n t i a l which produces e l a s t i c s c a t t e r i n g cross sections i n reasonable agreement with the measured cross sections. There are a number of such p o t e n t i a l s , but none of them reproduces with complete success the e l a s t i c cross sections of t h i s energy regime e n t i r e l y from f i r s t p r i n c i p l e s . As a r e s u l t , one must often resort to f i t t i n g the p o t e n t i a l parameters to experimental cross section data to a r r i v e at reasonable agreement between that data and the c a l c u l a t i o n . The s i t u a t i o n i s somewhat ameliorated i n the consideration of cross section r a t i o s , since the o p t i c a l model better predicts trends i n , rather than absolute, cross sections. In general, the closer that an o p t i c a l parameter finds i t s o r i g i n i n a f i r s t p r i n c i p l e , the les s i t should vary i n an o p t i c a l p o t e n t i a l f i t to cross section data. For example, the r e a l parts of the s and p wave sca t t e r i n g length should be close to t h e i r free nucleon values. In the f i t t i n g described i n t h i s t h e s i s , t h i s has for the most part been the case. An i n t e r e s t i n g r e s u l t common to the cross section f i t t i n g herein, however, i s an increase i n the magnitude of the r e a l part of the s wave sca t t e r i n g length, r e ( b Q ) , within the nucleus. This i s not s u r p r i s i n g , since b Q ( i e : B Q) i s an e f f e c t i v e s c a t t e r i n g length. Only s l i g h t l y repulsive when considering ir nucleon s c a t t e r i n g , the ( i s o s c a l a r ) s wave sca t t e r i n g becomes much more repulsive when the e f f e c t s of nuclear short range c o r r e l a t i o n s , i n p a r t i c u l a r P a u l i repulsion, are considered [ERI70]. The true absorption parameters, i n most cases, have been greatly increased i n magnitude, i n d i c a t i n g that a d d i t i o n a l studies of the e f f e c t s of true absorption i n the nucleus [M0I84] continue to be warranted. The s e n s i t i v i t y of the cross sections i n the Coulomb-nuclear and s-p interference regions to the E 2 L 2 parameter, X, can be seen i n a number of f i g u r e s , i n p a r t i c u l a r f i g u r e 4.12 (table 4.9). In order to determine t h i s parameter well, and hence avoid the e f f e c t s of some of i t s c o r r e l a t i o n s [BR081, SEK83, SEK83a] to the others (In p a r t i c u l a r the p wave sca t t e r i n g volumes), very small angle data on reference cross sections should be accumulated. In p a r t i c u l a r , the trend with nucleus i n t h i s angular region should be studied at energies such as 65 and 80 MeV with the aim of improving the o p t i c a l model d e s c r i p t i o n . The o p t i c a l model at 65 MeV appears to be encountering some problems. Experiments [JEN83, SOB84c] have indicated that A-hole processes should not contribute s i g n i f i c a n t l y , at t h i s energy, to the e l a s t i c s c a t t e r i n g . Some other processes may need to be considered to parameterise the o p t i c a l p o t e n t i a l s a t i s f a c t o r i l y i n t h i s regime. 5.3 1 2 C , n B Experiment The 1 2 C , 1 1 B experiment was the f i r s t of the i r + r a t i o experiments. There was, i n f a c t , one e a r l i e r experiment [DYT78], but i t suffered from poor s t a t i s t i c s and was not conceived, nor did i t attempt, to extract proton matter density d i s t r i b u t i o n information. In the current experiment, analysis was made to extract the RMS radius of proton matter i n 1 X B ( n r + ) r e l a t i v e to that i n 1 2 C ( 1 2 r + ) . A modified Gaussian (MG) form f o r the matter den s i t i e s was assumed. A rigourous analysis required the use of a r e l i a b l e pion o p t i c a l p o t e n t i a l ; the SMC79 p o t e n t i a l was chosen. Two parameter sets were used i n the analysis; the r e s u l t s were presented i n table 4.2. The r e s u l t s yielded a weighted average for 01r+-12r.(.) of -.064 (8). The unweighted value i s -.050 (28). The value ( n r + - 1 2 r + ) = -.064 (28) we then take as being the best value. The error r e f l e c t s , to some degree, both s t a t i s t i c a l and systematic u n c e r t a i n t i e s . A simple A1''3 behaviour [COL67a] of the RMS charge radius gives: < r Q > 1 / 2 = 0.82 (N+Z) 1 / 3 + 0.58 (5.1) which predicts a value for the matter radius change of Ar+=0.058 fm between the isotones. This agrees, within the experimental uncertainty, with the value measured i n t h i s experiment. A number of tests were performed to assure the i n t e g r i t y of the r a t i o method. These Included **r+ evaluations using various density d i s t r i b u t i o n s and o p t i c a l parameter sets, as well as various values for a few miscellaneous parameters. In each of these i t was found that the best f i t value of 1 1 r + , ( i . e . : * * ? ) , was not changed by more than the quoted s t a t i s t i c a l f i t t i n g e r rors. This i s not to say that n u l l e f f e c t s were seen. There e x i s t at present several other values of the charge radius of 1 1 B to which the r e s u l t s of t h i s experiment may be compared. (The charge radius, r e c a l l , i s obtained by the addition of the si n g l e proton radius to the nuclear proton matter radius i n quadrature.) Table 5.1 shows that the various values are within reasonable agreement, although that given by R i s k a l l a quotes no error. The r e s u l t of the present experiment i s the most precise to date. 196 Table 5.1 Measured Values of 1 1r+- 1 2r+ for U B Experiment Method 1 1 r + - 1 2 r + (fm) Present [BAR80] I T + r a t i o -.064(28) S t o v a l l et a l . t [ST066] (e",e-) -.05(13) Riskalat [RIS71] (e",e-) -.107 Alkazov and Domchenkovt [ALK83] (e",e-) -.032(33) Auerbach [AUE80] HF I -.048 HF II -.077 O l i n et a l . * [0LI81] u - atoms -.11(4) t Assuming 1 2 C charge radius of 2.470(5) fm [RUC82] and proton RMS radius of .7754 fm [CHA76]. * Assuming 1 2 C charge radius of 2.4832(18) fm [RUC82] 197 The present r e s u l t f o r the matter radius of ^B, r e l a t i v e to that of 1 2 C , leads to no discrepancies with e x i s t i n g charge radius information. This i s true at l e a s t to the p r e c i s i o n quoted. Note that the TRIUMF r e s u l t allows f o r , i n some measure, unce r t a i n t i e s i n parameter sets and the model used. The f i n a l quoted uncertainty i s l a r g e l y a function of a spread of the r e s u l t s over the data sets with energy (38.6, 47.7 MeV). This dependence i s apparently a random e f f e c t , e s p e c i a l l y given the r e s u l t s of the following sections. 5.4 CNO2 Experiments 5.4.1 1 8 0 , 1 6 0 Experiments In these experiments, the proton matter density of 1 8 0 was measured r e l a t i v e to that of 1 6 0 . The analysis incorporated a Fourier expansion f o r the density d i f f e r e n c e , applied to the proton matter d i s t r i b u t i o n of the 1 8 0 . The reference proton and the neutron de n s i t i e s were assumed to be of simple MG form with previously measured r a d i i . Calculations were performed with the SMC81 p o t e n t i a l . The best (average) value was found to be ( 1 8r+- 1 6r+)=.084±.008, with the error r e f l e c t i n g the spread i n the values seen i n table 5.2 Figures 4.13 and 4.14 presented derived proton density d i f f e r e n c e d i s t r i b u t i o n s : A p p ( r ) = 1 8 p p ( r ) - 1 6 p p ( r ) . Table 5.2 shows the RMS r a d i i . The T T + r e s u l t s agree with those of electron s c a t t e r i n g [MIS79, N0R82J and muonic atom [BAC80] experiments. There i s l i t t l e s e n s i t i v i t y to energy or assumed 1 8 0 neutron density. The proton charge form factor [CHA76] has been unfolded from the el e c t r o n scattering charge d i s t r i b u t i o n s . Table 5.2 Proton Matter Distribution Differences: 180-160 Energy Op t i c a l Parameter Set Density nmaxt,nx 6<r2>l/2 t t 62.8 E65 EIM65 E65 E65 FB FL F L t t t FL 4 0 3 0 3 0 3 0 .091(26) .079(12) .100(37) .097(28) 48.3 EIM50 EIM50 FL FB 3 0 3 0 .084(11) .087(15) Weighted Average Unweighted Average .084(7) .089(5) P r e c i s i o n E l e c t r o n Scattering [MIS79, NOR82] [SCH75] .077(5) .075(8) Muonic Atom Studies [BAC80] .079(5) SCSPP ( 1 6 0 + ( s d ) 2 ) ( 1 6 0 + (psd)) [BR079a [BR079a BR079b] BR079b] .011 .077 Semi Empirical Parameterisation [WES84] .107 Simple Parameterisation (equation 5.1) [C0L67a] .088 t T y p i c a l values: Rc=6.0 fm (FB); a=1.5 fm (FL) tt Errors shown i n parenthesis are s t a t i s t i c a l , t t t In t h i s case only, the 1 8 0 neutron matter RMS radius was set equal to i t s proton matter RMS radius. I t i s i n t e r e s t i n g to note [BR079a, BR079b], that the shape of the charge density d i s t r i b u t i o n differences between n u c l e i appears to be generated by the e f f e c t s of in t e r a c t i o n s i n v o l v i n g up to 6 - p a r t i c l e 4-hole configurations i n the 1 8 0 wave function. ( i e : The 1 6 0 core i s not i n e r t . ) The T T + data error envelopes encompass those r e s u l t i n g from a ±10% o p t i c a l parameter set uncertainty, provided that otherwise poorly determined imaginary terms i n the po t e n t i a l are derived from cross section f i t t i n g . The pion probes the nucleus at r a d i i greater than 1.5 fm, l i m i t e d by momentum transfer, and samples density differences with a r a d i a l r e s o l u t i o n characterised by X-1.5 fm. The r a d i a l s e n s i t i v i t y of the probe [FRI75, FRI83] i s determined by d e t a i l s of the in t e r a c t i o n ' s dependence on the nuclear d e n s i t i e s and t h e i r d e r i v a t i v e s . 5.4.2 CNO Experiments In these experiments the reference n u c l e i were taken as 1 2 C and 1 6 0 . the proton matter density d i s t r i b u t i o n .of ^ N was found with respect to these. The reference proton matter de n s i t i e s were of FL forms derived from p r e c i s i o n electron s c a t t e r i n g experiments. Neutron matter de n s i t i e s were derived from these with a simple assumption about Coulomb repulsion e f f e c t s which Increase the proton d i s t r i b u t i o n radius ( t y p i c a l l y by 1.5 %) r e l a t i v e to the neutron radius. Calculations were made with the SMC81 p o t e n t i a l . The r e s u l t s f o r the RMS r a d i i are retabulated i n table 5.3, along with r e s u l t s obtained with other probes. At 48.3 MeV, the i r + r a t i o r e s u l t s were extracted with a p o t e n t i a l parameter set f i t t e d simultaneously to 1 2 C and 1 6 0 e l a s t i c cross section d i s t r i b u t i o n s . The r e s u l t i n g values f o r ( l l + r + - 1 2 r + ) and ( 1 6 r + - 1 ' * r + ) are Table 5.3 Proton Matter Distribution Differences: 12C,1,fN,160 Experiment Energy O p t i c a l Parameter Set Ratio Densityt (nmax,nx) U r + - 1 2 r + (fm)tt 1 6 r + - 1 J * r + (fm)ft 48.3 62.8 62.8 EF50f EC65f EBLE65f l l*N: 1 2C FL(5,-3) (3,-3) (3,-3) .106(11) .002(94) .017(75) .148(15)* .252(95)* .237(75)* 48.3 62.8 62.8 EF50f E065f EBLE65f 1 6 0 : U N (4,-3) (4,-3) (3,-3) .092(15)* .056(52)* .097(17)* .162(11) .198(51) .157(12) Weighted Average Unweighted Average .098(8) .062(44) .159(7) .192(44) Muonic Atom [RUC82, SCH80] [SCH80, DUB74] .082(11) .159(24) El e c t r o n Scattering [SCH75] [ALK83]** .074(21) .139(33) .180(22) .116(34) Semi Empirical Parameterisation [WES84] .124 .099 Simple Parameterisation [COL67a] .106 .096 t T y p i c a l values: Rc=6.0 fm (FB); a=1.5 fm (FL) t t Errors shown i n parenthesis are s t a t i s t i c a l . * Value assumes the best value of 1 6 r + - 1 2 r + of .254(10) fm [KIM78, SIC80, RUC82]. ** Assumes 1 2 C charge radius of 2.470(5) fm [RUC72]. 1 6 0 charge radius of 2.709(9) fm [KIM78, SIC80] 201 i n t e r n a l l y consistent, and consistent with the r e s u l t s of the other probes. The c a l c u l a t e d r a t i o s f i t the data w e l l . The r e s u l t s at 62.8 MeV were derived with o p t i c a l parameter sets f i t t e d separately to 1 2 C and 1 6 0 e l a s t i c cross section d i s t r i b u t i o n s . The derived RMS r a d i i have larger errors than do those obtained at 48.3 MeV: t h i s i s merely a s t a t i s t i c a l e f f e c t . The form of the calculated r a t i o s do not f i t the r a t i o data as well at t h i s energy, probably because of inadequacies i n the o p t i c a l p o t e n t i a l . The best value f o r ( 1 6r +- l l»r+) i s .098 ± .044 fm and f o r ( 1 1 + r + - 1 2 r + ) , .159 ± .044 fm. The derived density diff e r e n c e d i s t r i b u t i o n s l t * p ( r ) - 1 2 p ( r ) are shown i n figures 4.15 through 4.18. The errors there do not r e f l e c t completeness e r r o r s . The s a l i e n t feature common to the density d i s t r i b u t i o n differences i s a s h i f t i n the proton matter density towards the i n t e r i o r of the nucleus, r e l a t i v e to that suggested by the a v a i l a b l e electromagnetic measurements. Such an e f f e c t could be created by d e t a i l s (such as might be produced by phenomena such as "core p o l a r i s a t i o n " ) i n the 1 I +N nuclear density which are orthogonal to the density form used i n the analyses of electron e l a s t i c s c a t t e r i n g from ^N. An analysis of 1 1 +N(e,e) 1 I +N experiments with a FB density form f o r the proton charge density (model independent analysis) might help to resolve the apparent discrepancy. Such an analysis would, however, be complicated by the spin and lack of s p h e r i c i t y of the 1 I +N nucleus. The T T + r a t i o derived de n s i t i e s agree with one another i n the r a d i a l region where momentum transfe r of the pion e l a s t i c s c a t t e r i n g process allows s e n s i t i v i t y to be exhibited: r ? 1.5 to 2 fm. Inside of t h i s , the 2 0 2 s t a t i s t i c a l e r r o r bands do not encompass the v a r i a t i o n s i n the derived d e n s i t i e s , as might be expected. The completeness error i s a s i g n i f i c a n t c o n t r i b u t i o n to the o v e r a l l error band e s p e c i a l l y at small nuclear r a d i i . Furthermore, dependence of the o p t i c a l p o t e n t i a l upon the density d e r i v a t i v e s may appear to enhance the s e n s i t i v i t y of the pion i n the center of the nucleus over what I t p h y s i c a l l y can be, as the de r i v a t i v e s of the Fourier components are t y p i c a l l y large i n t h i s region. 5.5 Summary The subject of t h i s thesis has been the study of the use of p o s i t i v e pion e l a s t i c s c a t t e r i n g d i f f e r e n t i a l cross section r a t i o s i n the determination of nuclear proton matter d i s t r i b u t i o n differences and t h e i r moments. The measurements have been r e l a t i v e to n u c l e i whose matter d i s t r i b u t i o n s and absolute cross sections have been used as references. The use of the r a t i o of cross sections, rather than the absolute cross sections themselves, minimises the e f f e c t s of uncertainties i n the understanding of the pion nuclear i n t e r a c t i o n i n our extraction of density d i f f e r e n c e information. Furthermore, the measurement of cross section r a t i o s i s i n s e n s i t i v e to many systematic experimental e f f e c t s encountered i n the measurement of the absolute cross sections. We have seen that RMS r a d i i extracted from the pion r a t i o s at low energy are consistent (within 1 standard deviation) with the r e s u l t s of other methods. The proton matter radius differences which we obtain are. as follows: ( n r + - 1 2 r + ) = -.064 (28) fm ( l t r + _ 1 2 r + ) = > 0 9 8 ( 4 4 ) f m ( 1 6 r + _ l t r + ) = a 5 9 ( 4 4 ) f m ( 1 8 r + _ 1 6 r + ) = > 0 8 4 ( 7 ) f m ( 1 80,16 0) {-.032(33) [ALK83] } { .074(21) [SCH75] } { .180(22) [SCH75] } { .077 (5) [MIS79] } The errors r e f l e c t s t a t i s t i c a l and, to a large extent, systematic un c e r t a i n t i e s i n the q u a n t i t i e s . The best e l e c t r o n s c a t t e r i n g r e s u l t s are shown i n braces. Analyses of the 1 8 0 , 1 6 0 experiments i n terms of Fourier proton matter d e n s i t i e s agree well with p r e c i s i o n "model independent" electron s c a t t e r i n g r e s u l t s , i n the region i n which the pion can be s e n s i t i v e to the nuclear proton matter d i s t r i b u t i o n s . S i milar analyses of the 1 2 C , 11+N, 1 6 0 experiments indi c a t e a s h i f t i n the proton matter density towards the nuclear center, r e l a t i v e to that suggested by ele c t r o n s c a t t e r i n g experiments. We r e i t e r a t e , however, that analyses of these electron s c a t t e r i n g experiments were not model independent, and may well generate d e n s i t i e s which are in c o r r e c t at small radius (high momentum transfer: r K. 2 fm.) 5.6 Epilogue We present these r e s u l t s as testimony to the a b i l i t y of the i r + to probe App(r) r e l i a b l y . We observe some o p t i c a l parameter s e n s i t i v i t y . This i s minimal, e s p e c i a l l y when cross section f i t t i n g i s used to determine the reference nucleus o p t i c a l parameters. S t a r t i n g with the el e c t r o n s c a t t e r i n g d e n s i t i e s one might have proceeded to study i s o t o p i c dependence i n the o p t i c a l p o t e n t i a l [DYT78, BLE83]. Without r e l i a b l e neutron density measurements such studies are of l i m i t e d scope. This strengthens the 204 o r i g i n a l motivation of t h i s work found i n the p r o v i s i o n of corroboration f o r the I T " neutron density measurements [J0H79, GYL85]. In f a c t , the i n d i c a t i o n of the present work i s that i t i s complications i n the o p t i c a l p o t e n t i a l due to changes i n the Coulomb interferences which generate much of the observed o p t i c a l model s e n s i t i v i t y , complications whose importance i s greatly diminished when negative pions are used i n the study of neutron d i s t r i b u t i o n d i f f e r e n c e s . This i s an e x c i t i n g prospect, as few other techniques access the moments of neutron d i s t r i b u t i o n s i n such an elegant manner. Nevertheless, i t i s clear that our understanding of the pion nucleus i n t e r a c t i o n i s s t i l l not f a r advanced from i t s infancy. In p a r t i c u l a r , higher order absorptive and c o r r e l a t i v e e f f e c t s must be rigorously q u a n t i f i e d . Experiments currently exploring more exotic reaction channels, such as those of double and si n g l e charge exchange [ALT85, C0084], i n e l a s t i c s c a t t e r i n g [TAC84], and t o t a l absorption cross section measurements [NAV83] may well provide e s s e n t i a l i n s i g h t i n these d i r e c t i o n s . Together with the high p r e c i s i o n electron s c a t t e r i n g charge density measurements now becoming a v a i l a b l e , r e l a t i v e neutron and proton density difference information derived from experiments such as described herein should aid the new generation of experiments to advance the study of many aspects of the nuclear i n t e r a c t i o n . 205 L i s t of References AJZ80 F. Ajzenberg-Selove, Nucl. Phys. A336 (1980) 1 AJZ82 F. Ajzenberg-Selove, Nucl. Phys. A375 (1982) 1 ALK83 G.D. Alkhazov and O.A. Domchenkov, Sov. 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L e t t . 63B (1976) 43 214 TH080 A.W. Thomas, private communication TH081 A.W. Thomas, Nucl. Phys. 354_ (1981) 51c WAD76 E.A. Wadlinger, Nucl. Inst. Meth. 134_ (1976) 243 WEA73 Handbook of Chemistry and Physics, ed. R.C. Weast (C.R.C. Press, Cleveland, 1973) p.B-248 WES84 E. Weslowski, J . Phys. G: Nucl. Phys. 10_ (1984) 321 WIE83 C. Wiedner, private communication WIE84 U. Wienands, pr i v a t e communication 215 Appendix I Subscripts and Superscripts Throughout t h i s thesis t r a i l i n g subscripts '+', and '0' r e f e r to proton, neutron and charge d i s t r i b u t i o n s r e s p e c t i v e l y . Leading superscripts indicate atomic number and leading subscripts are atomic charge (Z). Unless ambiguities r e s u l t , leading subscripts are omitted. The abbreviations r r )=<r 2> n 1 / 2 r e f e r to Root Mean Square (RMS) r a d i i weighted by the appropriate density: n e {+ or 'p', protons; - or 'n' ,neutrons; 0 or 'ch', charge}. This notation serves to remind one of the predominant s e n s i t i v i t y of TT+ to protons and TT- to neutrons. Appendix II Powder Boron Target Mass Measurements The measurement was performed by making use of the c o r r e l a t i o n between the range of the electrons emitted by a 8 - source and the mass thickness of the boron target. A ^^Ru source was chosen as appropriate for the purpose. 1 0 6 R u decays to 1 0 6 R h (t 1 / 2=367 days), which subsequently y i e l d s the high energy g - , s of i n t e r e s t (predominantly with t 1 / 2 ~ 3 0 seconds; end point energy: 3.53 MeV) [WEA73]. The apparatus used i s shown i n fig u r e A2.1. The source was fixed r e l a t i v e to the Geiger-Muller (G.M.) tube (bias=500 Volts) and p a r a l l e l to the target normal. To perform the i n i t i a l c a l i b r a t i o n , the source-tube axis was placed on a v e r t i c a l plane. Several mock targets were constructed of the same materials used for the o r i g i n a l target. These targets d i f f e r e d from the o r i g i n a l only i n t h e i r area (somewhat less i n the mock targets i n an e f f o r t to avoid weight deformations) and t h e i r mass thickness. The boron was of the same elemental composition as that In the o r i g i n a l target, and the range of mass thicknesses was chosen to span that expected i n the o r i g i n a l target. A number of runs were performed and r e s u l t s averaged to determine, for each mock target, the number of counts scaled from the G.M. tube. These runs were made at d i f f e r e n t points on the mock target, so the standard error i n the average r e f l e c t e d well both the s t a t i s t i c a l errors and those induced by l o c a l anisotropies i n the mock target thickness. The mock target data was then used to r e l a t e electron count rate to P~ SOURCE MOCK TARGET G.M. TUBE TIMER-SCALER Figure A2.1 Apparatus for determining the U B target mass thickness measurement of electron multiple scattering effects. target mass thickness. The r e l a t i o n was l i n e a r , so that a slope and intercept (as well as uncertainties i n those quantities) could be e a s i l y determined. To measure the mass thickness of the o r i g i n a l 1 1 B target, the apparatus was rotated so that i t s axis was i n a hori z o n t a l plane, thus enabling a measurement i n the in-beam o r i e n t a t i o n . Measurements were performed on a symmetrical g r i d of 9 locations on the target. A quantitative i n d i c a t i o n of target non-uniformities was obtained from v a r i a t i o n s i n count rate over the target. The boron target mass thickness was determined to be 377.6 ± 3.0 mg by performing a weighted average of the thicknesses measured on that target. The weighting factors were obtained from the beam p r o f i l e s ( f i g u r e 2.4). (Here, only, was an average value f o r the target normal angle assumed.) This r e s u l t was averaged f or the two energies, as the difference i n beam p r o f i l e s resulted i n , at most, ~0.5 mg/cm2 difference i n the target masses. Appendix III Decay Kinematics Two body kinematics have been calculated throughout t h i s thesis with the r e l a t i v i s t i c formulae of Baldin et a l . [BAL61]. Their conventions are e s s e n t i a l l y the same as i n the following, and are shown i n figure A3.1. (Notice, here, that CM quantities are denoted by the t i l d e overscore.) A3.1 Muon Cone Angle With M i l i n i t i a l l y at r e s t , we have: and: V = ElSlI l* 3' 1) V PI E l pi = = (A3.2) 61 P(EI+EII) The kinematics requires that the condition: 1 - V 2 t a n 2 91 < (A3.3) p i 2 - 1 must be f u l f i l l e d , and i t i s t h i s which determines the cone angle: 1 - V 2 9 = t a n - 1 [ ] 1 / 2 (A3.4) p l 2 - l The maximum cone has 9=90° ( i . e . : p l = l ) . It i s e a s i l y shown that, f o r a simple p a r t i c l e decay where one of the decay products i s massless, pl=l i s s a t i s f i e d when: MI 2-Ml 2 T l . = { [ 1 - ( ) ] _ 1 / 2 - 1 } MI (A3.5) m i n MI2+M12 L A B : CM ff*m, pi ''01 O—> <5-MI, PI Mil \ 0 2 <\MZ, P2 J*A\, P1 / e / J. MI, PI / MII.PII P2 Figure A3.1 D e f i n i t i o n of the kinematic observables i n the LABoratory (LAB) frame, and the Center of Momentum frame (CM). For pion decay t h i s gives T.-=5.445 MeV, below which energy no cone i s observed. A 3 . 2 Beam Attrition One may cal c u l a t e the f r a c t i o n , f, of pions which decay into muons and subsequently miss a plane c i r c u l a r detector centered on and perpendicular to the beam axis. Consider the geometry i n figure A3.2 with a p e n c i l beam of pions o r i g i n a t i n g at point 0. Now define s=tan9, d = ( l - V 2 ) 1 / 2 , and p=cos6. Then kinematics requires that: (1 - u 2 ) d 2 s 2 = (A3.6) (pl+u) 2 hence: - p i s 2 d 2 ( d 2 - s 2 ( p l 2 - l ) ) . V = ± [ ] 1 / 2 (A3.7) (s 2+d 2) ( s 2 + d 2 ) 2 Figure A3.3 i l l u s t r a t e s the values of u corresponding to values of s i n the lab. The f r a c t i o n of decaying p a r t i c l e s entering a (lab) cone of known vertex angle, 9 Q, i s proportional to the s o l i d angle subtended In the CM frame by that cone. We assume that T i r > T T f ( m i n ) , so that f o r 60 < e(cone)» /d(cos9) i s just: d 2 s 2 f« = [ 1 ( P l 2 - D ] 1 / 2 - (A3.8) (s 2+d 2) d 2 If 9o->9(cone)» then f=0. In short: f l = [ 1 " Tl ( P 1 2 ~ L ) H ( S ( c o n e ) - s ) ( A 3 ' 9 ) (s*+d £) d* Figure A3.2 A p e n c i l beam of pions incident upon a plane c i r c u l a r detector perpendicular to the beam. Mi = - A + B H-z = - A - B A = 0.1146 B = V0.6926 Figure A3.3 CM angle cosines corresponding to pions of LAB energy 43 MeV decaying into a cone with a vertex angle equal to one hal f of the muon cone angle. Decays ins i d e the shaded regions enter t h i s cone. 224 At a point x, a f r a c t i o n : f2 - E X P [ c l 7 t 7 ] 6 X P t c l T ? ] ( A 3 a o ) of the pion beam remains, (x i s the pion mean l i f e . ) Integrating along the beam, the f r a c t i o n of pions which decay in t o muons which miss the detector at D i s : s, J ( c o n e ) f l f 2 -± d s ( A 3.11) s 0 CYSTS 2 Writing " R0 A - CB exp [ — ] (A3.12) _d_ '( cone) B = [ __2 ]2 (A3.13) 5 A and C = -z (A3.14) CBYTS, . (cone) e = = (A3.15) (cone) i t follows that ! C/x ( l - x 2 ) 1 / 2 f - A / e H(l-e) dx (A3.16) £ x 2(x 2+B) The f r a c t i o n of pions which decay and miss a detector of the s p e c i f i e d geometry i s f . Equation 3.6, 1-f, i s then the f r a c t i o n of pions which either do not decay or decay into muons which subsequently h i t the detector. Appendix IV Core + Valence Matter Distributions A4.1 The Distributions The various d i s t r i b u t i o n s used i n section 4.3.4 to test f o r s e n s i t i v i t i e s to moments greater that < r 2 > 1 / 2 were: 1) Modified Gaussian (MG) 2) Modified Gaussian Core + Valence Nucleons (MGCV) 3) Constrained Fermi (CF) 4) Constrained Fermi Core + Valence Nucleons (CFCV) These d i s t r i b u t i o n s are summarised i n table A4.1 and i l l u s t r a t e d i n f i g u r e A4.1. The f i r s t of these (MG) was discussed i n section 4.3.3 and w i l l not be dealt with here. A4.1.1 The MGCV Distribution The MGCV d i s t r i b u t i o n i s s i m i l a r to that used by W. Gyles [GYL79a] i n his i n v e s t i g a t i o n s of the neutron radius of 1 3 C . Here i t i s generalised to allow multiple valence nucleons (or holes) i n eit h e r i s o s p i n state. A closed s h e l l core with known radius i s assumed to have the MG form. The i t h valence nucleon i s assumed to occupy a si n g l e o r b i t a l of a harmonic we l l , so that i t contributes an angle-averaged matter density of: Table A4.1 Summary of Characteristics of Matter Density Distributions Matter D i s t r i b u t i o n Parameters Constraints Degrees of Freedom MG <r 2> • • • 1 MGCV <r 2>, • • • 2 <r 2>(core) CF c,t c/t fixed 1 CFCV c(core), c(core),t(core) 1 t( c o r e ) , fixed by QD; <r 2> <r 2>(core) fixed 10 0 1 2 3 4 r ( F E R M I S ) Figure A4.1 Various density d i s t r i b u t i o n s p+(r) evaluated for f i v e nucleons with < r 2 > 1 / 2 - 2.25 fm. with the Rn£. given by [BER72] ln£ = N n £ v > " ' ~ "£+1/2 where: K o = " , ( v r * ) t / 2 e - w 2 / 2 L » / 0 (A4.2) L£+l/2 = j Q W V r 2 ) k < A 4' 3> N v! [ 2 £- l H' 2(2n+2£+l)!! ,1/2 n£ (2£+l)!! L nl 7 i J l M , * ; r _ f _ n k k n! (2£+l)!! °kn£ ^ i ; z k!(n-k)! (2£+2k+l)!! W-*) We r e c a l l that the angular momentum and p r i n c i p a l quantum numbers, £ and n, are non-negative integers. The well strength, v, i s just mco/n where m i s the reduced mass and a> i s the o s c i l l a t o r frequency. If there are j valence nucleons, then v i s given by: j Z (2n.+£.+ 3/2) i = l v = — (A4.6) t r 2 1 t o t a l t o t a l core core where the T's are numbers (N or Z) of nucleons ( t o t a l and core) and the r's are RMS matter r a d i i ( t o t a l and core) f o r a given i s o s p i n . The subscript t i s a reminder that valence nucleons of d i f f e r e n t i s o s p i n are to be dealt with separately. The values of £ and n are determined for each valence nucleon from the order of (2J+l)-degenerate l e v e l s predicted by an independent p a r t i c l e model [FEL53]. For a valence proton, one must allow f o r the e l e c t r o s t a t i c e f f e c t s of that nucleon. The i t h valence proton contributes a p o t e n t i a l , as a function of distance r from the center of the nucleus, of: * ( r > i = e ° 2 I L_-_.'l d- ( A 4 , 7 ) - e 02 ( f — 1 r . 2 d r , + j r t _ _ L _ _ _ i ] d r , ( M > 8 ) 0 r 0 r r The f i r s t term i n equation A4.8 i s dealt with a n a l y t i c a l l y and i s just: e 0 2 — v * I C k n £ C m n £ (m+kfl) !. (A4.9) m, k=0 The second term i s integrated with Simpson's Rule i n a manner i d e n t i c a l to that of the o r i g i n a l Krell-Thomas code. This i s a two parameter d i s t r i b u t i o n . The f i r s t parameter i s a core radius, the RMS radius of the MG core. The second parameter i s the actual RMS radius of the matter d i s t r i b u t i o n of nucleons of the appropriate i s o s p i n . By varying the core radius, the o v e r a l l matter d i s t r i b u t i o n may be al t e r e d while keeping the RMS radius constant. A4.1.2 The CF D i s t r i b u t i o n The Fermi d i s t r i b u t i o n i s of the usual form: I r _£ l 1 + e L a 1 p(r) = T p 0 0 — — — - - (A4.10) with the 90% to 10% skin thickness of: t = 4 (ln3) a ( A 4 . l l ) and has the (2m) t n moment: 230 . 2m. . 2m+3 rk n +^" n r r i l , , r . n! . n-r r, ~ - r . . <r > = 4TT P o oa { - ^ + z [[!+(-) ] _ _ _ k [ i - 2 ] r=0 + C(r+1)] + Cn} (A4.12) with 5 - I C - ) ^ 1 - ^ e " k J (A4.13) n J - l j " * 1 where k=c/a,<=Tr/k, and n=2(m+l). This defines <r 2> i n terms of c and a. ( p 0 0 i s defined by the requirement that <r°> = 1.) Hence the Fermi d i s t r i b u t i o n i s a two parameter d i s t r i b u t i o n . We note that the r a t i o QD=<r 4> 1 / 2/<r 2> defines the fourth moment of the d i s t r i b u t i o n r e l a t i v e to the second. If we wish to match t h i s to the MG value of QD, then equations A4.12 and A4.13 with values of the Riemann zeta function [HAY65] imply that: 1 3 49 31 7 [ _ U + K 2 H 5 } { I + 7 k 2 h Ki»H K 6 H g 6}]i/2 21 k 3 3 3 k 7 QD, , = (A4.14) fermi 1 10 7 5 — [ l + — K 2 H <-+ — ? J 5 3 3 k 5 so that with (see equation 4.21 of section 4.3.3): [15(2+3a)(2+7cO] 1 / 2 [ 5 T ( 7 T - 8 ) ] 1 / 2 QD , = = , (A4.15) gaussian 3 ( 2 + 5 a ) ( 5 T _ 4 ) QD . QD, (A4.16) gaussian fermi may be s a t i s f i e d . 231 Equation A4.14 i s plotted as a function of c/t i n f i g u r e A4.2 along with <r 2> 1 / 2/c. Solutions to equations A4.14 through A4.16 are given i n table A4.2. Matching moments i n t h i s way constrains c / t . The r e s u l t i s a one parameter density d i s t r i b u t i o n with <r 2> uniquely defined: the CF d i s t r i b u t i o n . An MG d i s t r i b u t i o n with the same <r 2> d i f f e r s i n even moments only i n those greater than the fourth. The CF d i s t r i b u t i o n then allows the te s t i n g f o r s e n s i t i v i t y to such moments. A4.1.3 The CFCV D i s t r i b u t i o n F i n a l l y , the CFCV density uses a CF core and adds harmonic o s c i l l a t o r valence nucleons. As Implemented here, i t i s a one parameter density; the core radius i s fixed to correspond to the measured value for the core nucleus ( i e : 1 2 C ) . Without constraints, i t i s a three parameter density. 232 0-6 H , , , , 1 0 I 2 3 4 5 C / T Figure A4.2 QD (dashed) and < r 2 > 1 / 2 as calculated for the Ferini d i s t r i b u t i o n as a function of c / t . Table A4.2 Solutions to Equation A4.16 for 2 through 8 Nucleons N c/t QD 2 1.01688 1.29099 3 1.12626 1.26948 4 1.23846 1.25000 5 1.23214 1.23718 6 1.38366 1.22836 7 1.43190 1.22198 8 1.47030 1.21716 Appendix V A Replacement for VIEWFIT Viewfit was o r i g i n a l l y written with the intent of eliminating normalisation e f f e c t s from the f i t t i n g to a cross s e c t i o n . This routine displayed x 2 and a v i s u a l plot of data and c a l c u l a t i o n to help i n determining a "good" f i t . Here we discuss a more sophisticated method of a r r i v i n g at the same r e s u l t . Normally, one defines: X 2 = I [ X i ( e x p ) - X K c a l c ) ] 2 ( A 5 < 1 ) 1 °i Suppose that, instead, we define: x2 = s [ X i ( e x p ) ~ 5 X i ( c a l c ) ] 2 ( A 5 > 2 ) i a± Then: x2 . E X ^ e x p ) _ u s x i ( e x p ) x i ( c a l c ) + ? 2 ^ x i ( c a l c ) ( A 5 > 3 ) i f f 2 1 cr2 1 ff2 This i s just a quadratic i n £; c l e a r l y there i s exactly one minimum for x2> occuring f o r : — — — = 0 = 2£ £ X 2 l< c a l c> - 2 E X i ( e x p ) X i ( c a l c ) ( A 5 > 4 ) 9 5 1 a i 1 a i or: £ X i ( e x p ) x i ( c a l c )  1 5 = 1 (A5.5) x 2 E i ( c a l c ) i 0< Defining x 2 i n t h i s manner removes a data set's s e n s i t i v i t y to i t s o v e r a l l normalisation r e l a t i v e to a t h e o r e t i c a l c a l c u l a t i o n , while not e x p l i c i t l y introducing an extra parameter into the t h e o r e t i c a l d e s c r i p t i o n of the process. 236 Appendix VI Spectrometer Transfer Coefficient Optimisation In i t s current implementation, the QQD spectrometer i s a software spectrometer. This means that, instead of measuring the d i s t r i b u t i o n of 6p (the f r a c t i o n a l v a r i a t i o n of a p a r t i c l e ' s momentum from the spectrometer's 'central momentum' as defined by the spectrometer's geometry and bending f i e l d ) , at the f o c a l plane of the spectrometer, the 6p are derived from p a r t i c l e t r a j e c t o r y information measured at the entrance to and e x i t from the spectrometer [BAN66]. The p a r t i c l e t r a j e c t o r i e s are characterised by the quantities X Q, Y Q, 9Q , C|)Q, and 6pHfi at the target plane, as i l l u s t r a t e d by figur e 3.12. The exi t coordinates, X4,Y4,X5,Y5, are related to these q u a n t i t i e s . The measured entrance parameters, however, are not the coordinates at the target plane, but rather the coordinates, X1,Y1,X3,Y3, at the spectrometer's front end. The e x i t coordinates may be written i n terms of the front end coordinates and 6p. For example: X u = P(m Q) + Q(m 1)6 + R(m 2)5 2, (A6.1) where P, Q, and R are polynomials of orders mQ, m-^, and m2, respe c t i v e l y , i n the entrance coordinates. Notice that as the orders m^  increase, the possible number of transfer c o e f f i c i e n t s r i s e s rather dramatically, as i l l u s t r a t e d i n table A6.1. For example, i f mQ=2, then: Table A6.1 Proliferation of Spectrometer Transfer Coefficients ORDER POSSIBLE DISTRIBUTION OF EXPONENTS AMONG (X1,Y1,X3,Y3) In Decreasing Order of exponent Number of Unique Permutations TOTAL NUMBER OF COEFFICIENTS 0 1 2 3 (0 0 0 O N v » > > I ( \ 0 0 O N \ » > » i (2 0 0 O N (\ 1 0 O N \ » » > / > \ * » » / ,3 0 0 O N (2 1 0 O N (1 1 1 O N / i * 0 0 O N /3 1 0 O N \ ) » » / » v . » » » / (2 2 0 O N Z-2 1 1 O N ^ ' ' , i 1 M N ' ' ' ; v » » I J (5 0 0 O N /-^  1 0 O N ,-3 2 0 O N Z-3 1 1 O N 2 V l ' 0 ' ( V l ' l ' l /6 0 0 O N /5 1 0 O N \ » » » / » \ » » » / (h 2 0 O N /•<+ 1 1 O N s V o ' o ' h V l ' O N /3 1 1 I N /2 2 2 O N V > > » I 1 4 4 9 6 10 4 » 12 4 20 4 > 12 6 » 12 1 25 4 > 12 12 > 12 56 12 > 4 4 > 12 12 > 12 84 6 > 24 4 > 4 6 P(m0=2) = c ^ X l 2 + a 2 Y l 2 + a 3X3 2 + a^Y3 2 + a 5 X l Y l + a 6XlX3 + a ?XlY3 + a 8X3Yl (A6.2) + a gX3Y3 + a 1 Q Y l Y 3 + a u X l + a 1 2X3 + a 1 3 Y l + a 1 1 +Y3 + a 1 5 f o r a t o t a l of 15 possible c o e f f i c i e n t s . The expression A6.1 may be meaningfully inverted to determine 6, once the c o e f f i c i e n t s are known (a s i m i l a r expression e x i s t s f o r X5). The corresponding expressions for Y4 and Y5, however, have at most weak dependence upon 6 and may not be used to determine that quantity. Notice that equation A6.2 i s l i n e a r i n the parameters a^, so those c o e f f i c i e n t s may be determined v i a a l i n e a r regression technique. An F-test [BEV69] was used to remove s t a t i s t i c a l l y i n s i g n i f i c a n t terms i n the expansions. The input data to the regression was calculated from array f i l e s containing 1 2 C data i n the mid angle range (90° to 110°). With the two d e l t a degrees of freedom a v a i l a b l e from that data (ground state and f i r s t excited, 4.44 MeV state of carbon), one may determine only the P and Q polynomials. R was set to zero; t h i s was a reasonable choice for the momentum spread encountered i n the experiments. C o e f f i c i e n t s were determined separately f o r the two data sets at d i f f e r e n t energies. A preliminary analysis f o r evaluation of 6 used TRANSPORT [BR080] c o e f f i c i e n t s . Each of about 2000 sc a t t e r i n g events was tagged as ground state event (tagG), 4.44 MeV i n e l a s t i c event (tagX), or unknown event (tagU). With t h i s tagging, equal numbers of tagX and tagG events were selected to allow even s t a t i s t i c a l weighting ( i n momentum). The c o e f f i c i e n t s determined i n t h i s manner are, appropriately, weighted by the beamspot ( f l u x d i s t r i b u t i o n on t a r g e t ) . The s t a t i s t i c s of the regression technique tend to conform the momentum d i s t r i b u t i o n to a Gaussian shape. The regression algorithm was performed by program QQDMP, which c a l l s IMSL routine [IMS82] to accomplish the actual regression and F-t e s t . Figure A6.1 i l l u s t r a t e s the sequence which QQDMP followed. Figure A6.2b i l l u s t r a t e s the improvement i n spectrometer r e s o l u t i o n with optimised c o e f f i c i e n t s over that obtained with TRANSPORT c o e f f i c i e n t s ( f i g u r e A6.2a). Figure A6.2c shows the ad d i t i o n a l Improvement achieved with the imposition of muon elimination cuts. 240 ( LOGON *) Found No Run MOLLI. Coll PRINTER(): Croote RUNnnnARR Yes t T > » (Run QQDMP) Choose: Energy, Mode Figure A6.1 Spectrometer c o e f f i c i e n t determination algorithm. Figure A6.2 Improvement of QQD re s o l u t i o n : 48.3 MeV Energy Spectrum, a) With transport c o e f f i c i e n t s and no muon cuts, b) With optimised c o e f f i c i e n t s and no muon cuts, c) With optimised c o e f f i c i e n t s and muon cuts on DDIF and ANGL. 242 Appendix VII Peak Fitting in the CNO2 Experiments A7.1 Introduction In Section 3.3.3 we discussed the motivation for and s p e c i f i c a t i o n s required of peak f i t t i n g i n the analysis of energy spectra from the QQD spectrometer. In t h i s appendix we discuss the peak shape and error analysis accompaning t h i s process. A7.2 The Peak Shape The peak shape of the energy spectra produced from the QQD spectrometer with optimised transfer c o e f f i c i e n t s i s a complicated function dependent upon many d e t a i l s of the channel-spectrometer configuration. We assume the basic shape to be of the form of two Gaussians (each referred to here as a component) on a common center. The r a t i o s of the halfwidths and of the r e l a t i v e heights of the components are assumed to define the peak shape at a given s c a t t e r i n g angle, regardless of target material. There i s one of these basic peaks for each s c a t t e r i n g state i n the energy spectrum to be analysed. The energy separations of the states are known [LED78] and can be adjusted f o r the varying s p e c i f i c energy loss of the pions whose energies d i f f e r upon leaving the target n u c l e i . The halfwidth of the fundamental (biggest) component i s assumed to be a constant, v a l i d f or a l l s c a t t e r i n g states at a given angle for a given target. The r e l a t i v e i n t e n s i t i e s of the d i f f e r i n g s c a t t e r i n g states, of course, d i f f e r f or each target and angle. The peak shape i s , a n a l y t i c a l l y : + 6 e ( N / i a )2 } (A7.1) where N, i n p a r t i c u l a r , i s a function of E. The d e f i n i t i o n s of the va r i a b l e s i n equation A7.1 are given i n table A7.1 The parameters 3, n, and i define the peak shape and were determined from carbon spectra at each angle. The e f f e c t i v e dispersion, n, was allowed to vary f o r each group of runs at a given angle s e t t i n g to allow for possible l i m i t a t i o n s i n the spectrometer c o e f f i c i e n t s due to neglected higher order terms i n 6p. The values of E^ were not v a r i e d . The values of E Q , a, a^, and N 0 were f i t t e d i n each run. It should be noted that the two-Gaussian p r e s c r i p t i o n for the peak shape at given e x c i t a t i o n energy E^ i s not s t r i c t l y necessary for a reasonable f i t to the observed spectra, as the second Gaussian i s meant to represent the muon decay background surrounding the main peaks. The i n c l u s i o n of the second Gaussian primarily allows degrees of freedom which f i t the decay contaminants not eliminated by the cuts and which may otherwise r e s u l t i n an overestimate of i n e l a s t i c contributions. Judicious use of the er r o r matrices from the peak shape determination and spectra f i t t i n g s allow the i n c l u s i o n of e f f e c t s i n the o v e r a l l uncertainty due to c o r r e l a t i o n s i n the extracted number of counts due to the peakshape (c o r r e l a t i o n s i n the parameters f i x e d from 1 2 C ) , as well as due to the c o r r e l a t i o n s i n the parameters (free parameters, f i t t e d to the i n d i v i d u a l energy spectra) themselves. Table A7.1 Definition of Peak Fitt i n g Parameters VARIABLE DEFINITION N { E - E 0 e i 0 - r\{\ZQ-Z±)<kl-eoi) } imax Number of excited states included i n f i t t i n g (i=0 corresponds to the ground state) 6 Relative strength of the second component T) E f f e c t i v e dispersion (n^l) E0 Energy centroid of the ground state peak E i E x c i t a t i o n energy of the i t n peak (i*0) e 0 i Kronecker function: e 0 0 = l , e g i = ^ ^ o r a Width of the fundamental (largest) component of the two Gaussian p r e s c r i p t i o n 1 Ratio of widths, a2/a of the 2 n d component Gaussian to the fundamental E Energy at which G(E), i s evaluated N0 Strength of the ground state peak a i Ratio of the strength of the it h i n e l a s t i c peak to that of the ground state peak A7.3 Mechanics of Fitting The actual f i t t i n g was c a r r i e d out with the routine QQDANA already referred to i n section 3.3.1 and appendix VI. This routine handled the data and parameter i n t e r f a c e to the IMSL routine ZXMIN, which was used to perform the nonlinear l e a s t squares parameter searches. IMSL routines LINV1P aided i n determining f i n a l f i t t i n g c o r r e l a t i o n matrices for the free parameters. Gaussian i n t e g r a t i o n was performed with the routine MERF [IMS82], which evaluates the sum of ERROR functions ( e r f ) appropriate to the defined peakshape over s p e c i f i e d energy regions. The f i t t i n g was accompanied by graphics p l o t s , generated with the a i d of CALCOMP type FORTRAN p l o t t i n g routines [KOS84], which aided i n the f i t t i n g and helped to ensure the i n t e g r i t y of the f i n a l f i t s . A7.4 Correlation Errors We wish to f i t an observed d i s t r i b u t i o n with a function of m free parameters and n fi x e d parameters. The fi x e d parameters are determined eit h e r ab i n i t i o ( f or example, the possible excited state energies for the element under consideration) or from f i t t i n g to an e a r l i e r run. For the fi x e d parameters b^ we have a c o r r e l a t i o n matrix N^£; the diagonal element, N^, i s the variance of b^ and Nj-^ i s the covariance between b^ and b j . R e c a l l that N i s symmetric, and \ £ - < < V V ( V V >• ( A 7 - 2 > Given the fi x e d parameters, a s o l u t i o n for the m free parameters (denoted by aj) e x i s t s , along with the associated c o r r e l a t i o n matrix Mjk [MAT65]. We r e c a l l that 246 M j k = < (a - a j ) ( a k - a k) > (A7.3) where I j i s the best value of aj which would r e s u l t from f i t t i n g s to a large number of s t a t i s t i c a l l y independent data sets. The c o r r e l a t i o n matrix, N, a f f e c t s our ce r t a i n t y i n the aj determinations. We define a matrix A j ^ = 9aj/9bi c, which t e l l s how much the derived value of aj varies with changes i n the value of b^. Suppose that a j i s the value of ( a j - a j ) when b^Ebj^. Then f o r b^b^: 9a . _ J (a - a ) - a + ( V V > ( A 7 ' 4 ) 9 b. k so that: < ( a . - a . X a ^ ) > s ^ . a ^ + A j k A i £ N k £ (A7.5) 5 M j i + A j k A i £ N k £ ( A 7 ' 6 ) = M .. (A7.7) J i and: 9a J _ _ < (a -a.)(b -b.) > - <(b -b. )(b -b )> (A7.8) k A J k N k i ( A ? - 9 ) The matrix A j ^ may be determined as follows. The parameters a± are determined from a regression analysis minimising x 2• The x 2 ( a i ) * s a hypersurface dependent upon the a-j. At a minimum, 3X2 3X 0 2 3 2 X 0 2 + I 6a. - 0 , (A7.10) 3 a i %a± j Sa^a.. where Xo.2 * s t n e value of x 2 at the minimum. The dependence of the second d e r i v a t i v e 3Xo 2/(Sa^Saj) upon the choice of b^ w i l l be small, but the change i n the gradient term i s larger, when b k v a r i e s . So: 8X 0 2 3 2X 0 2 3 a, k 3 a, 3 b, i i k 6b k ( A 7 . l l ) defines the minimum with a d i f f e r e n t value of b^. Combining these gives: 6a . A., s = - (M-l 0),, , (A7.12) when w r i t i n g : jk ~ jk' k 3 2X 0 2 3 2X 0 2 and M, . = (A7.13) l k 33,3b, 1 3 3a,3a. i k i j A7.5 Propagation of Err o r s Given energy spectra and f i t t e d curves for each of several targets, we may e s t a b l i s h the cross sections and r a t i o s of cross sections. Although the f i t t i n g allows easy extraction of the number of e l a s t i c events, i t also necessitates considerable care i n error a n a l y s i s . 248 The measured data include several groups of measurements; these groups d i f f e r i n spectrometer configuration ( i e : angle setting) or are separated i n time by more than several hours. Each group consists of at l e a s t one run from each target. Each group i s s t a t i s t i c a l l y independent, and has the following q u a n t i t i e s associated with each of i t s runs: i ) N f: The raw number of counts i n a state of i n t e r e s t within a given energy region s p e c i f i e d by a low and a high l i m i t . i i ) N: The peak f i t t e d number of counts i n that state integrated over a l l E, regardless of the region s p e c i f i e d . i i i ) N f t The peak f i t t e d t o t a l number of counts integrated over the region i n which N r was determined. The number of e l a s t i c events corresponding to the state i s then: N N = N . (A7.14) 6 1 1 Notice that a l l three of these q u a n t i t i e s , and i n p a r t i c u l a r t h e i r u n c e r t a i n t i e s a r a, and Oft> are strongly correlated. We note that N r and N f t d i f f e r only by the x 2 of the f i t [BEV69] within the s p e c i f i e d l i m i t s , so the appropriate c o r r e l a t i o n matrix f o r N e i i s just that f o r N. That i s , we may ignore the uncertainties i n N r and N^ t and concern ourselves only with those i n N e x p l i c i t l y . An approximation to equation A7.14 i s : N f + x 2 ( w i t h i n l i m i t s ) N - [ ] * N. (A7.15) N f t This expression was used to perform a small c o r r e c t i o n to the r a t i o values. The f u l l expression i n equation A7.14 was used i n the extraction of the carbon cross sectons, whereas the ground state to f i r s t excited state separation i n the case of 1 5 0 was large enough that d i r e c t i n t e g r a t i o n could be used i n that cross section e x t r a c t i o n . A7.5.1 Ratio Measurements We now discuss the propagation of errors i n functions of N, i n p a r t i c u l a r i n that i t concerns r a t i o measurements. Consider, f o r s i m p l i c i t y a case where each energy spectrum consists of two peaks. We choose f i v e v a r i a b l e s : h : Height of the e l a s t i c peak f : Ratio of the height of the i n e l a s t i c peak to that of the e l a s t i c peak W : Width common to both peaks u : Energy centroid of the e l a s t i c peak 6E: Energy separation of the i n e l a s t i c peak from the excited state peak We suppose that the data set consists of a number of groups of data: each having runs f o r 1 2 C , 1 6 0 and MT targets. The carbon and oxygen targets are assumed to generate beam energy losses and multiple scattering to the same extents, so that W i s the same for both targets. We construct a covariance supermatrix, as shown i n figu r e A7.1. There are two types of fi x e d v a r i a b l e here, 6E's are fi x e d to a p r i o r i known values, with a modest uncertainty due to the energy loss differences of d i f f e r e n t energy pions. The W i s determined i n a separate f i t t i n g to the carbon data, and i t s error 1 2c 160 Z3 O or o 12 T 3 O or o 16r 3 O -L_ 1 12, 16 0 a) 12, 16f V JE, u, h, f, « I!, >, b, f 2 V 11* «E,W !>,« <1» SEjV y2W f 2 H « 1 6EjV «E,' 0 0 0 <E,V »1 ».» 0 n " l »1« 0 I <1 0 U «E,H h,w SEjW « 2' 0 0 0 »J V,» »2« 0 " j h,» h,« 0 i '» 0 b) Figure A7.1 a) Covariance supermatrix constructed i n peak f i t t i n g to the energy spectra from the QQD. b) The c o r r e l a t i o n between GROUP 1 variables k and I, a k£, i s written 'k£' i n the matrix. The E's are the usual error matrices with free variables u,h, and f . 251 i s determined from that f i t . The MT target runs have no entries i n t h i s matrix. This i s because lack of s t a t i s t i c s p r o h i b i t s meaningful peak f i t t i n g f o r those runs. The MT co r r e c t i o n i s small, and i s taken d i r e c t l y from the raw number of counts i n the energy spectrum ( i e N r for the MT t a r g e t ) . Also, the peak shape rel a t e d fixed parameter entries occur twice i n each group (once for each t a r g e t ) . By considering the variables W^ , f o r example, as independent with perfect c o r r e l a t i o n s : CTw1w2=0w2» t n e counting w i l l be correct i n the propagation of errors that follows. To a r r i v e at the c o r r e l a t i o n between the numbers of integrated counts, N, f o r the group (written here as: 1 2N, 1 6N) that occurs through the c o r r e l a t i o n s i n the peak f i t t e d parameters, we pre- and post- multiply the group supermatrix by the diagonal matrix D, where D^i=di and d^  i s the vector: 3 1 2N 3 1 2N 3 1 2N 3 1 2N 3 1 2N 3 1 6N 3 1 6N 3 1 6N 3 1 6N 3 1 5N d = [ , , , , , , , , , ] (A7.16) 3W 36E 1 3y x 3h x 3f 1 3W 36E 2 3u 2 3h 2 3 f 2 The r e s u l t we c a l l N. At t h i s point, a group which has two subgroups for the same target may be combined into a si n g l e subgroup, as i l l u s t r a t e d i n figure A7.2. The ad d i t i o n Is element by element. The corresponding values of the integrated fluxes, f, f o r the runs are then added. What i s the uncertainty i n quantities derived from the N's? R e c a l l that a l l parameters may be considered, now, as separate and correlated, and one need not worry about double counting i n the propagation of e r r o r s . 252 c CO a COb C 0 Q O a OaOb C Ob OaOb Ob c C O a + C Ob C O a C Ob Oa+Ob 2 OaOb Figure A7.2 Reduction of supermatrix for a 'group' with two 'subgroups' corresponding to scattering from the same angle and target. 253 Furthermore, 1 6 N i s dependent upon a completely d i f f e r e n t set of var i a b l e s ( a l b e i t some p e r f e c t l y correlated) than i s 1 2 N . Then, for a function v ( 1 2 N , 1 6 N , 0 0 N ) = v ( n 1 , n 2 , n 3 ) = v ( n ) : 9v 9v 9v ° 2 = ) + Z l a [ ][ ] 0 0 3 n , i j 3x , 9 x . 3 J i 3 The second term i s : 9v 9 n 1 9v 9 n 2 9v 3n x 9v 9 n 2 E [ + ] [ + ] i , j 9 n 1 9x_L 9 n 2 9 x i 9 n 1 9X_. 9TI2 9X^ So, that: 9v 9v 9v 9n c i , j 3 T 1g(i) a V i ) (A7.17) (A7.18) (A7.19) where g(k) i s the subgroup (1 or 2) whose peak the k t n v a r i a b l e describes. (Subscripts and superscripts '00' re f e r to MT target q u a n t i t i e s ) . In the case that: v ( 1 2N, 1 6N, 0 0N) = (16 N/16f - 00 N/00 f)/(12 N/12 f _ 00 N / 0 0 f ) > ( A 7. 20) the following r e l a t i o n s r e s u l t : 9v 9 1 6N = [16f(12 N/12 f _ 0 0 N / 0 0 f ) ] - l (A7.21) 16 N 00 N 9v 9 1 2N = [ - l / l 2 f ] 16 f 00f 12 N 00 N (A7.22) iJ 12f 00 f 254 and: 3v 12, 00 N g 0 0 N 0 0 f 1 2 f 00< ] - M v - 1 ] (A7.23) 255 Appendix VIII Fourier Expansions of The Nuclear Density A8.1 Introduction It i s convenient for the purpose of numerical c a l c u l a t i o n to choose one of our a n a l y t i c forms, equation 4.18 or equation A4.10, to parameterise the nuclear proton and neutron d e n s i t i e s , p+(r) and p _ ( r ) , r e s p e c t i v e l y . Such a choice, however, r e s t r i c t s an analysis i n the extraction of nuclear s i z e information to the gross features of a d i s t r i b u t i o n . This may be a serious l i m i t a t i o n , e s p e c i a l l y i f the momentum transfer a v a i l a b l e i n e l a s t i c s c a t t e r i n g experiments allows the probing of the nuclear i n t e r i o r . A more serious l i m i t a t i o n , however, i s that the constraint imposed by an an a l y t i c form may r e s u l t i n underestimates of the uncertainties i n the derived moments. A8.2 Fourier Bessel Analyses The analysis of electron s c a t t e r i n g data [FRI75] frequently assumes an expansion of the nuclear charge density of the form: sin(mrr/R ) c p(r) = P l ( r ) + E a n » r < R c • n r = P l ( r ) + P p ( r ) , r < R c (A8.1) - P i ( r ) , r > R c with p i ( r ) a s t a r t i n g density, often of the Fermi or Modified Gaussian Form, and R c a cutoff radius. The d i s t r i b u t i o n of equation A8.1 has been used recently i n the an a l y s i s of neutron d i s t r i b u t i o n s from TT~ r a t i o experiments on isotopes of s u l f u r and magnesium. Variations have been used by others [BAR81] i n t h e i r analyses of nuclear matter d i s t r i b u t i o n s using hadronic probes. Many of the properties of the expansion found i n equation A8.1 that are relevent to the present analysis have been extensively discussed elsewhere [GYL84, FRI78], and w i l l not be repeated here. We r e c a l l that, as with the MGCV d i s t r i b u t i o n discussed e a r l i e r , we must deal properly with the e l e c t r o s t a t i c e f f e c t s of the added nucleon matter. This w i l l be discussed i n appendix IX, as i t was not a matter of importance In the analysis of [GYL84]. A8.3 Fourier Laguerre Expansions The choice of the FB expansion f o r our studies i s perhaps not the wisest choice. A more l o g i c a l and c a l c u l a t i o n a l l y convenient choice u t i l i s e s the harmonic o s c i l l a t o r solutions seen In appendix IV. With squaring, these give r i s e to de n s i t i e s of the form: A8.3.1 The Basis An orthonormal basis set of such densities on the i n t e r v a l (0,°°) i s : P(r) = a" 3 Z B n(r/a) 2n -(r/a)2 e (A8.2) N e m -(r/a)2 L i ' 2 [ 2 ( r / « ) 2 ] f (A8.3) 257 where 'a', corresponding to R c, i s a we l l depth parameter and the L^' are Laguerre polynomials. N m i s chosen to normalise the Jt m to a n x / 2 . w h e r e n x i s chosen f o r c a l c u l a t i o n a l convenience: a 3 [r(m+3/2)]3 4/2 r(m+l) from which: 1/2 N 2 * [ ] = a " n X (A8.4) m • 4/2 r(m+l) . / 0 , , , 2 i B = [ ] 1 / 2 e " ( r / a > L m/2[2(r/a)2] , a 3 _ n X [ r(m+3/2)] 3 (A8.5) a - ( r / a > 6 m 2k k = E L(m,k) (r/a) K 2 , (A8.6) (3-nx)/2 k=0 a where L(m,k)=0 for k > m. The values of L(m,k) are then uniquely defined by the recursion r e l a t i o n s [MOR53]: L j / 2 ( z ) = T(3/2) (A8.7) L } / 2 ( z ) = r(5/2)(3/2-z) (A8.8) 2n+l L l / 2 ( z ) I ( 2 n - l / 2 ) L 1 / 2 ( z ) - z L 1 / 2 ( z ) - (n-1/2) 2 L J/ 2(Z) } (A8.9) n n n - i n—i n-z 2n Now, - ( r / a ) 2 nmax n , P t, T (r) = E a I = E o £ L ( n , k ) 2 f c ( r / a ) Z l c (A8.10) F L n n n=0 n k=0 (3-nx)/2 258 nmax nmax E I a L(n,k) 2* k=0 n=k n = 0,. (r/a) 2k "(r/a)2 (3-nx)/2 (A8.ll) nmax £ 0,, (r/a) k=0 fc 2k "( r / a ) 2 (3-nx)/2 (A8.12) A8.3.2 Zero Sum Constraint We now evaluate: m nmax w _ 1 2k+3 J p F L ( r ) r 2 dr = E 8, a<3+nx)/2 { _ f r ] } 0 * L k=0 * 2 2 which must give zero contribution to the total density, as the density pj(r) has the normalisation T, the relevant number of nucleons. Hence the constraint: -1 nmax 2k+3 0O s 0 k r[ ] , T(3/2) k=l K 2 (A8.14) from which: nmax cc0 = [ e 0 - E a L(n,0) ] / L(0,0). k-1 (A8.15) This allows us to rewrite pp^ a s ; nmax P K T ( r ) - £ 0. { (r/a) F L k-1 k 2k 2k+3 F [ T " ] e ^ r / a > 2 } r(3/2) (3-nx)/2 (A8.16) A8.3.3 Derivatives of PpL(r) In order to evaluate uncertainties in the density due to uncertainties i n the f i t t e d density c o e f f i c i e n t s a n > we write the derivatives 3p,, T(r) , - ( r / a ) 2 FL 2 e v ' nmax and: { p(r) [(r/a) 2-(3-nx)/4] Z kg ( r / a ) 2 k } (A8.17) 3a a (3-nx)/2 k-1 k a 9 p F L ( r ) nmax 9p (r) 8 f 5k = £ [ — — ] (A8.18) 3a. k=l 36, 3a J i k i 2k+3 i T[~T] e " ( r / a > 2 = I L ( i , k ) 2 k { ( r / a ) 2 k - } (A8.19) k=l T(3/2) (3-nx)/2 3. A8.3.4 Radial Moments The r a d i a l moments for the Fourier Laguerre d i s t r i b u t i o n are calc u l a t e d from: to J o"[(Pi<0 + p F L ( r > ] r J + 2 d r < r J > = (A8.20) to Jo"[(Pi<r) + P F L ( r ) ] r 2 dr Given that p j ( r ) i s normalised to T, we have: 4 i r < r J > - < r 3 >. + — < r 3 >_. (A8.21) where < r J >j i s the j t n moment of the s t a r t i n g d i s t r i b u t i o n , P l ( r ) . Now, < r J >F T - Jo" Pp, (r) r 3 + 2 dr = - a*-<3+nx>/2 T\ TT{j,k) (A8.22) FL FL 2 k = 1 k 2 6 0 where the I T ( j , k . ) i s defined as: 2k+3 r( ) j+2k+3 2 j+3 r r ( j , k ) = { r ( ) r ( ) } ( A S . 2 3 ) 2 r ( 3 / 2 ) 2 The d e r i v a t i v e s of the r a d i a l moments are as follows: 9 < r J V 1 = - a * - ( 3 + n x ) / 2 r r ( j . k ) ( A 8 . 2 4 ) 3 0 k 8 < f J >FL (j+(3+nx)/2) M l + m 0 / 2 nmax aJ+U+nx;/^ z r r ( j , k ) (A8.25) 3a 2 k = 1 Equation A8.24, when combined with: 3 < r J > 3 < r j 30. FL nmax FL k = Z [ ] (A8.26) gives: 3a. k=l 38, 3a, i k i 3 < r J > , 1 .. / n , W n i a J + ( 3 + n x ) / 2 r ( r r ( j , k ) L ( i , k ) 2 k ) (A8.27) 3a. 2 k=l l A8.3.5 Uncertainties in ppi,(r) and the Radial Moments We wish to evaluate the unc e r t a i n t i e s i n ppL(r) and i t s derived moments. The nmax parameters p±, i e {0,1,2...,nmax) are Pg =a, P j ^ i i p 2=a 2, Pnmax = anmax« T h e covariance matrix a 2 ^ which represents the unc e r t a i n t i e s i n t h i s parameter set i s then O j k , where the values of j and k run over the same range as does i . 261 Then: 3 p F L ( r ) 8 p F L ( r ) °* ( r ) = 2 o l (A8.28) p F L ^ r ; i,j=0 1 J 3p. 3 P. i j with the d e r i v a t i v e s defined i n equations A8.17 and A8.19. S i m i l a r l y , using the derivatives i n equations A8.25 and A8.27, the uncer t a i n t i e s i n the r a d i a l moments are: 3 < r j > 3 < r j > „ nmax FL FL a2 - I CJ2 (A8.29) < r 3 > F L i,k=0 1 K 3 P i 3p k A8.3.6 The Densities Figure A8.1 shows the f i r s t 6 Fourier Laguerre d e n s i t i e s , with the zero net i n t e g r a l requirement defining the value of a Q . A l l other values of a± were set to zero, except for the corresponding to the order of the function i l l u s t r a t e d , i n which case i t was set to 0.01. The value of 'a' was 1.5 fm, a value near to that suggested by the strength of the nuclear p o t e n t i a l . CO C o o D C 262 0 1 2 3 4 5 6 0 1 2 3 4 5 6 O O O X 1 2 3 4 5 6 0 1 2 3 4 5 6 r ( fm) Figure A8.1 a) and b) The f i r s t s i x (zero norm) FL d e n s i t i e s . The t o t a l number of l o c a l minima and maxima on a given curve i s the order of the density. c) and d) Densities of a) and b) folded with the proton form f a c t o r . [a 0 normalisation] 263 Appendix IX Of Charge Densities and Matter Densities A9.1 Convolution of the Proton Form Factor By v i r t u e of the form of i t s i n t e r a c t i o n , the pion i s able to probe matter d e n s i t i e s d i r e c t l y . There i s a dependence upon the nuclear charge density as w e l l , but for the purposes of including Coulomb e f f e c t s i n our p o t e n t i a l , i t i s usually s u f f i c i e n t to assume the two i d e n t i c a l . The de facto standard f o r measuring nuclear sizes though, provides us with the nuclear charge density. To meaningfully compare our r e s u l t s with the standard, then, we must provide a p r e s c r i p t i o n for r e l a t i n g matter and charge d e n s i t i e s . There are a number of e f f e c t s which may be convoluted with a matter density to a r r i v e at a charge density. The CM motion of a nucleus i s i m p l i c i t l y folded i n t o our matter density measurement. Magnetic e f f e c t s are not usually seen i n the l i g h t e r n u c l e i , but rather i n heavier n u c l e i such as **8Ca [BER72], where high angular momentum o r b i t a l s are occupied and contribute s i g n i f i c a n t magnetically induced currents. The e f f e c t of the f i n i t e proton si z e i s the lar g e s t , and i s the one upon which our e f f o r t s here are concentrated. We write a convolution i n t e g r a l [BAR77] r e l a t i n g the charge and matter d e n s i t i e s as: p c h ( r ) = '° P m C l r - r ' l ] <V R '> D£' <A9A> 264 where the charge d ensity here i s Pch> the nuclear matter density p m ( i e : p+(r) or p_(r) ), and the s i n g l e proton matter density ppp. Chandra and Sauer [CHA76] suggest: 3 - ( r / r . ) 2 p p p ( r ) = I ^  [ITr^]— 2 e 1 (A9.2) where £^^=1 as a via b l e p r e s c r i p t i o n for the proton form f a c t o r , with the values of and r j 2 those given i n table A9.1. The f o l d i n g process guarantees that <r 2 c n> = <r 2 m> + <r 2pp>. A9.2 Folding P p p into a Fourier Laguerre Sum For a Fourier Laguerre d i s t r i b u t i o n , we may write: p . T ( r ) = a ^ - 3 > / 2 T % . ( r / a ) 2 3 e - ^ / a > 2 j-0 3 (A9.3) Performing the convolution and some algebraic manipulation y i e l d s : - ( r / r i ) 2 (nx+l)/2 3 ^ i 6 (B2/A ) nmax 6 j P F F L ( r ) = ± ± E e E * r i-1 r i / A i j-0 AJ+l/2 2j+l (2j+l)! B i . (k-1)!! E EVEN(k) [ J 2 * 1 - 1 - 1 1 (A9.4) k=0 k!(2j+l-k)! • A ± 2k/2+l where: EVEN(k) = 1 f o r k even (A9.5) = 0 fo r k odd , Table A9.1 Nucleon Form Factor Parameterlsatlon i i 2 3 < r 2 > + 1 .506373 .327922 .165705 .775425 PROTON .431566 .139140 1.525540 NEUTRON * i 1 -1 - 2x.7242/3 A .444136 .521464 -A i = [ 1 + - ] . and ar B . 1 A (A9.6) A9.3 Folding ppp into a Fourier Bessel Sum The FB fu n c t i o n a l form for p(r) does not allow evaluation of the folded charge density i n closed form. A numerical technique to accomplish the f o l d i n g numerically has been developed, but i s cumbersome and CPU-intensive. In the text of t h i s thesis where the FB matter density i s used i n o p t i c a l model analyses, the charge density of the nucleus was taken equal to the proton matter density, as the small difference between the two did not warrant s a t i s f y i n g the a d d i t i o n a l numerical demands. A9.4 Folding P p p into the Starting Density P1(r) In the Fourier analyses discussed i n the text, the s t a r t i n g d e n s i t i e s were always assumed to be of the MG form. It i s well known that a Gaussian p r e s c r i p t i o n for the p p p allows an a n a l y t i c form for the corresponding charge density to be written [ELT61]: 3 a 3 3(b2.-a2) a 2 r 2 - ( r / b ) 2 P c h ( r > l " T POO S +i - T [ 1 + T { — 5 — - + - — } ] e (A9.7) c n 1=1 b3. 2 b 2 b 2 b2. l i i i where b^ = a 2 + 2(r|)/3. A9.5 Evaluations of <fr0 In our discussion of core+valence nucleon d i s t r i b u t i o n s , we noted that evaluation of the nuclear e l e c t r o s t a t i c p o t e n t i a l , <|>(r), requires that we know: *o = 4 i r e o 2 /o r 2 dr. (A9.8) r We assumed the matter and charge d i s t r i b u t i o n s to be equal, i n that discussion. We w i l l now discuss the values of 4>0 f o r the folded charge d i s t r i b u t i o n s corresponding to the density forms referred to i n t h i s t h e s i s . A9.5.1 <t>0 for the Folded MG Charge density For the folded MG charge d i s t r i b u t i o n discussed immediately above, we have: 4-rreg2 a' 3 a 3 3T a a 5 0 2 (A9.9) A9.5.2 4>0 for the Unfolded FB Charge Density In the case of P p g ( r ) , i f we assume the Pch=Pm> w e have: c nmax n $0 = to Jo e 0 2 = 4Tre 0 2 [ ] I [ — ] • (A9.10) r T n=l n (odd) A9.5.3 $0 for the Folded FB Charge Density In the case of a F B d i s t r i b u t i o n , <|>Q may not be expressed i n a closed a n a l y t i c form, and must be evaluated numerically. A9.5.4 $0 for the Unfolded FL Charge Density When ppL(r) i s considered, then i n the case that charge and matter d e n s i t i e s are considered i d e n t i c a l ( i e , no fo l d i n g i s performed): • 0 = 27re 0 2 a ( n x + 1 ) / 2 T * k! 8, (A9 . l l ) k=0 k A9.5.5 4>0 for the Folded FL Charge Density If we choose to use the proper folded charge density to accompany a FL matter density, then: , 2 ( n x + 3 ) / 2 3 h n m a x 6 j y 1 ( 2 3 + 1 ) 1 1-1 ( a 2 + r 2 ) 1 / 2 j-0 , j k=0 k!(2j+l -k ) 1 A± (k-1)!! a EVEN(k) [{ — } 2 j _ k K j -k/2+1) ] (A9.12) 2k/2 r t A9.6 Unfolding of the Proton Form Factor from Electron Scattering Data To compare the matter d e n s i t i e s obtained i n these experiments with the charge d e n s i t i e s measured i n electron s c a t t e r i n g , the charge densities of Norum et a l . [NOR82] for 1 6 0 and 1 8 0 and Cardman [CAR80] for 1 2 C were f i t t e d with folded FL charge d e n s i t i e s , where the f i t t i n g parameters were a and of equation A8.10. The f a c t that the Fourier component must give a net zero contribution to the matter density of the nucleus means that the FL density and i t s folded counterpart must have the same RMS i n t e g r a l . We choose: B nmax a r , m a v = . (A9.13) nmax ' R(nmax,nmax) nmax-1 3 n - a R(nmax,0) l. a R(n,0) u nmax ' n=l n ' and: a Q = (A9.14) R(0,0) where: nmax-1 2k+3 " 2 W k T[ ] k - l 2 8 = (A9.15) IIIQ3 X 2nmax+3 nmax-1 R(nmax,k) 2k+3 (nmax r [ ] + Z [ k T ( )]) 2 k=l R(nmax,nmax) 2 nmax-1 k 2k+3 Z B k ( 1) r [ ] k=l nmax 2 eQ = (A9.16) T(3/2) R(j,k) = L(j , k ) 2 k (A9.17) and B, = 6, + a R(nmax,k) (A9.18) k k nmax which guarantees the FL de n s i t i e s to have zero RMS i n t e g r a l . The f i t t i n g was performed with the non-linear l e a s t squares f i t t i n g routine ZXMIN from the IMSL [IMS82] l i b r a r y . With the above choices f or ct0 and a n m a x , t h i s f i t t i n g was forced, by choosing appropriate P l ( r ) ' s , to give resultant matter d e n s i t i e s and charge d e n s i t i e s consistent with the experimental e l e c t r o n s c a t t e r i n g r e s u l t s . As the RMS charge r a d i i are the best determined quantity from such experiments, i t i s important to be consistent with respect to them. Figure A9.1 shows the experimental electron s c a t t e r i n g r e s u l t s f o r the charge d e n s i t i e s of 1 2 C and 1 6 0 , along with the s t a r t i n g (folded) densities corresponding to P l ( r ) . The FL Components representing the differences between the two are also i l l u s t r a t e d . The e f f e c t of unfolding the proton form f a c t o r from the proton matter d i f f e r e n c e , ( 1 8 p + ( r ) - 1 6 p + ( r ) ) , i s shown i n Figure A9.2. Figure A9.3 shows some of the zero-RMS-integral FL density functions. 100 T 1 1 1 r -10 100 1 2 3 4 5 6 0 1 2 3 4 5 6 -5 --10 0 1 2 3 4 5 6 ( f m ) Figure A9.1 1 6 0 and 1 2 C charge d e n s i t i e s , a) and b) The s o l i d curves are folded modified Gaussian den s i t i e s and the broken curves are the model independent den s i t i e s of [NOR82] and [CAR80]. c) and d) FL components representing the differ e n c e between the d i s t r i b u t i o n s i n a) and b). [a 0 normalisation] ro I 5.0 2.5 -0.0 -x E 2-2.5 CO Q ^ 5 . 0 • % 7 . 5 -10.0 0 1 2 3 4 5 6 RADIUS (fm) Figure A9.2 Proton matter ( s o l i d ) and charge (short dash) density differences between 1 8 » 1 6 0 . The unfolding was ca r r i e d out as described i n section A9.6. Also shown i s the charge density difference with MG forms with the RMS charge r a d i i of 1 8 0 and 1 6 0 (long dash). The dotted curve i s the charge density from a c a l c u l a t i o n of Brown et a l . [BR079a, BR079b]. -2 1 2 3 4 5 6 -2 J L J L 0 1 2 3 4 5 6 ( f m ) Figure A9.3 Matter d e n s i t i e s (a and b) and charge d e n s i t i e s (c and d) corresponding to the f i r s t 6 FL d e n s i t i e s , constrained to zero RMS i n t e g r a l and zero norm. [a 0 normalisation] 274 Appendix X Moments and the Folding Process A10.1 The Folding and the RMS Integrals The convolution i n t e g r a l used i n f o l d i n g the proton form factor with a matter density to a r r i v e at a charge density has been written: p c h ( r ) = / PmUr-jL'l) p P P ( r , ) d i ' ( A l o a ) where p m i s a matter d i s t r i b u t i o n and Ppp i s a form f a c t o r . Consider, now, the RMS i n t e g r a l : I = / P c h ( r ) r 2 dr = / [/ P m ( | r - r ' | ) r 2 dr ] P p p ( r ' ) dr' (A10.2) Changing i n t e g r a t i o n va r i a b les (rj-r_' ) -> _r' i n the inner i n t e g r a l and noting that terms containing odd powers of cos6'= _r»£'/(|r| |r'|) vanish i n the angular i n t e g r a t i o n , we a r r i v e at: I = ( 4TT f °° p (r) vh dr )( 4ir / °°p (r') r ' 2 dr' ) + f 4TT / °° p (r) r 2 dr )( 4ir f "p (r') r'1* dr' ) (A10.3) V J0 m 0 PP For unit normalised d i s t r i b u t i o n s , t h i s gives: < r 2 > = < r 2 > + < r 2 > (A10.4) ch m pp 275 A10.2 Other Moments via the Fourier Transform The s u b s t i t u t i o n above may be used to c a l c u l a t e the other even moments of the d i s t r i b u t i o n P ch( r)» but the c a l c u l a t i o n quickly becomes tedious. A more elegant way of a r r i v i n g at a general expression i s to note that the convolution i n t e g r a l A10.1 implies that: P c h ( q ) = p j q ) P p p ( q ) , (A10.5) where the p(q) are the Fourier transformed p ( r ) ' s : p(q) = / r sin(qr ) p(r) dr q 0 , x n . 2n v (-) <r > . = 4rr I [ q" n ] (A10.6) n=0 (2n)! (2n+l) The product i n equation A10.5 then gives: y 2 i v . 2 ( n - i ) v <r J> <r v J /> , „ n ? T V . i PP m < r c h > " ^ ^ 2n 1 < A 1 0' 7> c n j=0 Z n (2^1-1) Note that i f P p p ( q ) i s unit normalised (or consists of a sum of pieces that are un i t normalised) then p c n w i l l have the same normalisation as (or be the sum of pieces with the same normalisation as the sum of pieces comprising) p m« 

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