UBC Theses and Dissertations

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UBC Theses and Dissertations

Neutron radii from low energy pion scattering Gyles, William 1984

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C .1 NEUTRON RADII FROM LOW ENERGY PION SCATTERING b y W I L L I A M G Y L E S B . S c , T h e U n i v e r s i t y o f M a n c h e s t e r , 1976 M . S c , T he U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 979 A T H E S I S SUBMITTED I N P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PH I LOSOPHY i n THE F A C U L T Y OF GRADUATE S T U D I E S D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n q a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA A p r i l 1984 © W i l l i a m G y l e s , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r equ i r ements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Co lumb i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copy i ng o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s unde r s tood t h a t copy ing o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l owed w i thou t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 1956 Main Mall Vancouve r , Canada V6T 1Y3 i i Abstract Recent electron scattering measurements and muonic atom studies have allowed precise determinations of the charge d i s t r i b u t i o n s of n u c l e i . Measurements of the neutron d i s t r i b u t i o n s , however, have not progressed to th i s degree of s o p h i s t i c a t i o n , l a r g e l y because of the uncertainties i n the hadron-nucleus i n t e r a c t i o n . Charge d i s t r i b u t i o n measurements provide good tests of nuclear structure c a l c u l a t i o n s , but measurements of neutron d i s t r i b u t i o n s w i l l provide independent constraints on these c a l c u l a t i o n s and the potentials used. In t h i s experiment, %~ d i f f e r e n t i a l cross section r a t i o s were measured on pairs of isotopes ( 0 0 S , d ^ S ) , ( d 4 S , 3 2 S ) with 50 MeV pions and (26Mg,2l*Mg) with 45 MeV pions. Absolute d i f f e r e n t i a l cross sections were also measured for 3 2 S and 21+Mg. Magnetic spectrometers were used to c o l l e c t the data. The cross section r a t i o s were compared to o p t i c a l model calcu l a t i o n s i n which the parameters of a Fermi function representing the neutron d i s t r i b u t i o n of the larger isotope of each pair were varied. The rms radius difference between the two isotopes producing the best f i t was found to be independent of the d e t a i l s of the o p t i c a l p o t e n t i a l used, as long as the po t e n t i a l produced a f i t to the absolute cross sections. The neutron d i s t r i b u t i o n of the larger isotope was also represented as a Fermi function modified by a sum of spher i c a l Bessel functions, the c o e f f i c i e n t s of which were allowed to vary. The results for the rms radius differences were consistent with the Fermi function f i t s , except f or 3 4 S - 3 2 S , where the r e s u l t s d i f f e r e d by a f u l l standard deviation. T h e rms r a d i u s d i f f e r e n c e s f o u n d f o r t h e s u l f u r i s o t o p e s a g r e e d w i t h t h e r e s u l t s o f s h e l l - m o d e l c a l c u l a t i o n s b y H o d g s o n ( S t r 8 2 , H o d 8 3 ) . The e x t r a c t e d r m s r a d i u s d i f f e r e n c e o f t h e m a g n e s i u m i s o t o p e s was o n e s t a n d a r d d e v i a t i o n l e s s t h a n t h e s h e l l - m o d e l p r e d i c t i o n . The r e s u l t s f o r t h e F e r m i f u n c t i o n f i t s , F o u r i e r B e s s e l f i t s a n d t h e s i n g l e p a r t i c l e p o t e n t i a l ( S P P ) c a l c u l a t i o n s b y H o d g s o n ( H o d 8 3 ) a r e : F e r m i F o u r i e r B e s s e l SPP 36s - 3 2 S 0 . 1 3 5 ± 0 . 0 5 7 0 . 1 4 ± 0 . 0 7 0 . 1 7 1 ( f m ) 3hS - 3 2 S 0 . 1 0 3 ± 0 . 0 3 2 0 . 0 6 1 ± 0 . 0 3 5 0 . 0 9 1 ( f m ) 26 M g_24 M g o . 0 7 6 ± 0 . 0 4 3 0 . 0 7 7 ± 0 . 0 5 6 0 . 1 2 1 ( f m ) i v Table of Contents Abstract 11 Table of Contents iv List of Tables v i i List of Figures v i l l Acknowledgements x i i i Chapter I M e a s u r e m e n t o f N e u t r o n D i s t r i b u t i o n s 1 . 1 I n t r o d u c t i o n 1 1 . 2 N e u t r o n D i s t r i b u t i o n P r o b e s 5 1 . 2 . 1 P r o t o n s 6 1 . 2 . 2 a p a r t i c l e s 9 1 . 2 . 3 K a o n s 10 1 . 2 . 4 P i o n s 14 1 . 2 . 4 . 1 H i g h e n e r g y p i o n s 17 1 . 2 . 4 . 2 R e s o n a n c e e n e r g y p i o n s 18 1 . 2 . 4 . 3 S c a t t e r i n g o f l o w e n e r g y p i o n s 19 1.3" D i s c u s s i o n o f E x p e r i m e n t s P e r f o r m e d i n t h i s W o r k 21 Chapter II P i o n - N u c l e u s S c a t t e r i n g 2 . 1 I n t r o d u c t i o n 24 2 . 2 D e r i v a t i o n o f t h e P i o n - N u c l e u s O p t i c a l P o t e n t i a l 24 2 . 2 . 1 F i r s t o r d e r p o t e n t i a l 27 2 . 2 . 2 F i x e d s c a t t e r e r a p p r o x i m a t i o n 29 2 . 2 . 3 K i s s l i n g e r p o t e n t i a l 3 0 2 . 2 . 4 S e c o n d o r d e r p o t e n t i a l 31 2 . 2 . 5 A b s o r p t i o n 32 2 . 3 D e l t a - H o l e M o d e l 33 Chapter III The Experiments 3.1 I n t r o d u c t i o n 37 3.2 The 3 6 S , 3 2 S Experiment 43 3.2.1 The QD spectrometer 43 3.2.2 Wire chambers 46 3.2.3 Beam monitors 46 3.2.4 E l e c t r o n i c l o g i c 48 3.2.5 Data a c q u i s i t i o n 51 3.2.6 Targets 51 3.3 The 3 4 S , 3 2S and 2 6Mg, 2 4Mg Experiments 52 3.3.1 The QQD spectrometer 52 3.3.2 S c a t t e r i n g chamber 57 3.3.3 Target ladder 58 3.3.4 Wire chambers 58 3.3.5 Beam monitors 59 3.3.6 E l e c t r o n i c l o g i c 61 3.3.7 Background 61 3.3.8 Targets 63 Chapter IV A n a l y s i s 4.1 I n t r o d u c t i o n 65 4.2 QD Experiment 65 4.2.1 Cuts 65 4.2.2 Wire chamber e f f i c i e n c i e s 68 4.2.3 C a l c u l a t i n g the cross s e c t i o n r a t i o s 69 4.3 QQD Experiment 72 4.3.1 Cuts 72 4.3.2 Peak f i t t i n g 76 4.3.3 Cross s e c t i o n r a t i o s 76 4.4 C o r r e c t i o n f o r I s o t o p i c Composition 80 4.5 Absolute Cross Sections 84 Chapter V Density D i s t r i b u t i o n A n a l y s i s 5.1 I n t r o d u c t i o n 90 5.2 O p t i c a l P o t e n t i a l 90 5.2.1 F i t s to absolute cross s e c t i o n s 93 5.3 F i t s to Ratios of Cross Sections 93 5.3.1 Fermi f u n c t i o n f i t s 99 5.3.1.1 S e n s i t i v i t y to assumed nucleon d i s t r i b u t i o n s 99 5.3.1.2 S e n s i t i v i t y to o p t i c a l p o t e n t i a l parameters 100 5.3.1.3 3 6 S / 3 2 S r a t i o s 103 5.3.1.4 3 4 S / 3 2 S r a t i o s 104 5.3.1.5 2 6Mg/ 24Mg r a t i o s 104 v i 5 . 3 . 2 F o u r i e r B e s s e l a n a l y s i s 115 5 . 3 . 2 . 1 R a d i a l m o m e n t s 118 5 . 3 . 2 . 2 D e p e n d e n c e o n t h e c u t o f f r a d i u s , R c 1 2 0 5 . 3 . 2 . 3 N e u t r o n d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s 127 Chapter VI C o n c l u s i o n s 132 L is t of References 137 Appendix 1 F i n d i n g T r a n s f e r C o e f f i c i e n t s 144 Appendix 2 S u l f u r T a r g e t D e n s i t y M e a s u r e m e n t 147 Appendix 3 S p e c t r o m e t e r A c c e p t a n c e 152 Appendix 4 P e a k F i t t i n g 156 A 4 . 1 S u l p h u r D a t a 156 A 4 . 2 M a g n e s i u m D a t a 157 Appendix 5 R e p r i n t o f R e f e r e n c e ( J o h 7 9 ) 163 Appendix 6 T a r g e t L a d d e r 167 v i i L i s t of Tables TABLE I Specifications of QQD Spectrometer PAGE 53 TABLE II Cross Section Ratios of 3 6S/ 3 2S PAGE 71 TABLE III Cross Section Ratios of 3 4S/ 3 2S PAGE 82 TABLE IV Cross Section Ratios of 26Mg/21tMg PAGE 83 TABLE V Absolute Differential Cross Sections for 48.4 MeV %~ Scattering on 3 2S PAGE 88 TABLE VI Absolute Differential Cross Sections for 43.9 MeV i t - Scattering on 21tMg PAGE 89 TABLE VII Optical Potential Parameter Sets PAGE 92 TABLE VIII Parameter Sets used in Fitting the Data PAGE 94 TABLE IX Change in 6r n from 3kS/32S Data with' Variations of the Potential Parameters PAGE 105 TABLE X Fits to 3 6S/ 3 2S Data with Fermi Function Distribution Densities PAGE 106 TABLE XI Fits to 3kS/32S Data using Optical Potential SMC79 PAGE 107 TABLE XII Fits to 3hS/32S Data using Optical Potential SMC 81 PAGE 108 TABLE XIII Fits to 26Mg/24Mg Data with Fermi Function Densities PAGE 109 TABLE XIV Results of Fits to 3 6S/ 3 2S Data with Fourier Bessel Series for 3 6S Neutron Distribution PAGE 124 TABLE XV Results of Fits to 3 4S/ 3 2S Data using Fourier Bessel Series for 3kS Neutron Distribution PAGE 125 TABLE XVI Results of Fits to 26Mg/24Mg Data using a Fourier Bessel Series for 26Mg Neutron Distribution PAGE 126 L i s t of Figures FIGURE 1 PAGE 8 C o m p a r i s o n o f e r r o r b a n d o f p r o t o n d e n s i t y d i f f e r e n c e s b e t w e e n b t f F e - 4 0 C a o b t a i n e d f r o m a n a l y s i s o f 8 0 0 MeV p r o t o n e l a s t i c s c a t t e r i n g d a t a ( o u t e r b a n d ) w i t h t h a t o b t a i n e d b y e l e c t r o n s c a t t e r i n g e x p e r i m e n t s ( i n n e r c r o s s h a t c h e d b a n d ) . T h e f i g u r e was t a k e n f r o m R e f e r e n c e ( R a y 8 3 ) . FIGURE 2 PAGE 11 M a t t e r d e n s i t y d i s t r i b u t i o n p m d i f f e r e n c e o f 4 8 C a - 4 ° C a , e x t r a c t e d f r o m a n a l y s i s ( G i l 8 0 b ) o f 104 MeV a - p a r t i c l e s c a t t e r i n g . F i g u r e 2 a . s h o w s 4 i t r 2 6 p m a n d i s t a k e n d i r e c t l y f r o m r e f e r e n c e ( G i l 8 0 b ) . F i g u r e 2 b . s hows 6 p m a n d was o b t a i n e d b y d i v i d i n g F i g u r e 2 a . b y 4 n r 2 . FIGURE 3 PAGE 15 i t + p a n d ir~p t o t a l c r o s s s e c t i o n s . The c u r v e s a r e c a l c u l a t e d f r o m p h a s e s h i f t s w h i c h w e r e f i t t e d t o t h e w o r l d m - n u c l e o n d a t a u s i n g t h e p r o g r a m S A I D ( A r n 8 2 ) . FIGURE 4 PAGE 38 P e r c e n t a g e c h a n g e , 6 R , o f c a l c u l a t e d 3 6 S / 3 2 S c r o s s s e c t i o n r a t i o t o 15% i n c r e a s e i n o p t i c a l p o t e n t i a l p a r a m e t e r s . T he p a r a m e t e r s , o t h e r t h a n t h e o n e v a r i e d , a r e S e t 1 ( s e e T a b l e V I I ) . FIGURE 5 PAGE 39 P e r c e n t a g e c h a n g e o f c a l c u l a t e d 3 6 S / 3 2 S c r o s s s e c t i o n r a t i o t o 0 . 1 0 0 f m i n c r e a s e i n 3 6 c n , t h e F e r m i d i s t r i b u t i o n h a l f d e n s i t y r a d i u s p a r a m e t e r f o r t h e n e u t r o n d i s t r i b u t i o n o f 3 6 S . T h e p o t e n t i a l was SMC79 w i t h S e t 1 p a r a m e t e r s . A l s o s h o w n i s t h e f a c t o r E f f , d e f i n e d i n t h e t e x t . FIGURE 6 PAGE 42 M13 b e a m l i n e a n d QQD s p e c t r o m e t e r . BM1 a n d BM2 a r e beam m o n i t o r s B I a n d B2 d e s c r i b e d i n t h e t e x t . FIGURE 7 PAGE 44 S c h e m a t i c d i a g r a m o f t h e QD s p e c t r o m e t e r . FIGURE 8 PAGE 45 C o o r d i n a t e s u s e d i n d i s c u s s i o n o f t h e QD s p e c t r o m e t e r . FIGURE 9 PAGE 49 E l e c t r o n i c l o g i c f o r t h e QD e x p e r i m e n t . FIGURE 10 PAGE 50 T i m e - o f - f l i g h t s p e c t r u m s h o w i n g r e l a t i v e t r a v e l t i m e s a n d a m o u n t s o f IT'S, U'S , a n d e ' s down t h e M13 b e a m l i n e . T h i s d a t a i s f r o m t h e QQD e x p e r i m e n t a n d i s v e r y s i m i l a r t o t h a t o f t h e QD e x p e r i m e n t . FIGURE 11 PAGE 54 C o o r d i n a t e s u s e d i n d i s c u s s i o n o f QQD e x p e r i m e n t s . FIGURE 12 PAGE 56 V a r i a t i o n i n p o s i t i o n ( X 3 ) o f p i o n s a t w i r e c h a m b e r 3 w i t h momen t um . S p / p i s t h e p e r c e n t a g e v a r i a t i o n o f t h e momentum f r o m t h e n o m i n a l v a l u e o f 128 M e V / c . T h e d i s p e r s i o n d X 3 / d ( 6 p / p ) , i s g i v e n b y t h e s l o p e o f t h e c u r v e a n d i s s e e n t o b e a l m o s t c o n s t a n t b e t w e e n 6 p / p = -6% a n d 6 p / p = 4%. FIGURE 13 PAGE 60 C o m p a r i s o n o f c o u n t r a t e s o f beam m o n i t o r s B 1 » B 2 a n d j i l » u 2 . T h e p o i n t c i r c l e d s h o w s t h e e f f e c t o f o f f s e t t i n g t h e s l i t s b y 26 mm f r o m t h e b e a m l i n e c e n t r e . FIGURE 14 PAGE 62 E l e c t r o n i c l o g i c u s e d i n t h e QQD e x p e r i m e n t s . FIGURE 15 PAGE 64 D e n s i t y p l o t o f s c a t t e r e d p i o n s f r o m t h e e m p t y s t e e l t a r g e t f r a m e u s e d i n 3 l f S / 3 2 S e x p e r i m e n t . T h e p o s i t i o n s a t t h e t a r g e t a r e c a l c u l a t e d f r o m t h e m e a s u r e d p o s i t i o n s o f t h e p i o n s a t w i r e c h a m b e r s 1 a n d 2 . S u p e r i m p o s e d i s a d r a w i n g o f t h e t a r g e t f r a m e , t o s c a l e . FIGURE 16 PAGE 75 P o l a r a n g l e (w) b e t w e e n t h e m e a s u r e d a n d c a l c u l a t e d t r a j e c t o r i e s o f t h e p a r t i c l e a t t h e e x i t w i r e c h a m b e r s o f t h e QQD s p e c t r o m e t e r . T he d a s h e d l i n e r e s u l t s f r o m d i v i d i n g b y to. T h e p o s i t i o n o f t h e c u t made i s s h o w n . X FIGURE 17 PAGE 77 E n e r g y s p e c t r u m f r o m 3 4 S a t 1 0 0 ° w i t h f i t t e d p e a k s a t t h e g r o u n d s t a t e , a n d e x c i t e d s t a t e s a t 2 . 1 3 MeV a n d 4 . 5 M e V . FIGURE 18 PAGE 78 E n e r g y s p e c t r u m f r o m 2 6 M g a t 1 00 ° s h o w i n g f i t t e d g r o u n d s t a t e a n d 1 . 8 1 MeV s t a t e p e a k s . FIGURE 19 PAGE 95 4 8 . 4 MeV n~ o n 3 2 S a b s o l u t e d i f f e r e n t i a l c r o s s s e c t i o n s , c o m p a r e d t o c a l c u l a t i o n s w i t h t h e two p o t e n t i a l s SMC79 a n d S M C 8 1 . T h e e r r o r b a r s a r e s m a l l e r t h a n t h e d a t a p o i n t s . T h e p o t e n t i a l p a r a m e t e r s e t s a r e d e f i n e d i n T a b l e s V I I a n d V I I I . FIGURE 20 PAGE 96 4 3 . 9 MeV nT o n 2 4 M g a b s o l u t e d i f f e r e n t i a l c r o s s s e c t i o n s , c o m p a r e d t o c a l c u l a t i o n s w i t h t h e two p o t e n t i a l s SMC79 a n d S M C 8 1 . T h e p o t e n t i a l p a r a m e t e r s e t s : S e t 1 , S e t E , a n d S e t E f l O a r e d e f i n e d i n T a b l e s V I I a n d V I I I . FIGURE 21 PAGE 101 T h e e f f e c t o f c h a n g i n g t h e s h a p e o f t h e 3 4 S n e u t r o n d i s t r i b u t i o n w h i l e k e e p i n g t h e rms r a d i u s c o n s t a n t a t 3 . 2 5 9 f m . The p o t e n t i a l u s e d was SMC81 w i t h p a r a m e t e r S e t E f l . FIGURE 22 PAGE 110 3 6 S / 3 2 S c r o s s s e c t i o n r a t i o s . T he SMC79 p o t e n t i a l c u r v e s u s e d S e t 1 p a r a m e t e r s , d e f i n e d i n T a b l e V I I . T h e SMC81 p o t e n t i a l c u r v e u s e d S e t E f l p a r a m e t e r s . T he c u r v e l a b e l l e d NO F I T u s e d e q u a l n e u t r o n d i s t r i b u t i o n s h a p e s f o r 3 6 S a n d 3 2 S . FIGURE 23 PAGE 111 3 1 + S / 3 2 S c r o s s s e c t i o n r a t i o s f i t t e d w i t h t h e SMC79 p o t e n t i a l . T h e p a r a m e t e r s e t s i n d i c a t e d a r e d e f i n e d i n T a b l e s V I I a n d V I I I . S e t I f 2 h a d \ = 0 . The c u r v e l a b e l l e d NO F I T u s e d e q u a l n e u t r o n d i s t r i b u t i o n s h a p e s f o r 3 4 S a n d 3 2 S . FIGURE 24 PAGE 112 3 1 + S / 3 2 S c r o s s s e c t i o n r a t i o s f i t t e d w i t h t h e SMC81 p o t e n t i a l . T h e p a r a m e t e r s e t s i n d i c a t e d a r e d e f i n e d i n T a b l e s V I I a n d V I I I . T he c u r v e l a b e l l e d NO F I T u s e d e q u a l n e u t r o n d i s t r i b u t i o n s h a p e s f o r 3 t f S a n d 3 2 S . F I G U R E 2 5 P A G E 1 1 3 2 6 M g / 2 I f M g c r o s s s e c t i o n r a t i o s f i t t e d w i t h t h e SMC79 p o t e n t i a l . T h e p a r a m e t e r s e t s a r e d e f i n e d i n T a b l e s V I I a n d V I I I . T he c u r v e l a b e l l e d NO F I T u s e d e q u a l n e u t r o n d i s t r i b u t i o n s h a p e s f o r 2 6 M g a n d 2 l * M g . F I G U R E 2 6 P A G E 1 1 4 2 6 M g / 2 1 t M g c r o s s s e c t i o n r a t i o s f i t t e d w i t h t h e SMC81 p o t e n t i a l . T h e p a r a m e t e r s e t s a r e d e f i n e d i n T a b l e s V I I a n d V I I I . T he c u r v e l a b e l l e d NO F I T u s e d e q u a l n e u t r o n d i s t r i b u t i o n s h a p e s f o r 2 6 M g a n d 2 1 * M g . F I G U R E 2 7 P A G E 1 2 1 V a r i a t i o n o f y} o f t h e F o u r i e r B e s s e l f i t t o 3 1 t S / 3 2 S d a t a a s t h e c u t o f f r a d i u s , R c , i s c h a n g e d . The s t a r t i n g d e n s i t y f o r t h e 3 1 t S n e u t r o n d i s t r i b u t i o n was p r o p o r t i o n a l t o t h a t o f 3 2 S . F I G U R E 2 8 P A G E 1 2 2 V a r i a t i o n o f % 2 o f t n e F o u r i e r B e s s e l f i t t o 3 t f S / 3 2 S d a t a a s t h e c u t o f f r a d i u s , R c , i s c h a n g e d . T h e s t a r t i n g d e n s i t y f o r t h e 3 t f S n e u t r o n d i s t r i b u t i o n was t h e b e s t f i t F e r m i d i s t r i b u t i o n t o t h e d a t a . F I G U R E 2 9 P A G E 1 2 9 R e s u l t s o f F o u r i e r B e s s e l a n a l y s i s o f 3 6 S / 3 2 S c r o s s s e c t i o n r a t i o s . T h e d a s h e d c u r v e s a r e t h e s t a t i s t i c a l e r r o r e n v e l o p e a n d t h e s o l i d l i n e h a s t h e c o m p l e t e n e s s e r r o r a d d e d . T h e SMC81 p o t e n t i a l was u s e d w i t h p a r a m e t e r s e t E f l . T he s o l i d d o t s a r e t h e r e s u l t s o f SPP ( H o d 8 3 ) c a l c u l a t i o n s , w h i c h w e r e e v a l u a t e d e v e r y 0 . 5 f m . F I G U R E 3 0 P A G E 1 3 0 R e s u l t s o f F o u r i e r B e s s e l a n a l y s i s o f 3 l * S / 3 2 S c r o s s s e c t i o n r a t i o s . T h e d a s h e d l i n e i s t h e s t a t i s t i c a l e r r o r e n v e l o p e a n d t h e s o l i d l i n e h a s t h e c o m p l e t e n e s s e r r o r a d d e d . The SMC81 p o t e n t i a l was u s e d w i t h p a r a m e t e r s e t E f l . T h e s o l i d d o t s a r e t h e r e s u l t s o f SPP ( H o d 8 3 ) c a l c u l a t i o n s , w h i c h w e r e e v a l u a t e d e v e r y 0 . 5 f m . FIGURE 31 PAGE 131 R e s u l t s o f F o u r i e r B e s s e l a n a l y s i s o f 2 6 M g / 2 4 M g r a t i o s . T he d a s h e d l i n e i s t h e s t a t i s t i c a l e r r o r e n v e l o p e a n d t h e s o l i d l i n e h a s t h e c o m p l e t e n e s s e r r o r a d d e d . The SMC81 p o t e n t i a l w a s u s e d w i t h p a r a m e t e r S e t E f l O . T h e s o l i d d o t s a r e t h e r e s u l t s o f SPP ( H o d 8 3 ) c a l c u l a t i o n s , w h i c h w e r e e v a l u a t e d e v e r y 0 . 5 f m . FIGURE 32 PAGE 150 C a l i b r a t i o n c u r v e f o r m e a s u r i n g s u l f u r t a r g e t t h i c k n e s s b y a t t e n u a t i o n o f e l e c t r o n s f r o m a c o l l i m a t e d 1 0 6 R u s o u r c e . T h e p o i n t f o r t h e 3 4 S t a r g e t h a s b e e n a d j u s t e d t o a c c o u n t f o r t h e h i g h e r a t o m i c w e i g h t o f 3 4 S . FIGURE 33 PAGE 151 V a r i a t i o n o f 3 2 S a n d 3 4 S t a r g e t d e n s i t i e s a s m e a s u r e d f r o m t h e a t t e n u a t i o n o f e l e c t r o n s f r o m a 1 0 6 R u s o u r c e . T he d e n s i t i e s w e r e m e a s u r e d a t t h e p o i n t s o f i n t e r s e c t i o n o f t h e l i n e s . T h e mean t a r g e t d e n s i t i e s w e r e 3 6 5 m g / c m 2 f o r 3 2 S a n d 2 9 0 m g / c m 2 f o r 3 4 S . FIGURE 34 PAGE 154 D i s t r i b u t i o n o f Q' ( d e f i n e d i n t h e t e x t ) f r o m t h e w h o l e c a r b o n t a r g e t . FIGURE 35 PAGE 155 A c c e p t a n c e o f t h e s p e c t r o m e t e r , h o r i z o n t a l l y a c r o s s t h e t a r g e t p o s i t i o n . T h i s wa s c a l c u l a t e d a t e a c h p o s i t i o n o n t h e t a r g e t f r o m t h e d i s t r i b u t i o n o f t r a j e c t o r i e s o f t h e d e t e c t e d p a r t i c l e s . Acknowledgement s Producing a thesis requires putting a large number of words down onto paper. They must, of course, be the righ t words i n the r i g h t order. In every part of the production of this thesis, I have had invaluable help and support from my friends and colleagues. Indeed, the most rewarding experience i n the purely l i t e r a r y part of this exercise was to see how r e a d i l y many of my friends gave of t h e i r time when I needed i t . Choosing the r i g h t words begins with choosing the righ t experiment and acquiring the data. My thesis supervisor, Prof. Richard Johnson, provided the d i r e c t i o n for t h i s work and a large part of the energy for keeping everything, including myself, moving i n that d i r e c t i o n . I am indebted to him also f or teaching me most of what I know about experimental physics, and I hope I can achieve his a b i l i t y of viewing experimental problems from the correct perspective, thus reducing t h e i r magnitude to manageable proportions. Perhaps most importantly, I have also enjoyed working on experiments with Dick because of his very i n d i v i d u a l sense of humour. Many people contributed to acquiring the data and I would l i k e to express my gratitude to a l l of them, including the ones I don't mention by name. In p a r t i c u l a r I would l i k e to thank Dave G i l l for the tremendous amount of work and i n s p i r a t i o n he contributed to these experiments and for his cheerful optimism throughout the runs. I thank him also, along with Randy Sobie, Sig Martin, Cris Wiedner, Roman Tacik and Hans Roser, for commissioning the QQD spectrometer i n a time that must q u a l i f y f or some kind of record. I would also l i k e to thank Christopher Stevens for so often quickly producing p r e c i s i o n work from my l a s t minute sketchy drawings. Getting the large number of words onto paper i n the correct order proved the most demanding task i n terms of physical endurance and ingenuity. This task was s i m p l i f i e d to some degree by the use of the AES word processing system at TRIUMF. I would l i k e to thank George Ludgate and Visha Saravanbawan, for coupling me r e l a t i v e l y p a i n l e s s l y to this system. It was heartwarming to see how many of my friends responded to my requests for help i n meeting various deadlines, by working through the night on many occasions. My good f r i e n d , Chris Oram, performed a miracle i n t h i s stage by restoring order a f t e r I had given i n to entropy. Bruce Barnett, Pat B e l l , John Bailey, Radha Jhappan and Nancy McMaster also deserve my sp e c i a l thanks here. I would also l i k e to thank Radha and Nancy for being such good friends and understanding house mates and Radha for many of the f i g u r e s . Special mention and place of honour i n th i s category, however, must go to L i z Hewetson for keeping me a l i v e and r e l a t i v e l y sane whilst persevering with a word processing system, that she never made friends with, for up to 48 hours at a s t r e t c h . L a s t l y I would l i k e to thank Bruce Barnett for his contributions of d i l i g e n t hard work, bright ideas and i l l u m i n a t i n g discussions throughout this work. I am indebted to him for volunteering so much of his time and energy on so many occasions. 1 CHAPTER I Measurement of Neutron Distributions 1.1 Introduction T h e f i r s t e x p e r i m e n t a l d e t e r m i n a t i o n o f a n u c l e a r r a d i u s was made u s i n g a s t r o n g l y i n t e r a c t i n g p r o b e b y R u t h e r f o r d i n 1929 ( R u t 2 9 ) , i n h i s a n a l y s i s o f a l p h a - p a r t i c l e s c a t t e r i n g a n d a l p h a - d e c a y o f h e a v y n u c l e i . I t was n o t u n t i l 1 9 5 1 t h a t L y m a n , H a n s o n a n d S c o t t ( L y m 5 1 ) made t h e f i r s t d e t e r m i n a t i o n s o f n u c l e a r c h a r g e r a d i i f r o m e l e c t r o n s c a t t e r i n g . S i n c e t h e n , e l e c t r o n s c a t t e r i n g e x p e r i m e n t s a n d a n a l y s e s , c o u p l e d w i t h m u o n i c a t o m m e a s u r e m e n t s , h a v e d e v e l o p e d t o a h i g h d e g r e e o f s o p h i s t i c a t i o n , a l l o w i n g p r e c i s e r o o t mean s q u a r e r a d i i a n d d e t a i l e d s h a p e s o f t h e c h a r g e d i s t r i b u t i o n s t o be m e a s u r e d ( D r e 7 4 , F r i 7 3 , F r i 7 5 , N e g 7 9 a ) . C r o s s s e c t i o n s down t o a b o u t 1 0 - 1 0 f m 2 / s r a r e m e a s u r e d a n d r o o t mean s q u a r e ( r m s ) c h a r g e r a d i i w i t h a p r e c i s i o n o f a f e w t h o u s a n d t h s o f a fm may b e e x t r a c t e d . M e a s u r e m e n t s o f t h e m a t t e r o r n e u t r o n d i s t r i b u t i o n s , o n t h e o t h e r h a n d , r e q u i r e s t r o n g l y i n t e r a c t i n g p r o b e s a n d h a v e b e e n i m p e d e d b y t h e u n c e r t a i n t i e s i n t h e p r o b e - n u c l e u s i n t e r a c t i o n s . A s w e l l a s b e i n g i n t r i n s i c a l l y i n t e r e s t i n g q u a n t i t i e s , t h e n e u t r o n r a d i i o f n u c l e i a r e r e l e v e n t t o t h e u n d e r s t a n d i n g o f i s o t o p e s h i f t s ( p a r t i c u l a r l y o f t h e c a l c i u m i s o t o p e s ) , C o u l o m b e n e r g y d i f f e r e n c e s o f m i r r o r p a i r s ( N o l a n - S c h i f f e r ( N o l 6 9 ) a n o m a l y ) , t h e r e n o r m a l i s a t i o n o f t h e e f f e c t i v e i n t e r a c t i o n ( V a r 7 8 ) a n d e v e n t h e e x i s t e n c e o f a c h a r g e s y m m e t r y b r e a k i n g i n t e r a c t i o n ( S a t 7 6 ) . T h e s t r o n g i n t e r a c t i o n h a s b e e n f o u n d t o be c h a r g e s y m m e t r i c down t o t h e l e v e l o f a f e w p e r c e n t ( S a l 8 3 , D a v 8 1 ) , i m p l y i n g t h a t t h e n - n a n d p - p i n t e r a c t i o n s a r e a p p r o x i m a t e l y e q u a l a p a r t f r o m e l e c t r o m a g n e t i c e f f e c t s . I t i s clear that t h i s must only be an approximation, considering the neutron-proton mass d i f f e r e n c e . Without the Coulomb i n t e r a c t i o n , therefore, one might expect the neutron and proton d i s t r i b u t i o n s i n a nucleus with Z=N, to be equal. The Coulomb p o t e n t i a l causes a mutual repulsion between the protons which from a nai've viewpoint could be thought to increase the radius of the proton d i s t r i b u t i o n . However, from a less naive viewpoint, since the neutrons and protons are constrained to have equal binding energy, fo s t a b i l i t y against 6-decay, the increased depth of the proton p o t e n t i a l well causes the proton o r b i t s to be more constrained and hence the radius of the proton d i s t r i b u t i o n to be narrower. This roughly cancels the e f f e c t of the Coulomb repulsion. In addition, the nucleon-nucleon force i s momentum dependent and the Coulomb i n t e r a c t i o n causes neutrons and protons of equal binding energy to have d i f f e r e n t momenta i n the nucleus. The Pauli exclusion p r i n c i p l e i n h i b i t s i n t e r a c t i o n between l i k e p a r t i c l e s i n the nucleus. In a nucleus with N>Z, each proton has more unlike p a r t i c l e s with which to i n t e r a c t than the neutrons do, leading to an enhancement of the a t t r a c t i v e proton p o t e n t i a l by an amount proportional to (N-Z)/A. This part of the nuclear p o t e n t i a l i s known as the symmetry term. Differences between neutron and proton d i s t r i b u t i o n s are seen to a r i s e , therefore, through the e f f e c t s of the symmetry p o t e n t i a l and the Coulomb p o t e n t i a l . I t i s clear that c a l c u l a t i o n of the nucleon d i s t r i b u t i o n s requires a thorough treatment of the complex i n t e r p l a y between these d i f f e r e n t e f f e c t s . The Coulomb energy d i f f e r e n c e , E^, i s the difference between a state i n a nucleus (Z,N) and the i s o b a r i c analogue state i n the nucleus 3 ( Z + 1 , N - 1 ) . T h e N o l a n - S c h i f f e r a n o m a l y r e f e r s t o a d i s a g r e e m e n t b e t w e e n e x p e r i m e n t a l l y m e a s u r e d a n d t h e o r e t i c a l l y c a l c u l a t e d E^. A n a s s u m p t i o n i n t h e a n a l y s i s o f t h e C o u l o m b e n e r g y d i f f e r e n c e i s t h a t t h e d i s t r i b u t i o n o f t h e e x t r a p r o t o n i n t h e i s o b a r i c s t a t e i s t h e same a s t h a t o f t h e e x t r a ( N - Z ) n e u t r o n s i n t h e p a r e n t s t a t e . T h e m a i n c o n t r i b u t i o n t o E^ i s t h e n g i v e n b y t h e C o u l o m b i n t e r a c t i o n o f t h i s e x t r a p r o t o n w i t h t h e c o r e o f p r o t o n s . A d d i t i o n a l c o r r e c t i o n s due t o e x c h a n g e , e l e c t r o m a g n e t i c s p i n -o r b i t i n t e r a c t i o n , n - p m a s s d i f f e r e n c e a n d t h e T h o m a s - E h r m a n s h i f t ( T h o 5 1 , T h o 5 2 , E h r 5 1 ) a r e t h e n made ( N o l 6 9 ) . N o l a n a n d S c h i f f e r f o u n d t h a t t h e e x p e r i m e n t a l C o u l o m b e n e r g y d a t a c o u l d n o t b e r e p r o d u c e d u n l e s s t h e rms r a d i u s o f t h e n e u t r o n e x c e s s d i s t r i b u t i o n was o n l y s l i g h t l y g r e a t e r t h a n t h a t o f t h e d i s t r i b u t i o n o f t h e c o r e . The c a l c u l a t e d C o u l o m b e n e r g y i s t o o s m a l l i f a W o o d s - S a x o n w e l l w h i c h f i t s t h e e m p i r i c a l c h a r g e r a d i u s i s u s e d t o c a l c u l a t e t h e n e u t r o n e x c e s s w a v e f u n c t i o n s . I n c o m p a r i n g i s o t o p e s o f c a l c i u m , f o r i n s t a n c e , i t was p r e d i c t e d b y t h i s m e t h o d ( S c h 6 9 ) t h a t t h e n e u t r o n r a d i u s i s n e a r l y c o n s t a n t up t o 4 8 C a . T h i s i s i n d i s a g r e e m e n t w i t h r e c e n t p r o t o n ( C h a 7 8 , R a y 8 1 ) a n d a l p h a p a r t i c l e ( G i l 8 0 ) s c a t t e r i n g r e s u l t s a n d a l s o w i t h H a r t r e e - F o c k c a l c u l a t i o n s ( s e e R a y 8 1 ) . T h e m e a s u r e d C o u l o m b e n e r g y d i f f e r e n c e s c o u l d be e x p l a i n e d b y c o r e p o l a r i s a t i o n i f t h e d i f f e r e n c e r n - r p w a s i n d e e d s m a l l . I n t h i s c a s e t h e v a l e n c e n e u t r o n s w i l l p u s h i n t h e c o r e n e u t r o n s a n d i n t h e i s o b a r i c a n a l o g u e n u c l e u s t h e v a l e n c e p r o t o n s w i l l c o m p r e s s t h e p r o t o n c o r e p r o d u c i n g a n i n c r e a s e i n e l e c t r o s t a t i c e n e r g y . H a r t r e e - F o c k c a l c u l a t i o n s ( G a i 7 1 , N e g 7 1 , A u e 7 4 ) t a k i n g i n t o a c c o u n t i s o s p i n i m p u r i t i e s i n t h e c o r e a n d p o l a r i s a t i o n o f t h e c o r e b y t h e e x t r a n u c l e o n s u g g e s t e d t h a t t h e p o l a r i s a t i o n o f t h e c o r e i s o f n e g l i g i b l e i m p o r t a n c e a n d t h e c o r r e c t i o n d u e t o i s o s p i n i m p u r i t y i s n o t l a r g e e n o u g h t o r e m o v e t h e a n o m a l y . V a r m a 4 a n d Z a m i c k ( V a r 7 8 ) , a l s o , c a l c u l a t e t h a t a l a r g e c o r e p o l a r i s a t i o n c a u s e s p r o b l e m s w i t h t h e e f f e c t i v e p a i r i n g i n t e r a c t i o n b e t w e e n i d e n t i c a l n u c l e o n s . T h e y c o n c l u d e t h a t t h e N o l a n - S c h i f f e r a n o m a l y c a n n o t b e e x p l a i n e d b y c o r e p o l a r i s a t i o n . S a t o ( S a t 7 6 ) s u g g e s t s t h a t t h e a n o m a l y c a n b e e x p l a i n e d b y a c h a r g e s y m m e t r y b r e a k i n g f o r c e w h i c h i s n o t i n c o n s i s t e n t w i t h n u c l e o n - n u c l e o n d a t a . S i n c e t h e C o u l o m b e n e r g y d i f f e r e n c e i s i n t i m a t e l y c o n n e c t e d t o t h e d i s t r i b u t i o n o f t h e e x t r a c o r e n u c l e o n , m e a s u r e m e n t s o f n e u t r o n d i s t r i b u t i o n s s h o u l d s h e d some l i g h t o n t h e a n o m a l y . P r e c i s e m e a s u r e m e n t s o f t h e c h a r g e d i s t r i b u t i o n s o f n u c l e i a r e d e r i v e d f r o m a n a l y s i s o f e l e c t r o n s c a t t e r i n g a n d m u o n i c a t o m d a t a ( F r i 7 5 , N e g 7 9 a ) . T h e s e c h a r g e d i s t r i b u t i o n s a r e r e p r o d u c e d q u i t e w e l l w i t h H a r t r e e - F o c k c a l c u l a t i o n s ( N e g 7 9 b ) a f t e r m a k i n g c o r r e c t i o n s f o r t h e p r o t o n f o r m f a c t o r a n d n e u t r o n e l e c t r o m a g n e t i c s p i n - o r b i t i n t e r a c t i o n a n d f o r m f a c t o r ( B e r 7 2 ) . P a r a m e t e r s a r e a d j u s t e d i n t h e H a r t r e e - F o c k c a l c u l a t i o n s ( N e g 7 0 , C a m 7 2 , D a v 7 4 ) t o g i v e a g r e e m e n t w i t h s i n g l e p a r t i c l e e x c i t a t i o n e n e r g i e s o r n u c l e o n r e m o v a l e n e r g i e s . E x t r a p a r a m e t e r s a r e n e e d e d t o r e d u c e t h e c a l c u l a t e d c h a r g e d e n s i t i e s i n t h e n u c l e a r c e n t r e b y a b o u t 10% t o a g r e e w i t h t h e m e a s u r e d v a l u e s ( D a v 7 4 ) . E q u i v a l e n t l y , t h e s e p a r a m e t e r s may be f i t t o r e p r o d u c e t h e m e a s u r e d c h a r g e r a d i i o f t h e s p h e r i c a l n u c l e i . I t i s e s s e n t i a l l y t h e d e n s i t y d e p e n d e n c e o f t h e e f f e c t i v e i n t e r a c t i o n w h i c h i s a d j u s t e d h e r e a n d n e u t r o n d e n s i t y m e a s u r e m e n t s s h o u l d p r o v i d e a c h e c k o n t h i s a d j u s t m e n t . C o m p a r i s o n o f t h e c a l c u l a t i o n s t o m e a s u r e d n e u t r o n d i s t r i b u t i o n s o r r a d i i s h o u l d b e s e n s i t i v e t o t h o s e e f f e c t s o f t h e C o u l o m b p o t e n t i a l a n d t h e s y m m e t r y t e r m i n t h e n u c l e a r p o t e n t i a l , w h i c h a r e n o t c l e a r l y s e p a r a t e d f r o m t h e e f f e c t s o f o t h e r p a r t s o f t h e p o t e n t i a l i n c o m p a r i n g t o 5 proton d i s t r i b u t i o n s only. Indeed, as pointed out above, Hartree-Fock c a l c u l a t i o n s which predict the charge d i s t r i b u t i o n s well are unable to resolve the Nolan-Schiffer anomaly. 1.2 Neutron D i s t r i b u t i o n Probes Unfortunately the experimental measurements of neutron d i s t r i b u t i o n s are sparse and imprecise, compared to those of the charge d i s t r i b u t i o n , and subject to model dependency. The model dependency i s a consequence of the necessity to use strongly i n t e r a c t i n g probes. The probe-nucleus i n t e r a c t i o n must be described i n terms of the known probe-nucleon in t e r a c t i o n s and the nucleon d i s t r i b u t i o n s inside the nucleus. The usual procedure here i s to develop an o p t i c a l p o t e n t i a l from a multiple scattering s e r i e s . Convergence of the series i n a few terms requires a large mean free path of the probe inside the nucleus and hence few scatterings from the nucleons. Penetration of the probe to the nuclear i n t e r i o r also requires a mean free path comparable to or larger than the nuclear radius. Protons, alpha ( a - ) p a r t i c l e s and pions have been used as probes of the matter or neutron d i s t r i b u t i o n s . P o s i t i v e kaons have been suggested but precise experiments await the a r r i v a l of a kaon fac t o r y . We s h a l l discuss b r i e f l y the advantages and disadvantages of each probe. None of the probes could be described as i d e a l but by combining measurements from d i f f e r e n t probes perhaps the advantages of each can be added. In any case i t i s useful to compare measurements from the various probes as a test of model dependency. Other possible probes (Bar77) such as the (p,n) reaction, pion charge exchange, pion production, beta decay and regenerative neutral kaon scattering are not discussed here. 6 1.2.1 Protons T h e t o t a l c r o s s s e c t i o n f o r p - p s c a t t e r i n g ( s e e C h a 7 8 ) h a s a m i n i m u m a t a r o u n d 4 0 0 MeV a n d a r a p i d r i s e t o a p l a t e a u a b o v e a b o u t 7 0 0 M e V . A g o o d c h o i c e f o r p r o t o n - n u c l e u s e x p e r i m e n t s i s t h e r e f o r e a r o u n d 1 GeV w h e r e t h e c r o s s s e c t i o n i s f l a t a n d e x p a n s i o n s o f t h e t - m a t r i x a b o u t t h e o n -s h e l l p o i n t s h o u l d w o r k w e l l . The c r o s s s e c t i o n a t t h i s e n e r g y i s a b o u t 50 mb w h i c h s u g g e s t s a m e a n f r e e p a t h i n t h e n u c l e u s o f a b o u t 1 f m , t h e r e f o r e s e c o n d o r d e r t e r m s i n t h e m u l t i p l e s c a t t e r i n g s e r i e s s h o u l d b e s i g n i f i c a n t . A n a l y s i s i s g e n e r a l l y i n t e r m s o f a K e r m a n , McManus a n d T h a l e r ( K e r 5 9 ) o p t i c a l p o t e n t i a l . T h e s i m p l e s t f i r s t o r d e r KMT p o t e n t i a l i s g i v e n b y •j— U 1 = t p + t p A - l P P n ^n w h e r e p^ ^ i s t h e F o u r i e r t r a n s f o r m o f t h e p o i n t p r o t o n , n e u t r o n g r o u n d s t a t e d e n s i t y a n d t i s t h e r e s p e c t i v e p r o t o n , n e u t r o n t - m a t r i x a n d A i s p , n t h e a t o m i c n u m b e r . T h e F e r m i m o t i o n o f t h e n u c l e o n s i s o f t e n i g n o r e d , w h i c h i s r e a s o n a b l e s i n c e t h e c r o s s s e c t i o n i s f l a t , a n d t h e t p > n a r e a s s u m e d t o be f u n c t i o n s o n l y o f e n e r g y a n d momentum t r a n s f e r . W i t h t h e s e a s s u m p t i o n s U 1 b e c o m e s a l o c a l p o t e n t i a l i n c o o r d i n a t e s p a c e ; V k r ) = I J d r ' P i ( r ' ) t . ( r - r ' ) i = p , n The s e c o n d o r d e r p o t e n t i a l c o n t a i n s c o r r e c t i o n s ( K e r 5 9 , B o r 7 7 , C h a 7 8 ) f o r t h e e f f e c t s o f c o r r e l a t i o n s a n d s p u r i o u s c e n t r e o f m a s s m o t i o n . T h e a d d i t i o n o f t h e s e c o r r e c t i o n s h a s a s m a l l e f f e c t o n t h e c a l c u l a t i o n s b u t m o v e s t h e c a l c u l a t i o n s away f r o m t h e d a t a ( C h a 7 8 ) . T h e e x t r a c t e d d i f f e r e n c e s i n s i z e s o f t h e c a l c i u m i s o t o p e s a l s o c h a n g e w i t h t h e 7 i n c l u s i o n o f t h e s e s e c o n d o r d e r t e r m s ( C h a 7 8 ) , h o w e v e r , a n d t h e r e f o r e t h e s e t e r m s n e e d t o be b e t t e r u n d e r s t o o d . T h e q u a l i t y o f t h e r e s u l t s o b t a i n e d o n t h e n e u t r o n d i s t r i b u t i o n i s d e m o n s t r a t e d i n t h e a n a l y s i s ( R a y 8 1 ) o f d a t a ( I g o 7 9 ) o n t h e c a l c i u m i s o t o p e s . A r e a s o n a b l e a g r e e m e n t b e t w e e n t h e n e u t r o n d e n s i t y d i f f e r e n c e s e x t r a c t e d f r o m a m o d e l i n d e p e n d e n t a n a l y s i s o f t h e 8 0 0 MeV d a t a a n d D e n s i t y M a t r i x E x p a n s i o n (DME) ( N e g 7 2 ) p r e d i c t i o n s was o b t a i n e d , e x c e p t i n t h e n u c l e a r i n t e r i o r , a l t h o u g h t h e rms n e u t r o n d i s t r i b u t i o n r a d i u s d i f f e r e n c e s o f 4 8 C a - ' t 0 C a f o u n d i n t h e a n a l y s i s ( 0 . 1 6 ± 0 . 0 4 fm) a n d t h e DME c a l c u l a t i o n ( 0 . 2 6 fm ) d i f f e r . M o d e l i n d e p e n d e n t h e r e m e a n s o n l y t h a t t h e d e n s i t y d i s t r i b u t i o n was r e p r e s e n t e d a s a sum o f o r t h o g o n a l f u n c t i o n s , n o t i n d e p e n d e n c e f r o m t h e m o d e l u s e d t o d e s c r i b e t h e p r o t o n - n u c l e u s i n t e r a c t i o n . R a y a n d H o f f m a n ( R a y 8 3 ) t e s t e d t h e a c c u r a c y o f t h e d e t e r m i n a t i o n o f n e u t r o n d e n s i t i e s b y e x t r a c t i n g t h e p r o t o n d e n s i t y d i f f e r e n c e b e t w e e n 4 8 C a a n d 5 4 F e f r o m 8 0 0 MeV p r o t o n e l a s t i c s c a t t e r i n g d a t a ( H o f 7 8 , R a y 8 1 ) . T h i s d i f f e r e n c e i s k n o w n f r o m e l e c t r o n s c a t t e r i n g m e a s u r e m e n t s . U s i n g a s p i n d e p e n d e n t , l o c a l s e c o n d o r d e r KMT o p t i c a l p o t e n t i a l m o d e l , t h e y f i n d a n rms p r o t o n r a d i u s d i f f e r e n c e , 5 4 F e - 4 8 C a , o f 0 . 2 6 ± 0 . 0 7 fm a s c o m p a r e d t o 0 . 2 0 ± 0 . 0 1 f m f r o m e l e c t r o n s c a t t e r i n g a n d m u o n i c a t o m s ( W o h 8 0 , W o h 8 1 , E m r 8 3 ) . A c o m p a r i s o n o f t h e i r p r o t o n d e n s i t y d i f f e r e n c e w i t h t h a t f r o m e l e c t r o n s c a t t e r i n g i s s h o w n i n F i g u r e 1 . I t c a n be s e e n t h a t t h e a g r e e m e n t i s g o o d a l t h o u g h t h e e r r o r s i n t h e n u c l e a r c e n t r e a r e v e r y l a r g e a n d a t t h e s u r f a c e t h e y a r e 5 t o 10 t i m e s t h o s e o f t h e e l e c t r o n s c a t t e r i n g . T h e e r r o r s i n t h e n u c l e a r c e n t r e a r e d o m i n a t e d b y s t a t i s t i c s a n d m o d e l d e p e n d e n c y a n d r e f l e c t t h e s h o r t mean f r e e p a t h o f t h e p r o t o n s i n t h e n u c l e u s . 8 i i i i i i i i PROTON ISOTONIC DENSITY DIFFERENCE r(fm) FIGURE 1 Comparison of e r r o r band of proton d e n s i t y d i f f e r e n c e s between 5 t*Fe - l f 8Ca obtained from a n a l y s i s of 800 MeV proton e l a s t i c s c a t t e r i n g data (outer band) with that obtained by e l e c t r o n s c a t t e r i n g experiments (inner crosshatched band). The f i g u r e was taken from Reference (Ray83). 9 1.2.2 a particles B e i n g i s o s c a l a r , a p a r t i c l e s i n t e r a c t e q u a l l y w i t h p r o t o n s a n d n e u t r o n s a n d a r e s e n s i t i v e t o t h e m a t t e r d i s t r i b u t i o n , p m > o f n u c l e i . Due t o t h e p r o b l e m s i n c o n s t r u c t i n g a m i c r o s c o p i c t h e o r y o f t h e s c a t t e r i n g o f a c o m p o s i t i v e p a r t i c l e f r o m a n u c l e u s , t h e a n a l y s i s o f a p a r t i c l e s c a t t e r i n g i s l a r g e l y p h e n o m e n o l o g i c a l . A n o p t i c a l p o t e n t i a l i s g e n e r a t e d b y f o l d i n g e i t h e r ( J a c 6 9 , B e r 6 9 , G i l 8 0 a ) a n oc-nucleon p o t e n t i a l w i t h p m , o r ( M a j 7 8 ) a n u c l e o n - n u c l e o n p o t e n t i a l w i t h p a a n d p m . T h e i m a g i n a r y p o t e n t i a l , r e s p o n s i b l e f o r a p a r t i c l e a b s o r p t i o n , i s p h e n o m e n o l o g i c a l a n d i s a l a r g e p a r t o f t h e p o t e n t i a l , a l t h o u g h t h e e x t r a c t e d n e u t r o n d e n s i t i e s a r e n o t s e n s i t i v e t o t h e p a r t i c u l a r f o r m u s e d . B e c a u s e a p a r t i c l e s a r e s t r o n g l y a b s o r b e d a t a l l e n e r g i e s , t h e s c a t t e r i n g i s s e n s i t i v e m a i n l y t o t h e n u c l e a r e x t e r i o r . T h i s i s u s e f u l f o r e x t r a c t i n g t h e d e n s i t y d i s t r i b u t i o n a t l a r g e r a d i i a n d a n rms r a d i u s may be e v a l u a t e d b y a s s u m i n g a f u n c t i o n a l f o r m f o r t h e d e n s i t y d i s t r i b u t i o n . I n a n y c a s e t h e rms r a d i u s i s s e n s i t i v e m a i n l y t o d e n s i t i e s a t r a d i i b e y o n d t h e h a l f d e n s i t y r a d i u s . By e x t e n d i n g t h e d a t a t o a n g l e s b e y o n d t h e d i f f r a c t i v e r e g i o n i n t o t h e ' r a i n b o w s c a t t e r i n g ' , t h e i n t e r i o r o f t h e n u c l e u s i s p r o b e d ( G o l 7 2 ) . G i l s e t a l . ( G i l 8 0 b ) e x t r a c t e d t h e m a t t e r d e n s i t y d i s t r i b u t i o n d i f f e r e n c e , 6p m , o f 4 8 C a - 4 ° C a f r o m 104 MeV a p a r t i c l e s c a t t e r i n g . T h e p o t e n t i a l u s e d i s p h e n o m e n o l o g i c a l a n d 3 p a r a m e t e r s a r e f i t t o t h e a - 4 0 C a d i f f e r e n t i a l c r o s s s e c t i o n . I t was f o u n d t h a t t h e e f f e c t i v e i n t e r a c t i o n m u s t i n c l u d e s a t u r a t i o n a t h i g h d e n s i t i e s ( n u c l e a r i n t e r i o r ) t o r e p r o d u c e t h e s c a t t e r i n g a t l a r g e a n g l e s . B u d z a n o w s k i e t a l . ( B u d 7 9 ) s u g g e s t t h a t e x c h a n g e e f f e c t s i n t h e n u c l e u s - n u c l e u s c o l l i s i o n s a r e i m p o r t a n t a n d t h a t t h e s e w i l l v a r y a s t h e n e u t r o n n u m b e r i s i n c r e a s e d . T h e c o m p l i c a t e d d e n s i t y d e p e n d e n c e o f t h e i n t e r a c t i o n m i g h t l i m i t t h e r e l i a b i l i t y o f a s c a t t e r i n g f o r m e a s u r i n g n u c l e a r m a t t e r d i f f e r e n c e s , w i t h o u t some u n d e r s t a n d i n g o f t h e f o r m o f t h i s d e p e n d e n c e . G i l s e t a l . d i s p l a y t h e r e s u l t s a s 4 i t r 2 6 p m w h i c h i s r e p r o d u c e d h e r e i n F i g u r e 2 . A l s o i n F i g u r e 2 i s t h e d i s t r i b u t i o n 6 p m > o b t a i n e d b y d i v i d i n g t h e p u b l i s h e d c u r v e b y 4 i t r 2 . I t c a n be s e e n t h a t t h e u n c e r t a i n t y i n c r e a s e s r a p i d l y i n s i d e o f 3 f m . T h e a u t h o r s p o i n t o u t t h a t , t o g e t t h e g o o d s e n s i t i v i t y a t r a d i i down t o t h i s v a l u e , i t i s i m p o r t a n t t o t a k e d a t a o v e r t h e w h o l e a n g u l a r r a n g e . 1.2.3 Kaons B e c a u s e t h e p o s i t i v e k a o n i s t h e l i g h t e s t p a r t i c l e w i t h s t r a n g e n e s s S = + 1 , i t c a n o n l y d e c a y v i a t h e w e a k i n t e r a c t i o n . A s t h e r e a r e no S = +1 b a r y o n s , t h e c a n n o t be a b s o r b e d o n a p a i r o f n u c l e o n s . F u r t h e r m o r e , t h e r e a r e no S = +1 r e s o n a n c e s o r p a r t i c l e s n e a r t h e K"1" m a s s o f 4 94 M e V . A s t h e K + - n u c l e o n c r o s s s e c t i o n ( C h a 8 3 ) i s a b o u t 10 mb ( a n d c o n s t a n t ) a t e n e r g i e s l e s s t h a n a b o u t 4 5 0 MeV , t h e K"*" mean f r e e p a t h i n t h e n u c l e u s i s a b o u t 5 f m . The c o m b i n a t i o n o f l o n g mean f r e e p a t h a n d n o n - r e s o n a n t i n t e r a c t i o n m e a n s t h a t t h e f i r s t o r d e r o p t i c a l p o t e n t i a l s h o u l d w o r k w e l l w i t h a f e w s m a l l c o r r e c t i o n s . A l s o t h e p - w a v e i s m u c h l e s s i m p o r t a n t f o r k a o n s t h a n f o r p i o n s , r e d u c i n g t h e i m p o r t a n c e o f t h e L o r e n t z - L o r e n z e f f e c t ( S e c t i o n 2 . 2 . 4 ) a n d t h e g r a d i e n t t e r m ( S e c t i o n 2 . 2 . 3 ) i n t h e p o t e n t i a l . M u c h i n t e r e s t h a s b e e n p a i d t o t h e k a o n a s a n u c l e a r s t r u c t u r e p r o b e ( T r i 8 1 , D o v 8 2 ) w i t h t h e s u g g e s t i o n s o f t h e b u i l d i n g o f k a o n f a c t o r i e s . I t m u s t b e r e m a r k e d , t h o u g h , t h a t a s much i n t e r e s t was p a i d t o t h e p i o n a s a n u c l e a r s t r u c t u r e p r o b e b e f o r e t h e b u i l d i n g o f p i o n f a c t o r i e s . 11 r(fm) (b) FIGURE 2 M a t t e r d e n s i t y d i s t r i b u t i o n p m d i f f e r e n c e o f 4 8 C a - 4 ° C a , e x t r a c t e d f r o m a n a l y s i s ( G i l 8 0 b ) o f 104 MeV a - p a r t i c l e s c a t t e r i n g . F i g u r e 2 a . s h o w s 4 i r r 2 6 p m a n d i s t a k e n d i r e c t l y f r o m r e f e r e n c e ( G i l 8 0 b ) . F i g u r e 2 b . s h o w s 6 p m a n d was o b t a i n e d b y d i v i d i n g F i g u r e 2 a . b y 4 i r r 2 . T h e K - n u c l e o n p h a s e s h i f t s h a v e r e c e n t l y b e e n d e t e r m i n e d b y t h e B o l o g n a , G l a s g o w , Roma a n d T r i e s t e c o l l o b o r a t i o n ( B G R T ) ( G i a 7 0 , G i a 7 4 ) , M a r t i n ( M a r 7 5 ) , M a r t i n a n d O a d e s ( M a r 8 0 ) a n d W a t t s ( W a t 8 0 ) . T h e K+ e l a s t i c , i n e l a s t i c a n d t o t a l c r o s s s e c t i o n s o n 1 2 C a n d l + 0 C a a t 8 0 0 M e V / c w e r e m e a s u r e d b y t h e C M U - H o u s t o n - B N L c o l l a b o r a t i o n ( M a r 8 2 ) a t t h e B r o o k h a v e n i r / K s p e c t r o m e t e r . F i r s t o r d e r o p t i c a l p o t e n t i a l c a l c u l a t i o n s ( R o s 8 0 , M a r 8 2 ) e m p l o y i n g t h e a m p l i t u d e s f r o m M a r t i n a g r e e q u i t e w e l l w i t h t h e d a t a , b u t o t h e r s u s i n g t h e a m p l i t u d e s f r o m BGRT p r o d u c e a p o o r f i t . The i d ' - n u c l e o n d a t a may t h u s b e u s e d t o h e l p d e t e r m i n e t h e K ^ - n u c l e u s i n t e r a c t i o n ; t h e q u a l i t y o f t h e a g r e e m e n t u s i n g t h e M a r t i n a m p l i t u d e s s u g g e s t s t h a t t h e h i g h e r o r d e r c o r r e c t i o n s t o t h e p o t e n t i a l a r e n o t l a r g e . I t i s p o s s i b l e , h o w e v e r , t h a t t h i s a g r e e m e n t i s f o r t u i t o u s . U n t i l t h e I C ^ - n u c l e o n d a t a a r e i m p r o v e d a n d p r o d u c e a u n i q u e s e t o f p h a s e s h i f t s , n o c o n c l u s i o n c a n be d r a w n o n t h e r e l i a b i l i t y o f t h e f i r s t o r d e r K + -n u c l e u s o p t i c a l p o t e n t i a l . A s t u d y o f t h e s e n s i t i v i t y o f k a o n s t o t h e n e u t r o n d i s t r i b u t i o n d i f f e r e n c e o f 4 8 C a - 1 + 0 C a was made b y C o t a n c h ( C o t 8 1 ) . He c o n c l u d e s t h a t t h e l a r g e s t u n c e r t a i n t y a t p r e s e n t c omes f r o m t h e l a c k o f k n o w l e d g e o f t h e K + - n e u t r o n i n t e r a c t i o n . H o w e v e r , b y f i x i n g t h e K + - n u c l e u s i n t e r a c t i o n b y f i t t i n g t o a b s o l u t e c r o s s s e c t i o n s , i t i s p o s s i b l e t o o b t a i n t h e i s o t o p i c n e u t r o n d e n s i t y d i f f e r e n c e s . U n f o r t u n a t e l y , C o t a n c h d o e s n o t e x p l i c i t l y d e m o n s t r a t e t h e r a d i a l s e n s i t i v i t y o f t h e K+ a s a n e u t r o n d i s t r i b u t i o n p r o b e . A l s o , t h e s e n s i t i v i t y t o t h e p r o t o n d i s t r i b u t i o n may be s i g n i f i c a n t , a s t h e K + p i n t e r a c t i o n i s l a r g e r t h a n t h e K + n i n t e r a c t i o n a t l o w e n e r g i e s . K r e l l a n d Thomas ( K r e 8 3 ) a l s o s t u d i e d t h e s e n s i t i v i t y o f K + s c a t t e r i n g t o i s o t o p i c d i f f e r e n c e s i n n e u t r o n d i s t r i b u t i o n s . T h e y u s e t h e n u c l e i i S 0 a n d i b 0 f o r t h e i r c a l c u l a t i o n s a n d f i n d a l a r g e a n g l e d e p e n d e n t s e n s i t i v i t y , o f t h e c r o s s s e c t i o n r a t i o s o f 1 8 0 / 1 6 0 , t o a 0 . 2 fm v a r i a t i o n i n t h e n e u t r o n r a d i u s o f 1 6 0 . F o r 50 MeV k a o n s a t 9 0 ° t h e r e i s a 20% d i f f e r e n c e b e t w e e n t h e c r o s s s e c t i o n r a t i o s c a l c u l a t e d w i t h t h e two 1 6 0 r a d i i . T h i s s e n s i t i v i t y i n c r e a s e s a s t h e e n e r g y i s i n c r e a s e d . F o r a p u r e l y l o c a l i n t e r a c t i o n , t h e n u c l e a r p o t e n t i a l i s p r o p o r t i o n a l t o t h e d e n s i t y ; t h e s c a t t e r i n g a m p l i t u d e i n t h e f i r s t B o r n A p p r o x i m a t i o n i s j u s t t h e F o u r i e r t r a n s f o r m o f t h e d e n s i t y d i s t r i b u t i o n . A t 50 MeV t h e 1 6 0 c r o s s s e c t i o n i s c a l c u l a t e d ' t o be 3 m b / s r a t 9 0 ° a n d f a l l s t o 0 . 3 m b / s r a t 1 5 0 " , w h e r e t h e momentum t r a n s f e r i s 4 26 M e V / c . F o r a k a o n e n e r g y o f 200 M e V , t h e momentum t r a n s f e r i s 426 M e V / c a t 5 4 " ; a t a n e n e r g y o f 3 5 0 MeV t h a t a n g l e i s 3 6 . 5 " . I n c l u d i n g t h e e f f e c t s o f t h e t r a n s f o r m a t i o n o f t h e s o l i d a n g l e f r o m t h e c e n t r e o f m a s s t o t h e l a b o r a t o r y f r a m e , t h e c r o s s s e c t i o n s a t t h e s e a n g l e s f o r 2 0 0 MeV a n d 3 5 0 MeV k a o n s b e c o m e 0 . 3 4 m b / s r a n d 0 . 3 5 m b / s r r e s p e c t i v e l y . T o g e t a r e a s o n a b l e c o u n t r a t e f o r a n e x p e r i m e n t , t h e a n g u l a r r a n g e w i l l be m o r e r e s t r i c t e d a s t h e e n e r g y i s i n c r e a s e d , a n d p r e c i s e m e a s u r e m e n t s o f a n g l e s a n d e n e r g i e s w i l l be n e c e s s a r y . F o r a s e c o n d a r y b e a m - l i n e , o p e r a t i n g w i t h a l a r g e p h a s e s p a c e a c c e p t a n c e t o p r o v i d e a d e q u a t e f l u x , p o s i t i o n m e a s u r i n g d e t e c t o r s i n t h e b e a m - l i n e w i l l b e n e c e s s a r y . T h i s p r o b l e m w i l l be m o r e s e v e r e f o r l a r g e r n u c l e i , w h e r e t h e d i f f r a c t i o n m i n i m a move t o s m a l l e r a n g l e s . 1.2 .4 Pions O v e r t h e l a s t d e c a d e a l a r g e a m o u n t o f p i o n - n u c l e u s d a t a h a s b e e n a m a s s e d . M o s t o f t h e a t t e n t i o n o n t h i s d a t a h a s f o c u s s e d o n t h e u n d e r s t a n d i n g a n d d e s c r i p t i o n o f t h e p i o n - n u c l e u s i n t e r a c t i o n . T h i s i n t e r a c t i o n i s o f i n t r i n s i c i n t e r e s t a n d h a s i m p l i c a t i o n s o n n u c l e a r s t r u c t u r e c a l c u l a t i o n s ( H u b 7 9 , R i s 8 3 ) , a n d t h e u n d e r s t a n d i n g o f h e a v y i o n c o l l i s i o n s ( M i s 8 0 ) . H o w e v e r , p a r t o f t h e d r i v e t o b u i l d t h e m e s o n f a c t o r i e s ( T R I U M F , L A M P F , S I N ) w h i c h g e n e r a t e d t h i s d a t a , was t h e p o s s i b i l i t y o f u s i n g p i o n s t o s t u d y n u c l e a r s t r u c t u r e . The p i o n - n u c l e o n ( u - N ) i n t e r a c t i o n i s s t r o n g l y i s o s p i n d e p e n d e n t . F i g u r e 3 c o m p a r e s t h e %+p c r o s s s e c t i o n w i t h t h e i t - p c r o s s s e c t i o n a t l o w e n e r g i e s . T he l a r g e bump i n t h e T i + p c r o s s s e c t i o n a t a r o u n d 195 MeV i s d u e t o t h e p r e s e n c e o f t h e l o w e s t e n e r g y p i o n - n u c l e o n r e s o n a n c e , t h e A"1-*" o f ma s s 1 232 MeV , i s o s p i n 1 = 3 / 2 a n d s p i n J = 3 / 2 . T h i s r e s o n a n c e i s w i d e , T = 115 M e V , a n d t h e r e f o r e a f f e c t s t h e p i o n - n u c l e o n i n t e r a c t i o n down t o much l o w e r p i o n e n e r g i e s . T h e p i o n - n u c l e o n s y s t e m may be d e c o m p o s e d - i n t o i s o s p i n s t a t e s : |TT +,P> o r | ix~,n> = | I = 3 / 2 > a n d |TT,P> o r | n + , n > = /T73|I=3/2> + V2/3\ I - l / 2 > . A s s u m i n g t h a t a t t h e r e s o n a n c e t h e i s o s p i n 1/2 i n t e r a c t i o n i s much s m a l l e r t h a n t h e i s o s p i n 3 / 2 , o n e w o u l d e x p e c t t h e r a t i o o f c r o s s s e c t i o n s n + p / n ~ p t o b e 9 . F r o m i s o s p i n i n v a r i a n c e t h e r a t i o o f c r o s s s e c t i o n s T c _ n / i t - p i s e q u a l t o t h a t o f T t + p / n T p . A t t h e r e s o n a n c e , t h e r e f o r e , %~ w i l l b e m o r e s e n s i t i v e t o t h e n e u t r o n s i n t h e n u c l e u s t h a n t o t h e p r o t o n s . T h i s i m m e d i a t e l y s u g g e s t s t h e p o s s i b i l i t y o f u s i n g n e g a t i v e p i o n s a t r e s o n a n c e e n e r g i e s t o s e p a r a t e n e u t r o n a n d p r o t o n m a t r i x e l e m e n t s i n n u c l e a r t r a n s i t i o n s , o r t h e p r o t o n a n d n e u t r o n d e n s i t i e s . ir +p and ir~p t o t a l cross sections. The curves are calculated from phase s h i f t s which were f i t t e d to the world u-nucleon data using the program SAID (Arn82). T h e p i o n - n u c l e u s i n t e r a c t i o n i s n o t p e r f e c t l y u n d e r s t o o d a n d i s c o m p l i c a t e d b y s e v e r a l f a c t o r s . T h i s i n t e r a c t i o n i s u s u a l l y d e s c r i b e d b y a n o p t i c a l p o t e n t i a l ( s e e C h a p t e r I I ) , f r o m w h i c h c r o s s s e c t i o n s may be c a l c u l a t e d f o r c o m p a r i s o n t o e x p e r i m e n t a l d a t a . T h e mean f r e e p a t h o f t h e p i o n i n t h e n u c l e u s v a r i e s b e t w e e n l e s s t h a n a F e r m i , a t r e s o n a n c e e n e r g i e s , t o a b o u t 5 fm a t l o w e n e r g i e s (~50 M e V ) . S i n c e n u c l e a r r a d i i a r e t y p i c a l l y s e v e r a l f m , t h e p i o n w i l l h a v e a r e a s o n a b l e c h a n c e o f i n t e r a c t i n g w i t h m o r e t h a n o n e n u c l e o n i n t h e n u c l e u s . T h e r e f o r e s e c o n d o r d e r t e r m s i n a m u l t i p l e s c a t t e r i n g e x p a n s i o n o f t h e p i o n - n u c l e u s t - m a t r i x a r e s i g n i f i c a n t . T h e s i z e s o f t h e s e s e c o n d o r d e r t e r m s ( L a n 8 0 , T h o 8 0 ) a r e d e p e n d e n t o n two n u c l e o n c o r r e l a t i o n s i n t h e n u c l e u s , w h i c h a r e n o t w e l l u n d e r s t o o d t h e m s e l v e s , a p a r t f r o m t h e p r o b l e m o f i n c l u d i n g t h e m i n a p o t e n t i a l . I n c o o r d i n a t e s p a c e t h e n u c l e o n - n u c l e o n c o r r e l a t i o n s l e a d t o t h e n u c l e a r L o r e n t z - L o r e n z e f f e c t ( E r i 6 6 ) . The s t r e n g t h , X , o f t h e L o r e n t z - L o r e n z e f f e c t ( t h e s y m b o l , X , i s u s e d t h r o u g h o u t t h i s t e x t ) i s a f f e c t e d b y t h e r a n g e o f t h e TI-N i n t e r a c t i o n ( F a " 1 7 0 , E i s 7 3 , I a c 7 4 ) , P a u l i c o r r e l a t i o n s ( D e l 7 6 ) a n d t h e % c o u p l i n g t o t h e p m e s o n ( B a y 7 5 , M i l 7 8 ) . T h e p r e s e n c e o f t h e p i o n - n u c l e o n r e s o n a n c e , t h e A ( 1 2 3 2 ) , m e a n s t h a t t h e p i o n may f o r m a A i n t h e n u c l e u s , w h i c h t h e n i n t e r a c t s w i t h t h e r e s t o f t h e n u c l e u s . T h e A may l o s e a l l o f i t s e n e r g y t o n u c l e o n s , i n w h i c h c a s e t h e p i o n h a s b e e n a b s o r b e d b y t h e n u c l e u s , o r a p i o n may be r e -e m i t t e d . T h e A p r o p a g a t i o n i n t h e n u c l e u s may be t r e a t e d e x p l i c i t l y ( M o n 8 1 ) o r a s h i g h e r o r d e r t e r m s i n t h e p i o n - n u c l e u s o p t i c a l p o t e n t i a l ( L a n 8 0 ) . T h e e f f e c t o f t h e A b e c o m e s i n c r e a s i n g l y i m p o r t a n t a t p i o n e n e r g i e s n e a r t h e r e s o n a n c e ( 1 9 4 M e V ) , b u t b e c a u s e t h e A r e s o n a n c e i s w i d e , i t a f f e c t s p i o n s c a t t e r i n g down t o l o w e n e r g i e s a n d , a s a r e s u l t , t h e p i o n - n u c l e o n s c a t t e r i n g a m p l i t u d e s a r e e n e r g y d e p e n d e n t . T h i s c o m p l i c a t e s t h e a v e r a g i n g o f t h e s c a t t e r i n g a m p l i t u d e s o v e r t h e F e r m i momen t a o f t h e n u c l e o n s i n t h e n u c l e u s . I n a d d i t i o n , t h e p i o n c a n be a b s o r b e d b y t h e n u c l e u s . P i o n a b s o r p t i o n i s n o t w e l l u n d e r s t o o d , a n d i s u s u a l l y i n c o r p o r a t e d p h e n o m e n o l o g i c a l l y i n t o a p o t e n t i a l m o d e l a s a p 2 t e r m ( E r i 6 6 , T h o 8 0 ) . The u n c e r t a i n t y i n t h e p i o n - n u c l e u s i n t e r a c t i o n l i m i t s t h e u s e o f t h e p i o n a s a n u c l e a r s t r u c t u r e p r o b e , s i n c e n u c l e a r s t r u c t u r e e f f e c t s may be m i m i c k e d b y c h a n g e s i n t h e d e s c r i p t i o n o f t h e p i o n - n u c l e u s i n t e r a c t i o n . H o w e v e r , i t i s s t i l l p o s s i b l e t o u s e t h e p i o n t o s t u d y c h a n g e s i n n u c l e a r s t r u c t u r e b e t w e e n i s o t o p e s , s i n c e t h i s p u t s a f a r l e s s s t r i n g e n t r e q u i r e m e n t o n t h e k n o w l e d g e o f t h e i n t e r a c t i o n . F o r c o n v e n i e n c e , t h r e e e n e r g y r e g i o n s f o r i t - n u c l e u s s c a t t e r i n g e x p e r i m e n t s may be d e f i n e d : l o w e n e r g y (<80MeV ) w h e r e t h e p i o n h a s a h i g h p r o b a b i l i t y o f p a s s i n g t h r o u g h t h e n u c l e u s w i t h o n l y o n e i n t e r a c t i o n w i t h a n u c l e o n ; r e s o n a n c e e n e r g y (~ 120 t o ~ 240 M e V ) , w h e r e t h e p i o n s e e s t h e n u c l e u s e s s e n t i a l l y a s a b l a c k d i s k w i t h r a d i u s l a r g e r t h a n t h e h a l f d e n s i t y r a d i u s ; h i g h e n e r g y (~ 1 G e V ) , w h e r e t h e p i o n - n u c l e o n c r o s s s e c t i o n i s a g a i n s m a l l (~ 40 mb) b u t r e s o n a n t , a n d t h e n - p i n t e r a c t i o n i s g r e a t e r t h a n t h e 7t + p i n t e r a c t i o n . T h e r e i s , o f c o u r s e , a g r a d u a l t r a n s i t i o n f r o m o n e r e g i m e t o a n o t h e r a s t h e p i o n e n e r g y i s c h a n g e d . 1.2.4.1 High energy pions T h e e a r l i e s t p i o n - n u c l e u s e x p e r i m e n t t o be u s e d t o m e a s u r e n u c l e a r + r a d i i was t h e s t u d y o f t h e r a t i o o f r e a c t i o n c r o s s s e c t i o n s o n l e a d , a„, f o r i t - a n d n + s c a t t e r i n g a t 0 . 8 4 GeV a n d 1 . 2 4 G e V , b y A b a s h i a h ( A b a 5 6 ) i n 1 9 5 6 . T h i s d a t a was f o u n d ( A u e 6 7 ) t o be c o m p a t i b l e w i t h a s m a l l n e g a t i v e v a l u e f o r 6 n p = ( ^ r ^ ~ <r^> ' ) • M o r e p r e c i s e e x p e r i m e n t s o n C , A l , C a , N i , S n , Ho a n d Pb b y A l l a r d y c e e t a l . ( A 1 1 7 3 ) g a v e v a l u e s o f 6 r c o n s i s t e n t w i t h z e r o t o w i t h i n 0 . 1 f m . np 1.2.4.2 Resonance energy pions F i t s t o t h e e l a s t i c s c a t t e r i n g d a t a i n t h e r e s o n a n c e e n e r g y r e g i o n a r e g e n e r a l l y g o o d . T h i s i s d ue t o t h e s t r o n g a b s o r p t i o n a n d l a r g e l y d i f f r a c t i v e n a t u r e o f t h e s c a t t e r i n g . The p i o n h e r e d o e s n o t p e n e t r a t e d e e p l y i n t o t h e n u c l e u s a n d s o i s n o t s e n s i t i v e t o t h e d e t a i l e d n a t u r e o f t h e i n t e r a c t i o n . Some a t t e m p t s ( E g g 7 7 , E g g 7 9 , J a n 7 8 ) h a v e b e e n made t o e x p l o i t t h e d i f f r a c t i v e n a t u r e o f t h e s c a t t e r i n g b y s t u d y i n g t h e a n g l e c h a n g e b e t w e e n •jt+ a n d i t - o f t h e f i r s t d i f f r a c t i v e m i n i m u m o n i s o t o p e s . I n c o m p a r i n g two i s o t o p e s , t h e a n a l y s i s i s l a r g e l y i n d e p e n d e n t o f t h e C o u l o m b i n t e r a c t i o n , w h i c h a f f e c t s n + a n d %~ d i f f e r e n t l y . S t e r n h e i m a n d Y o o ( S t e 7 8 ) p o i n t o u t t h e d a n g e r o f t h i s a p p r o a c h . T h e p r o d u c t q R , w h e r e q i s t h e momentum t r a n s f e r a n d R i s t h e e f f e c t i v e r a d i u s a t w h i c h t h e s c a t t e r i n g t a k e s p l a c e , e s s e n t i a l l y d e t e r m i n e s t h e p o s i t i o n o f t h e d i f f r a c t i o n m i n i m u m . T h e e f f e c t i v e v a l u e o f q , h o w e v e r , i s d e p e n d e n t o n t h e p o t e n t i a l , a m o r e a t t r a c t i v e p o t e n t i a l p r o d u c i n g a h i g h e r q . D i f f e r e n c e s i n t h e p o t e n t i a l b e t w e e n 1 6 0 a n d 1 8 0 d u e t o t h e p r e s e n c e o f t h e e x t r a n e u t r o n s w o u l d t h e r e f o r e c h a n g e t h e p o s i t i o n o f t h e m i n i m u m e v e n w i t h e q u a l r a d i i . T h e v a l u e o f R e x t r a c t e d i s w e l l i n t o t h e n u c l e a r s u r f a c e . A v a l u e o f p ( r ) / p Q w h e r e p Q i s t h e c e n t r a l d e n s i t y , a t t h i s r a d i u s may be c a l c u l a t e d u s i n g a n o p t i c a l p o t e n t i a l m o d e l . I n f o r m a t i o n o n a n i n t e g r a l q u a n t i t y , s u c h a s a n rms r a d i u s , c a n o n l y be o b t a i n e d b y t h e n a s s u m i n g a f o r m f o r t h e d e n s i t y d i s t r i b u t i o n . T h e v a l u e o f p ( r ) / p Q i t s e l f i s , i n f a c t , d e p e n d e n t o n t h e d e t a i l s o f t h e o p t i c a l p o t e n t i a l . T h e r e s u l t o f t h i s c a l c u l a t i o n d e p e n d s o n how a b s o r p t i v e t h e o p t i c a l p o t e n t i a l i s ; a m o r e a b s o r p t i v e p o t e n t i a l p r e d i c t s a l o w e r d e n s i t y a t w h i c h t h e n u c l e u s i s ' b l a c k ' a n d a l l t h e p i o n s a r e a b s o r b e d . I v e r s o n e t a l . ( I v e 7 9 ) a n a l y s e a s i m i l a r e x p e r i m e n t o n 1 8 0 a t 164 MeV w i t h a n o p t i c a l m o d e l . T h i s w i l l b e m o r e r e l i a b l e t h a n j u s t s t u d y i n g t h e s h i f t i n t h e d i f f r a c t i v e m i n i m a , a s t h e d i f f e r e n c e i n t h e s t r o n g i n t e r a c t i o n a m p l i t u d e s c a u s e d b y t h e e x t r a n e u t r o n s i s t a k e n i n t o a c c o u n t . H o w e v e r , t h e d a t a i s s t i l l o n l y s e n s i t i v e t o t h e d e n s i t i e s a t l a r g e r a d i i . 1.2.4.3 Scattering of low energy pions A t l o w e n e r g i e s t h e p i o n - n u c l e o n i n t e r a c t i o n i s much w e a k e r t h a n i t i s n e a r t h e r e s o n a n c e s o t h a t , a t 50 MeV , t h e t o t a l n + p a n d T i + n c r o s s s e c t i o n s a r e r e s p e c t i v e l y 12 mb a n d 8 . 5 m b . The mean f r e e p a t h i n t h e n u c l e u s i s g i v e n b y 1 / a p , w h e r e a i s t h e t o t a l c r o s s s e c t i o n p e r n u c l e o n a n d p i s t h e a v e r a g e n u m b e r o f n u c l e o n s p e r u n i t v o l u m e , a b o u t 0 . 2 f m - 3 . U s i n g t h e f r e e p i o n - n u c l e o n t o t a l c r o s s s e c t i o n f o r a g i v e s a mean f r e e p a t h i n t h e n u c l e u s o f a b o u t 5 f m . T a k i n g t h e c r o s s s e c t i o n p e r n u c l e o n a s 0 ^ , / A , w h e r e o^ i s t h e p i o n - n u c l e u s t o t a l c r o s s s e c t i o n , a n d A i s t h e a t o m i c n u m b e r , s u g g e s t s t h a t t h e mean f r e e p a t h d o e s n o t r i s e s o r a p i d l y a s t h e e n e r g y i s l o w e r e d . C a r r o l l ( C a r 7 6 ) m e a s u r e d p i o n t o t a l c r o s s s e c t i o n s o n a r a n g e o f n u c l e i a t e n e r g i e s f r o m 65 MeV t o 3 2 0 M e V . The t o t a l i t + c r o s s s e c t i o n ( t h e n + a n d n~ t o t a l c r o s s s e c t i o n s a r e v e r y s i m i l a r i n t h i s w o r k ) o n i r o n i s a b o u t 1 750 mb o r 3 1 m b / n u c l e o n , a t 6 5 M e V . T h i s g i v e s 1 . 6 fm f o r t h e mean f r e e p a t h . T h i s e s t i m a t e o f t h e mean f r e e p a t h i s a l o w e r b o u n d s i n c e t h e P a u l i p r i n c i p l e r e d u c e s t h e a v e r a g e c r o s s s e c t i o n f o r n u c l e o n s i n t h e i n t e r i o r . T h e same a n a l y s i s w i t h t h e . c a r b o n c r o s s s e c t i o n g i v e s 1 . 9 fm f o r t h e mean f r e e p a t h . T h e m e a n f r e e p a t h i s l a r g e r t h a n t h a t a t t h e r e s o n a n c e . A p e r h a p s s u r p r i s i n g f e a t u r e o f t h e l o w e n e r g y p i o n - n u c l e o n i n t e r a c t i o n i s t h e g r e a t i s o s p i n s e l e c t i v i t y . I n f a c t , a t 50 MeV t h e r a t i o o f p - w a v e n f n / i t ~ p s c a t t e r i n g a m p l i t u d e s ( i n t h e n u c l e u s ) i s a b o u t 1 0 / 1 ( s e e T a b l e V I I ) . L a r g e d i f f e r e n c e s i n t h e %~ e l a s t i c s c a t t e r i n g c r o s s s e c t i o n a n g u l a r d i s t r i b u t i o n among i s o t o p e s a r e s e e n , t h e r e f o r e , b e c a u s e o f t h e r e s u l t i n g s e n s i t i v i t y t o t h e c h a n g e i n t h e n e u t r o n n u m b e r . T h i s s e n s i t i v i t y h a s a l r e a d y b e e n e x p l o i t e d i n t h e m e a s u r e m e n t o f rms r a d i u s d i f f e r e n c e s o f t h e i s o t o p e s 1 3 C , 1 2 C a n d 1 8 0 , 1 6 0 ( ( J o h 7 9 ) , i n c l u d e d h e r e a s A p p e n d i x 5 ) . I n t h a t w o r k i n d e p e n d e n c e o f t h e r e s u l t s t o d e t a i l s o f t h e o p t i c a l m o d e l a n d a l a c k o f s e n s i t i v i t y t o t h e f o r m o f t h e d e n s i t y d i s t r i b u t i o n w e r e d e m o n s t r a t e d . I t i s d i f f i c u l t t o d e m o n s t r a t e i n d e p e n d e n c e f r o m t h e f o r m o f t h e o p t i c a l m o d e l i n a n u n a m b i g u o u s w a y . A b e t t e r t e s t o f t h e m e t h o d f o r m e a s u r i n g n e u t r o n d i s t r i b u t i o n r a d i i i s t o t a k e a d v a n t a g e o f t h e i s o s p i n i n v a r i a n c e o f t h e i n t e r a c t i o n a n d m e a s u r e k n o w n p r o t o n d i s t r i b u t i o n r a d i i o f i s o t o n e s u s i n g i t + . I f t h e p r o t o n r a d i i t h u s m e a s u r e d a g r e e w i t h t h o s e m e a s u r e d b y e l e c t r o n s c a t t e r i n g o r m u o n i c a t o m s , t h i s i s a g o o d i n d i c a t i o n t h a t t h e n e u t r o n r a d i i m e a s u r e d i n a n a n a l o g o u s m a n n e r w i t h t h e %~ a r e r e l i a b l e . T h e p r o t o n r a d i i d i f f e r e n c e b e t w e e n a n d * 2 C w e r e m e a s u r e d ( B a r 8 0 ) i n t h i s w a y . The d i f f e r e n c e i n rms c h a r g e r a d i i , 1 2 C - 1 1 B , was m e a s u r e d t o b e 0 . 0 7 2 ± 0 . 0 2 f m . A s s u m i n g a 1 2 C c h a r g e r a d i u s o f 2 . 4 7 2 fm ( C a r 8 0 ) t h i s r e s u l t s i n a ^ B c h a r g e r a d i u s o f 2 . 4 0 0 f m . T h i s c o m p a r e s w e l l w i t h t h e e l e c t r o n s c a t t e r i n g m e a s u r e m e n t s o f 2 . 4 2 ± 0 . 1 2 fm ( S t o 6 6 ) a n d 2 . 3 7 fm ( R i s 7 1 ) , a l t h o u g h t h e u n c e r t a i n t i e s o f t h e e l e c t r o n s c a t t e r i n g m e a s u r e m e n t s a r e l a r g e . A m u o n i c X - r a y m e a s u r e m e n t ( 0 1 i 8 1 ) g a v e f o r t h e 1 1 B c h a r g e r a d i u s a v a l u e o f 2 . 3 8 ± 0 . 0 4 f m . F u r t h e r ir m e a s u r e m e n t s a r e b e i n g p e r f o r m e d ( B a r 8 3 ) o n 1 2 C , l 4 N , 1 6 0 a n d 1 8 0 , w h e r e t h e p r o t o n r a d i i , a n d h e n c e t h e r a d i i d i f f e r e n c e s , a r e b e t t e r k n o w n . 1.3 Discussion of Experiments Performed i n this Work I n a n e a r l i e r e x p e r i m e n t , t h e n e u t r o n r a d i i ( t h e r a d i i o f t h e d i s t r i b u t i o n o f n e u t r o n s ) d i f f e r e n c e s o f i s o t o p e s o f c a r b o n a n d o x y g e n w e r e m e a s u r e d u s i n g l o w e n e r g y p i o n s ( A p p e n d i x 5 , ( J o h 7 9 ) ) . R a n g e t e l e s c o p e s w e r e u s e d t o d e t e c t t h e p i o n s . A s t r o n g s e n s i t i v i t y t o t h e n e u t r o n r a d i u s a n d a n i n d e p e n d e n c e f r o m t h e m o d e l o f t h e p i o n - n u c l e u s i n t e r a c t i o n w e r e s h o w n . T h e i n t r i n s i c r e s o l u t i o n o f t h e r a n g e t e l e s c o p e s i s p o o r ( a b o u t 3 M e V ) , h o w e v e r , s o t h a t many n u c l e i w i t h l o w l y i n g e x c i t e d s t a t e s c a n n o t be s t u d i e d t h i s w a y . T h e c o m m i s s i o n i n g o f l o w e n e r g y p i o n s p e c t r o m e t e r s ( S o b 8 3 ) a t TR IUMF a l l o w e d t h e e x t e n s i o n o f t h i s m e t h o d t o l a r g e r n u c l e i , m a g n e s i u m a n d s u l f u r , w h e r e h i g h e r r e s o l u t i o n i s n e e d e d . I n t h e s e e x p e r i m e n t s t h e d i f f e r e n t i a l c r o s s s e c t i o n r a t i o s o f i s o t o p e p a i r s 3 6 S , 3 2 S ; 3 4 S , 3 2 S a n d 2 6 M g , 2 4 M g w e r e m e a s u r e d . T h e d i f f e r e n t i a l c r o s s s e c t i o n s o f 3 2 S a n d 2 4 M g , n o r m a l i s e d t o t h e k n o w n n + p c r o s s s e c t i o n , w e r e a l s o m e a s u r e d . T h e i n c r e a s e i n n u c l e a r s i z e b r i n g s t h e e l a s t i c s c a t t e r i n g i n t o t h e d i f t r a c t i v e r e g i m e . T h i s i s d i s t i n c t l y d i f f e r e n t t o t h e c a s e o f t h e c a r b o n a n d o x y g e n i s o t o p e s . T h e d i f f i c u l t y o f u s i n g d i f t r a c t i v e s c a t t e r i n g o f p i o n s t o m e a s u r e n u c l e a r s i z e s was p o i n t e d o u t b y S t e r n h e i m a n d Y o o ( S t e 7 8 ) a n d i s d i s c u s s e d i n S e c t i o n 1 . 2 . 4 . 2 . I n t h i s w o r k we c o n c e n t r a t e o n m e a s u r i n g d i f f e r e n c e s i n n e u t r o n r a d i i b e t w e e n i s o t o p e s . I t w o u l d s e em t h a t d i f t r a c t i v e s c a t t e r i n g w o u l d be s u i t a b l e f o r t h i s a s t h e p o t e n t i a l p a r a m e t e r s may b e f i x e d b y f i t t i n g t o t h e s m a l l e r i s o t o p e . Some a s s u m p t i o n m u s t b e made a b o u t t h e n e u t r o n d i s t r i b u t i o n o f t h i s i s o t o p e ; v a r i a t i o n s i n t h i s d i s t r i b u t i o n w i l l r e s u l t i n v a r i a t i o n s i n t h e o p t i c a l p o t e n t i a l p a r a m e t e r s , a n d h e n c e , v a r i a t i o n s i n t h e r a d i i o f t h e l a r g e r i s o t o p e s . The r e s u l t i n g u n c e r t a i n t y o f t h e d i f f e r e n c e s i n t h e r a d i i b e t w e e n t h e i s o t o p e s , h o w e v e r , w i l l b e l e s s t h a n t h a t f o r t h e r a d i u s o f t h e l a r g e r i s o t o p e . C a l c u l a t i o n s w e r e made o f t h e s e n s i t i v i t y o f t h e c r o s s s e c t i o n r a t i o s t o v a r i a t i o n s i n t h e o p t i c a l p o t e n t i a l p a r a m e t e r s , u s i n g t h e o p t i c a l p o t e n t i a l o f S t r i e k e r , M cManu s a n d C a r r ( S t r 7 9 ) . T h e s e c a l c u l a t i o n s a r e d i s c u s s e d i n S e c t i o n 3 . 1 . I t was f o u n d t h a t b e y o n d a b o u t 1 0 0 " t h e c a l c u l a t e d c r o s s s e c t i o n r a t i o s w e r e v e r y d e p e n d e n t u p o n t h e o p t i c a l p o t e n t i a l p a r a m e t e r s u s e d . T h e d a t a w e r e t h e r e f o r e l i m i t e d t o a n g l e s l e s s t h a n 1 0 0 " f o r t h e s u l f u r e x p e r i m e n t s a n d 1 0 5 " f o r t h e m a g n e s i u m e x p e r i m e n t . T h e d a t a w e r e f i t w i t h o p t i c a l p o t e n t i a l c a l c u l a t i o n s i n w h i c h t h e n e u t r o n d i s t r i b u t i o n o f t h e l a r g e r i s o t o p e was v a r i e d . I t w a s f o u n d t h a t t h e r e s u l t s f o r t h e d i f f e r e n c e , 6 r n , i n rms n e u t r o n r a d i i b e t w e e n t h e two i s o t o p e s wa s i n d e p e n d e n t o f t h e p a r a m e t e r s o r f o r m o f t h e o p t i c a l p o t e n t i a l . Two f o r m s o f t h e n e u t r o n d e n s i t y w e r e u s e d . I n t h e f i r s t c a s e , t h e d e n s i t y d i s t r i b u t i o n was r e p r e s e n t e d w i t h a t w o p a r a m e t e r F e r m i f u n c t i o n . G o o d f i t s t o t h e d a t a w e r e o b t a i n e d , w i t h much l o w e r x 2 t h a n c o u l d be o b t a i n e d i n f i t t i n g t h e a b s o l u t e c r o s s s e c t i o n s . T h i s i s i n d i c a t i v e o f b o t h a l a c k o f m o d e l d e p e n d e n c e i n t h e c a l c u l a t i o n o f r a t i o s , a n d a n a b s e n c e o f s y s t e m a t i c e r r o r s i n t h e d a t a . The g o o d f i t s o b t a i n e d h e r e a r e a l s o a n i n d i c a t i o n t h a t t h e d a t a a r e n o t v e r y s e n s i t i v e t o h i g h o r d e r d e t a i l s o f t h e d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s . I n t h e s e c o n d c a s e t h e n e u t r o n d e n s i t i e s w e r e r e p r e s e n t e d b y a F e r m i f u n c t i o n m o d i f i e d b y a sum o f B e s s e l f u n c t i o n s . D i f f e r e n t r e s u l t s w e r e o b t a i n e d f o r 6r n i n t h e two c a s e s , a l t h o u g h t h e r e s u l t s w e r e c o n s i s t e n t w i t h i n t h e e r r o r s , e x c e p t f o r 3 4 S - 3 2 S . N e u t r o n d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s w e r e a l s o e x t r a c t e d f r o m t h e d a t a w i t h t h e F o u r i e r B e s s e l f i t s . T he l o w momentum t r a n s f e r was n o t a g r e a t r e s t r i c t i o n a s t h e d i s t r i b u t i o n d i f f e r e n c e s h a v e l i t t l e s t r u c t u r e ( a c c o r d i n g t o s i n g l e P a r t i c l e P o t e n t i a l [ S P P ] c a l c u l a t i o n s ( S t r 8 2 , H o d 8 3 ) ) a n d c a n be r e p r e s e n t e d w e l l b y a f e w F o u r i e r B e s s e l t e r m s . T h e r e s u l t s f o r t h e F e r m i f u n c t i o n f i t s , F o u r i e r B e s s e l ( F B ) f i t s a n d t h e SPP c a l c u l a t i o n s b y H o d g s o n ( H o d 8 3 ) a r e F e r m i FB SPP 3 6 S - 3 2 S 0 . 1 3 5 ± 0 . 0 5 7 0 . 1 4 ± 0 . 0 7 0 . 1 7 1 f m . 3 4 S - 3 2 S 0 . 1 0 3 ± 0 . 0 3 2 0 . 0 6 1 ± 0 . 0 3 5 0 . 0 9 1 f m . 2 6 M g - 2 4 M g 0 . 0 7 6 ± 0 . 0 4 3 0 . 0 7 7 ± 0 . 0 5 6 0 . 1 2 1 f m . CHAPTER II Pion-Nucleus Scattering 2.1 Introduction The u s u a l m e t h o d o f d e s c r i b i n g t h e p i o n - n u c l e u s i n t e r a c t i o n i s t h r o u g h t h e u s e o f a n o p t i c a l p o t e n t i a l . A n o p t i c a l m o d e l d e s c r i b e s t h e s c a t t e r i n g o f t h e i n c i d e n t p a r t i c l e f r o m a n u c l e a r p o t e n t i a l w h i c h i s i n d e p e n d e n t o f t h e c o o r d i n a t e s o f t h e i n d i v i d u a l n u c l e o n s . I n t h e s t a n d a r d o p t i c a l m o d e l t h e n u c l e u s i s t r e a t e d a s a c o n t i n u o u s m e d i u m o f r e f r a c t i v e i n d e x | i , w h e r e t h e i n c i d e n t p a r t i c l e h a s a w a v e n u m b e r k o u t s i d e t h e n u c l e u s a n d uk i n s i d e , i n a n a l o g y t o t h e s c a t t e r i n g o f l i g h t f r o m a d i s p e r s i v e m e d i u m . A m o r e g e n e r a l t r e a t m e n t i s t o c o n s t r u c t t h e p o t e n t i a l f r o m a c o n s i d e r a t i o n o f t h e m u l t i p l e s c a t t e r i n g o f t h e i n c i d e n t p a r t i c l e f r o m t h e n u c l e o n s i n t h e n u c l e u s . T h i s p r o c e d u r e e s s e n t i a l l y a v e r a g e s o u t i n t e r a c t i o n s f r o m t h e i n d i v i d u a l n u c l e o n s t o g i v e a p o t e n t i a l f o r t h e n u c l e u s . T h i s p o t e n t i a l i s now n o t r e s t r i c t e d t o h a v e t h e same f o r m a s t h e n u c l e o n d i s t r i b u t i o n . A b r i e f d e s c r i p t i o n o f t h e d e r i v a t i o n o f p i o n -n u c l e u s o p t i c a l p o t e n t i a l s i s g i v e n h e r e . P a r t i c u l a r e m p h a s i s i s p l a c e d o n t h e d e v e l o p m e n t o f t h e c o o r d i n a t e s p a c e p o t e n t i a l o f t h e k i n d u s e d i n t h e a n a l y s i s i n S e c t i o n 5 . 2.2 Derivation of the Pion-Nucleus Optical Potential T h e p i o n - n u c l e u s o p t i c a l p o t e n t i a l i s d e r i v e d b y m e a n s o f a m u l t i p l e s c a t t e r i n g e x p a n s i o n o f t h e p i o n n u c l e a r t - m a t r i x i n t e r m s o f t h e p i o n -n u c l e o n t - m a t r i x . T h i s p r o c e d u r e wa s d e v e l o p e d b y W a t s o n a n d c o l l a b o r a t o r s ( F r a 5 3 , F r a 5 6 , W a t 5 7 , H U f 7 5 ) . T h e L i p p m a n - S c h w i n g e r e q u a t i o n f o r t h e p i o n - n u c l e u s t - m a t r i x i s T ( E ) = V + V G Q ( E + ) T ( E ) (1) where the propagator G G ( E + ) = (E+-1C-HN)- 1 and E + = E + ±n E i s the t o t a l energy of the it-nucleus system k i s the pion k i n e t i c energy operator A V i s the pion-nucleus p o t e n t i a l = ^ v , where the sum i s over the 1=1 nucleons, and A A H M = I Ik- + I u } i s the f u l l nuclear Hamiltonian assuming two body i 1 j<i J i n t e r a c t i o n s , u-^j, only. A solution of equation (1) i s : A T ( E ) = I T 1(E ) f J, ±(E) (2) T ± ( E ) = v ± + v I G Q ( E ) T 1 ( E ) (3) Cb. ( E ) = 1 + G ( E ) I x (E)cb ( E ) The T-L are not the two body pion-nucleon t-matrices, as the propagator G Q contains the f u l l nuclear Hamiltonian. The pion-nucleon t-matrix, t ^ , i s defined by t ± ( e ) = v ± + v i g i ( e ) t 1 ( e ) = v i ( l + g i ( e ) t i ( e ) ), and g ±(e) = ( e - k - K i ) ~ 1 , where i s the k i n e t i c energy operator of the i ' t h nucleon. Therefore v i = t i ( e ) " \S±U)t±(e) and from (3) we have (omitting the energy arguments) •c.(E) = t (1+g t J - ^ l + G T . ) 1 i i i o i = t 1 ( l + g 1 t 1 ) ( l + g 1 t i ) - l ( l - r G O T I ) - t i g 1 t i ( l + g 1 t i ) - l ( H G O T I ) • ti (1- fGoTi ) - V i V ^ i V ^ i V ' ^ o V = t, + t.G T. -i i o 1 " fci + ti(Go-«l)Ti We now have a multiple s c a t t e r i n g expansion of the pion-nuclear t -matrix i n terms of the pion-nucleon t-matrix. The on-shell pion-nucleon t-matrix i s determined from pion-nucleon experiments but extrapolation to off the energy s h e l l i s dependent on a model(Fea83). The o p t i c a l p o t e n t i a l , V, i s defined by % i s the k i n e t i c energy operator of the centre of mass motion of the nucleus, and P Q i s the pro j e c t i o n operator onto the nuclear ground state, i . e . T(E) = V + Vg Q(E)T(E) where gQ = P o/(E - k - V , P = I tb >< cb I . o 1 o ^o 1 Solving for V, V(E) - T(E) - Vg Q(E)T(E) = T(E) - T(E) g Q(E)T(E) + . . . (4) Combining equations (2), (3) and (4) A V(E) = I {tAe) + I t (e)(G (E)-g (E)) t (e)} i-1 j * i J J A + I {^(eX G o ( E ) - g o ( E ) - 8 1 ( e ) t 1 ( e ) ) } + (order t 3 ) i=l The parametric energy, e, i s s t i l l a r b i t r a r y , e = E i s usu a l l y chosen. 2.2.1 F i rs t order potential The f i r s t order p o t e n t i a l for e l a s t i c s c a t t e r i n g , V*(E), i s defined by the matrix element A V^k',k;E) = ^ ^ n ' j O ^ t.(E)|k;n;0> i - 1 1 where the expectation value i s taken between nuclear ground states, and |k;n,i> represents a state with pion of momentum k and a nucleus with centre of mass momentum n and i n t e r n a l state i . Writing the momentum space wave function of the nucleus as <J> (p.. ,p ,... ,p ,.. .p ) we have s i n , . ^ v r d ^ d ^ 2 d ~ ? ± ^ ^ A t , - -V!(k' ,k;E) "Ij . . . . x * (p ,p .. i-1 (2n) 3 (2n) 3 (2TI) 3 (27t) 3 (2p) 3 ° Z» P i ' » " P A ) * <k',P 1'|t ±(E) |k,pi> x <t>0(p1,P2,.. .p ±,. • . ? A ) . Notice that the scattering on each nucleon takes place with momentum value appropriate to that nucleon and, therefore, the nucleon momenta are averaged over the momentum d i s t r i b u t i o n of the nucleus. Ignoring spin and isospi n e f f e c t s we have 28 V ^ k ' . k j E ) = / p(p',p") <k',p'|t (E)|k,p> -^1 (2n)3 <2i0 3 where the nuclear momentum space s i n g l e nucleon d e n s i t y i s P ( P ' » P ) = I . J <i>0 (P 1- • • PI-I»P'»Pi+i» • • • P A ) X A - - - - d p \ . . d P i - i d p i + i . . d h 0 1 1 1 1 + 1 ' A (2,1)3 (2*) 3 ( 2 7 T ) 3 ( 2 W ) 3 Thus the general form of the f i r s t order o p t i c a l p o t e n t i a l contains the nuclear s t r u c t u r e i n f o r m a t i o n through the de n s i t y m a t r i x , p. Furthermore, i t contains the pion-nucleon i n t e r a c t i o n through the o f f -s h e l l t r a n s i t i o n m a t r i x . Neither of these q u a n t i t i e s i s determined from experiments i n a model independent way. The o n - s h e l l t - m a t r i x , t(k',k;E) with |k'| = |k| = /(2M E) i s measured i n pion-nucleon experiments. The nuclear ground s t a t e form f a c t o r , P(q) = J P(P + q, P) = / e i ( ^ * r ) p ( r ) d 3 r , (2rc) 3 i s measured with e l e c t r o n s c a t t e r i n g . An approximation known as the f a c t o r i s a t i o n approximation i s g e n e r a l l y used to reduce the o p t i c a l p o t e n t i a l to a c a l c u l a b l e q u a n t i t y . The p (nucleon momentum) dependence i s removed from the t-ma t r i x , a l l o w i n g the i n t e g r a l to be f a c t o r i s e d to give 1 vHk'.k.E) = <k' |t(E)|k> p(q"), (5) where momentum conservation i n performing the i n t e g r a l over p has been used. 2.2.2 Fixed scatterer approximation The fixed scatterer approximation (FSA) assumes that the nucleons i n the nucleus are f i x e d i n the nuclear centre of mass system and so ignores the Fermi motion and nuclear binding. The n-N t-matrix i s assumed to be that of the free nucleon; E i n (5) above i s then the pion-nucleon t o t a l energy corresponding to a free nucleon. This i s not a good approximation for pions because of the rapid v a r i a t i o n of the it-N s c attering amplitude with energy. P a r t i a l wave it-N amplitudes are functions of k, k' and /s, where k and k' are the r e l a t i v e momenta of the pion before and a f t e r the c o l l i s i o n and /s i s the t o t a l energy a v a i l a b l e i n the n;-N centre of mass system: + -Q2/{2(m N + V } where: a) = (k + m ) it o Tt Bj i s the binding energy of the struck nucleon and Q i s the momentum of the it-N system i n the it-nucleus centre of mass frame. In evaluating the o p t i c a l p o t e n t i a l , the amplitudes need to be averaged over the binding energies. L i u and Shakin showed (Liu79) that for low energy pions (<50 MeV) the f i r s t order c a l c u l a t i o n with FSA agrees well with a f u l l dynamical c a l c u l a t i o n i f the energy at which the amplitudes are evaluated i s lowered. A large second order p o t e n t i a l i s s t i l l needed, however, so i t i s not reasonable to f i t the energy parameter to e l a s t i c s c a t t e r i n g data. 2.2.3 Kissl inger potential A s i m p l e f i r s t o r d e r p i o n - n u c l e u s o p t i c a l p o t e n t i a l i n t h e c o o r d i n a t e s p a c e r e p r e s e n t a t i o n was d e v e l o p e d b y K i s s l i n g e r ( K i s 5 5 ) i n 1 9 5 5 . T h i s f o r m s t h e b a s i s o f m o s t o p t i c a l p o t e n t i a l s i n u s e . A b r i e f d e s c r i p t i o n o f t h i s p o t e n t i a l w i l l be g i v e n h e r e . The t r a n s i t i o n m a t r i x may be w r i t t e n g e n e r a l l y a s < k ' | t ( E ) | k > = b ( E ; k ' , k ) + c ( E ; k ' , k ) k ' « k 1 Tl It Tt I f t h e d e p e n d e n c e o f t h e a m p l i t u d e s b a n d c o n t h e o f f - s h e l l m o m e n t a i s a s s u m e d t o b e n e g l i g i b l e , t h i s may be s i m p l i f i e d t o < k ' | t ( E ) l k > = b ( E ) + c ( E ) k ' - k Tt Tt Tt The l o w e s t o r d e r o p t i c a l p o t e n t i a l i s < k ' IV 1 |*c> = A < k ' | t ( E ) | k > / e i ( k ~ k ' ) , r p ( r ) d r T h e R H . S . may b e w r i t t e n a s • w ^ x t - i k * » r , N i k » r . r - - i k ' » r . . - i k » r . -A b ( E ) J e p ( r ) e d r + A c ( E ) J ( V e ) » p ( r ) ( V e ) d r . Tl It I n t e g r a t i n g t h e s e c o n d t e r m b y p a r t s g i v e s , , . r - i k ' » r . - . i k » r - , . f - i k ' » r r - - i k * r -A b ( E ) J e p ( r ) e d r - A c ( E ) J e ( V « p ( r ) V ) e d r Tt M J w h e r e e a c h g r a d i e n t o p e r a t o r now a c t s o n a l l f u n c t i o n s o f f t o t h e r i g h t o f i t . R e m o v i n g t h e p l a n e w a v e t e r m s we h a v e V k r ) = A b ( E ) p ( r ) - A c ( E ) V « p ( r ) V ( 6 ) Tl 11 T h i s i s t h e K i s s l i n g e r p o t e n t i a l . T he k ' » k t e r m i n t h e t r a n s i t i o n m a t r i x r e s u l t s , t h e r e f o r e , i n a n o n - l o c a l p o t e n t i a l i n w h i c h g r a d i e n t o p e r a t o r s a c t o n t h e p i o n w a v e f u n c t i o n w h e n t h e p o t e n t i a l i s u s e d i n t h e w a v e e q u a t i o n . 2 . 2 . 4 Second o r d e r p o t e n t i a l T h e s e c o n d o r d e r p o t e n t i a l ( E r i 6 6 , E r i 7 0 ) d e s c r i b e s s c a t t e r i n g f r o m two n u c l e o n s i n s e q u e n c e , w i t h t h e n u c l e u s i n a n e x c i t e d s t a t e i n t h e i n t e r m e d i a t e s t a g e . The n u c l e a r m e d i u m i s n o t h o m o g e n e o u s , b u t g r a n u l a r , a n d t h e p i o n f i e l d a t a n u c l e o n i s m o d i f i e d b y t h i s . T he n u c l e o n c o r r e l a t i o n s e f f e c t i v e l y k e e p t h e n u c l e o n s s e p a r a t e d a n d t h e r e f o r e e a c h n u c l e o n may be c o n s i d e r e d t o b e a t t h e c e n t r e o f a h o l e , i n w h i c h t h e r e a r e no n u c l e o n s . T h i s h a s t h e e f f e c t o f r e d u c i n g t h e p i o n f i e l d a t t h e n u c l e o n , t h e r e b y r e d u c i n g t h e p i o n - n u c l e u s p o t e n t i a l . T he r e p u l s i v e s -wave p o t e n t i a l i s p r e d o m i n a n t l y p r o d u c e d b y t h e s e c o n d o r d e r t e r m a s t h e f i r s t o r d e r s - w a v e p o t e n t i a l i s a l m o s t z e r o . The p - w a v e p o t e n t i a l i s m o d i f i e d t o 4TT A p ( r ) c 1 + ( A - l ) p ( r ) c V. T h i s m o d i f i c a t i o n i s k n o w n a s t h e L o r e n t z - L o r e n z e f f e c t b e c a u s e o f i t s s i m i l a r i t y t o t h e c l a s s i c a l L o r e n t z - L o r e n z e f f e c t f o r t h e d i e l e c t r i c c o n s t a n t o f a c r y s t a l . T he L o r e n t z - L o r e n z p a r a m e t e r , X, i s i n c l u d e d t o a c c o u n t f o r t h e f i n i t e r a n g e o f t h e n - n u c l e o n i n t e r a c t i o n . T h i s g i v e s t h e n f o r t h e p - w a v e p o t e n t i a l V 1% W>c V. ( 7 ) 1 + - | 2 L X ( A - l ) p ( r ) c The v a l u e o f X d e p e n d s o n t h e r a n g e o f t h e it-N i n t e r a c t i o n . S t u d i e s b y F a l d t ( F a " 1 7 0 ) , E i s e n b e r g e t a l . ( E i s 7 3 ) a n d I a c h e l l o a n d L a n d e ( I a c 7 4 ) i n d i c a t e t h a t X i s r e d u c e d t o a b o u t 0 . 2 b y a f i n i t e r a n g e i n t e r a c t i o n . H o w e v e r , D e l o r m e a n d E r i c s o n ( D e l 7 6 ) f i n d t h a t P a u l i c o r r e l a t i o n s i n c r e a s e \ . F i t s t o p i o n i c a t o m l e v e l s a n d e l a s t i c s c a t t e r i n g h a v e n o t d e t e r m i n e d t h e n e c e s s i t y o f t h i s t e r m . 2.2.5 Absorption P i o n s may b e a b s o r b e d i n t h e n u c l e u s a n d h a v e t h e i r r e s t m a s s d i s t r i b u t e d among t h e n u c l e o n s a s k i n e t i c e n e r g y . A p i o n d e p o s i t i n g a l l o f i t s e n e r g y t o o n e n u c l e o n w o u l d i n c r e a s e t h e k i n e t i c e n e r g y o f t h a t n u c l e o n b y a t l e a s t 139 MeV w h i l e a d d i n g l i t t l e momentum ( z e r o i n t h e c a s e o f a p i o n a t r e s t ) t o t h e n u c l e u s a s a w h o l e . T h e momentum i n c r e a s e o f t h e a b s o r b i n g n u c l e o n w o u l d t h e r e f o r e h a v e t o b e o b t a i n e d f r o m i n t e r a c t i o n w i t h t h e r e s t o f t h e n u c l e u s . T h i s momentum i n c r e a s e i s a p p r o x i m a t e l y e q u a l t o t h e momentum o f a n u c l e o n w i t h k i n e t i c e n e r g y o f 139 MeV o r p=/2m m =500 M e V / c . S i n c e t h e F e r m i momentum i n a l a r g e Tt p n u c l e u s i s t y p i c a l l y 250 M e V / c , i t i s e x t r e m e l y u n l i k e l y t h a t a n y n u c l e o n c o u l d r e c e i v e a momentum o f 5 0 0 M e V / c t h r o u g h c o l l i s i o n s w i t h o t h e r n u c l e o n s . T h e e n e r g y o f t h e p i o n i n a b s o r p t i o n i s , t h e r e f o r e , u s u a l l y s h a r e d a m o n g s t two o r m o r e n u c l e o n s . T h e momentum m a t c h i n g c a n t h e n be b r o u g h t a b o u t b y t h e p i o n s c a t t e r i n g o f f i t s momentum s h e l l f r o m o n e n u c l e o n b e f o r e b e i n g a b s o r b e d o n a n o t h e r . I f t h e momentum f r o m t h e f i r s t s c a t t e r i n g i s l a r g e r t h a n t h e i n i t i a l momen tum, t h e n t h e momentum t o b e s u p p l i e d b y t h e n u c l e u s i n - a b s o r p t i o n o f t h e p i o n i s r e d u c e d . P i o n a b s o r p t i o n i s u s u a l l y i n c l u d e d i n t h e o p t i c a l p o t e n t i a l i n a p h e n o m e n o l o g i c a l f o r m . The f o r m o f t h i s p o t e n t i a l i s s i m i l a r t o t h e f i r s t o r d e r p o t e n t i a l b u t w i t h p ( r ) r e p l a c e d b y p 2 ( r ) . T h e d e p e n d e n c e o n t h e s q u a r e o f t h e d e n s i t y r e p r e s e n t s t h e n e e d f o r two n u c l e o n s i n t h e a b s o r p t i o n p r o c e s s . T h i s f o r m d o e s n o t d i s c r i m i n a t e b e t w e e n a b s o r p t i o n o n n n , pp o r np p a i r s . I f , f o r i n s t a n c e , t h e a b s o r p t i o n i s p r i m a r i l y o n np p a i r s , o r q u a s i d e u t e r o n s , t h e n p n p p s h o u l d be u s e d . T h e two new p a r a m e t e r s B a n d C ( a n a l o g o u s t o b a n d c i n e q u a t i o n 6 ) may be f i t t e d t o p i o n i c a t o m l e v e l ( S t r 8 0 ) w i d t h s . T h i s , o f c o u r s e , c a n o n l y f i x I m ( B ) a n d I m ( C ) a t z e r o e n e r g y . The e n e r g y v a r i a t i o n o f t h e s e p a r a m e t e r s may be a s s u m e d t o be t h e same a s t h a t f o r p i o n a b s o r p t i o n o n a d e u t e r o n . A l t e r n a t i v e l y , v a l u e s a t h i g h e r e n e r g i e s may be o b t a i n e d f r o m f i t t i n g s t o e l a s t i c s c a t t e r i n g d a t a ( S t r 7 9 ) . T h e r e i s s t r o n g c o r r e l a t i o n o f I m ( B ) a n d I m ( C ) w i t h t h e i m a g i n a r y p a r t s o f b a n d c . To s e p a r a t e t h e c o n t r i b u t i o n s f r o m t h e d i f f e r e n t i m a g i n a r y t e r m s i t i s t h e r e f o r e n e c e s s a r y t o f i t ( C a r 8 2 ) t h e a b s o r p t i o n a s w e l l a s t h e q u a s i - e l a s t i c s c a t t e r i n g . The r e a l , d i s p e r s i v e , p a r t s o f B a n d C a r e u s u a l l y a l s o i n c l u d e d . T h e s e a r e f i t t e d t o t h e e l a s t i c s c a t t e r i n g d a t a o r f i x e d a t t h e n e g a t i v e o f t h e r e s p e c t i v e i m a g i n a r y t e r m s , ( s e e H U f n e r ( H U f 7 5 ) ) . 2.3 Delta-Hole Model T h e 7 c - n u c l e o n a n d i t - n u c l e u s i n t e r a c t i o n s a r e d o m i n a t e d a t r e s o n a n c e e n e r g i e s b y t h e e f f e c t s o f t h e A ( 1 2 3 2 ) r e s o n a n c e a n d t h e r e f o r e t h e d y n a m i c s o f t h e i s o b a r p r o p a g a t i o n i n t h e n u c l e u s h a v e a s i g n i f i c a n t i n f l u e n c e o n p i o n s c a t t e r i n g . T h e e f f e c t s o f i s o b a r p r o p a g a t i o n a r e n o t a c c o u n t e d f o r i n t h e s t a n d a r d p i o n - n u c l e u s o p t i c a l p o t e n t i a l w h e n t h e f a c t o r i s a t i o n a p p r o x i m a t i o n i s m a d e . T h e A - h o l e m o d e l p r o v i d e s a n a l t e r n a t e m e t h o d o f s u m m i n g t h e m u l t i p l e s c a t t e r i n g s e r i e s b y e x p l i c i t l y i n c l u d i n g t h e e f f e c t s o f t h e i s o b a r d e g r e e o f f r e e d o m . T h e A - h o l e s t a t e s a r e n u c l e a r s t a t e s c o n s i s t i n g o f a n i s o b a r p l u s o n e n u c l e o n h o l e , a n d a r e t h e ' d o o r w a y s t a t e s ' t h r o u g h w h i c h a l l o t h e r s t a t e s , o t h e r t h a n t h a t o f a p i o n p l u s g r o u n d s t a t e n u c l e u s , a r e r e a c h e d . The d o o r w a y s t a t e h y p o t h e s i s w a s f i r s t a p p l i e d t o l o w - e n e r g y n u c l e a r r e a c t i o n s b y F e s h b a c h , K e r m a n a n d Lemmer ( F e s 6 7 ) a n d d e v e l o p e d b y K i s s l i n g e r a n d Wang ( K i s 7 3 , K i s 7 6 ) f o r p i o n n u c l e u s s c a t t e r i n g . W i t h i n t h e d o o r w a y s t a t e f r a m e w o r k we may w r i t e t h e o p t i c a l p o t e n t i a l a s Vopt( E) = V A(E) + V P D ( E ) [ E - Hn(E)]- 1 V D P ( E ) where V ^ E ) i s the non-resonant i n t e r a c t i o n , Hj)(E) i s the Hamiltonian of the doorway state, Vpn(E) and Vnp(E) are the inte r a c t i o n s coupling the e l a s t i c and doorway states; V ^ E ) = PHP, Vp D(E) = PHD, V D P ( E ) = DHP, where P and D project onto the spaces of the e l a s t i c TEA channel and the doorway states r e s p e c t i v e l y . The doorway states are chosen to diagonalise (E - H D ( E ) ) . The p o t e n t i a l matrix for e l a s t i c s c a ttering i s then |<t> •><<!>. I <k'|v o p t|k> = <k',o|V l|k,o> + < k ' , o|y p D T 1 J g _ J y T v D p | k , o > The f i r s t term i s the non-resonant i n t e r a c t i o n and the second term represents the coupling to the doorway states ( ^j)* The numerator may be evaluated by r e l a t i n g i t to the resonant n-nucleon amplitude i n the impulse approximation. Second order contributions, such as Pau l i b l o c k i n g , may a l s o be t a k e n i n t o a c c o u n t ( H i r 7 9 ) . T h e i s o b a r w a v e f u n c t i o n s may be e x p a n d e d i n a c o m p l e t e s e t o f o s c i l l a t o r w a v e f u n c t i o n s ; a n a p p r o x i m a t i o n made i s t o c u t o f f a t some o s c i l l a t o r e x c i t a t i o n e n e r g y t o l i m i t t h e d i m e n s i o n a l i t y o f t h e d o o r w a y s p a c e . T h e c a l c u l a t i o n i s s i m p l i f i e d b y a s s u m i n g a s p e c i f i c m o d e l f o r t h e d o o r w a y s t a t e s ; i n t h e A - h o l e m o d e l t h e y a r e t a k e n t o be p a r t i c l e h o l e s t a t e s o b t a i n e d b y e x c i t i n g a n u c l e o n t o a A , i n a s i n g l e p a r t i c l e s t a t e , c o u p l e d t o t h e r e m a i n i n g s i n g l e h o l e i n t h e t a r g e t n u c l e u s . T h e A - h o l e p r o p a g a t o r i s d i a g o n a l i s e d i n a s h e l l m o d e l b a s i s t o g i v e A - h o l e e i g e n s t a t e s a n d e i g e n e n e r g i e s . The r e s u l t i n g p i o n - n u c l e u s p a r t i a l w a v e a m p l i t u d e s a r e f o u n d t o be d o m i n a t e d b y one o r two c o l l e c t i v e i s o b a r - h o l e s t a t e s ( H i r 7 7 , H i r 7 9 ) . T h e s e c o l l e c t i v e s t a t e s t h e n f o r m t h e b a s i s s e t f o r t h e A - h o l e c a l c u l a t i o n . T h e e f f e c t s o f p i o n a b s o r p t i o n a r e a c c o u n t e d f o r b y i n c l u d i n g a c o m p l e x p h e n o m e n o l o g i c a l ' s p r e a d i n g p o t e n t i a l ' i n t o t h e A - h o l e i n t e r a c t i o n , f i t t i n g t h e p a r a m e t e r s t o e l a s t i c s c a t t e r i n g ; H i r a t a e t a l . ( H i r 8 3 ) s t u d y t h e t h e o r e t i c a l b a s i s f o r t h i s s p r e a d i n g p o t e n t i a l . A s p i n o r b i t c o m p o n e n t i n t h e 7 t - n u c l e o n i n t e r a c t i o n h a s a l s o b e e n i n t r o d u c e d ( H o r 8 0 ) , t o i m p r o v e a g r e e m e n t w i t h e l a s t i c s c a t t e r i n g r e s u l t s . W i t h A - n u c l e u s i n t e r a c t i o n p a r a m e t e r s d e t e r m i n e d f r o m e l a s t i c s c a t t e r i n g , t h e A - h o l e m o d e l h a s b e e n u s e d t o c a l c u l a t e t h e e x c i t a t i o n o f d i s c r e t e s t a t e s ( L e n 8 2 ) a n d q u a s i - f r e e s c a t t e r i n g ( B a u 8 2 , T h i 8 2 ) , s h o w i n g g o o d a g r e e m e n t w i t h e x p e r i m e n t . T h i s i s a g o o d i n d i c a t i o n t h a t t h e s t r e n g t h s a r e b e i n g c o r r e c t l y a p p o r t i o n e d t o t h e v a r i o u s t e r m s i n t h e A -n u c l e u s i n t e r a c t i o n . T h e A - h o l e m o d e l i s , h o w e v e r , l e s s a p p r o p r i a t e a t l o w e n e r g i e s w h e r e t h e A ( 1 2 3 2 ) r e s o n a n c e d o e s n o t d o m i n a t e t h e s c a t t e r i n g T h i s i s d e m o n s t r a t e d b y (TC,H 'Y) e x p e r i m e n t s a t 65 a n d 90 MeV ( S o b 8 4 ) , 36 where the magnetic substate populations do not indicate delta contributions. Hence the A-hole model i s not used i n the analysis of t h i s work. CHAPTER I I I The E x p e r i m e n t s 3.1 I n t r o d u c t i o n To m i n i m i s e t h e m o d e l d e p e n d e n c y o f t h e s e m e a s u r e m e n t s , c a l c u l a t i o n s w e r e made o f t h e s e n s i t i v i t y o f t h e c r o s s s e c t i o n r a t i o s t o v a r i a t i o n s i n t h e o p t i c a l p o t e n t i a l p a r a m e t e r s . U s i n g t h e o p t i c a l p o t e n t i a l o f S t r i e k e r , M cManu s a n d C a r r a n d S e t 1 ( S t r 7 9 ) p a r a m e t e r s ( s e e T a b l e V I I ) c r o s s s e c t i o n s w e r e c a l c u l a t e d f o r t h e 50 MeV iC o n 3 6 S a n d 3 2 S a n d t h e r a t i o o f t h e c r o s s s e c t i o n s R was f o u n d . One p o t e n t i a l p a r a m e t e r was t h e n i n c r e a s e d b y 15% a n d t h e c a l c u l a t i o n s r e p e a t e d . T h e p e r c e n t a g e c h a n g e i n t h e c r o s s s e c t i o n r a t i o a t e a c h a n g l e was t h e n c a l c u l a t e d . T h i s was r e p e a t e d f o r e a c h p o t e n t i a l p a r a m e t e r . The r e s u l t s a r e s h o w n i n F i g u r e 4 . F i g u r e 5 i s t h e p e r c e n t a g e c h a n g e i n t h e c r o s s s e c t i o n r a t i o f o r a n i n c r e a s e b y . 1 0 0 f m o f t h e n e u t r o n h a l f d e n s i t y r a d i u s , c n o f 3 6 S . I t i s c l e a r f r o m F i g u r e 4 t h a t t h e max imum s e n s i t i v i t y t o t h e p o t e n t i a l p a r a m e t e r s i s v e r y c l o s e t o t h e p o s i t i o n o f t h e d i f f r a c t i v e m i n i m u m a t 1 2 0 " . B e l o w 1 0 0 " t h e r e i s l i t t l e d e p e n d e n c e o n t h e p o t e n t i a l p a r a m e t e r s . T h e d a t a f o r t h e s u l f u r e x p e r i m e n t s , t h e r e f o r e , w e r e o n l y t a k e n up t o a n a n g l e o f 1 0 0 " ; t h e d i f f r a c t i o n m i n i m u m f o r m a g n e s i u m i s a t a b o u t 1 2 5 " a n d d a t a w e r e t a k e n up t o 1 0 5 " . I n a n y c a s e , t h e d i f f e r e n t i a l c r o s s s e c t i o n f a l l s r a p i d l y t o w a r d s t h e d i f f r a c t i o n m i n i m u m a n d a l t h o u g h t h e s e n s i t i v i t y t o t h e n e u t r o n r a d i u s i n c r e a s e s h e r e i t e v e n t u a l l y b e c o m e s l e s s e f f i c i e n t i n t e r m s o f e x p e r i m e n t a l t i m e t o t a k e d a t a a t l a r g e r a n g l e s . 38 20 AO 60 80 100 120 MO C M . ANGLE (degrees) FIGURE 4 Percentage change, 6R, of calculated 3 6S/ 3 2S cross section ratio to 15% increase in optical potential parameters. The parameters, other than the one varied, are Set 1 (see Table VII). 3 9 FIGURE 5 Percentage change of c a l c u l a t e d 3 6 S / 3 2 S cross s e c t i o n r a t i o to 0.100 fm increase i n 3 6 c n , the Fermi d i s t r i b u t i o n h a l f d e n s i t y radius parameter f o r the neutron d i s t r i b u t i o n of 3 6 S . The p o t e n t i a l was SMC79 with Set 1 parameters. Also shown i s the f a c t o r E f f , defined i n the t e x t . 40 We d e f i n e t h e s e n s i t i v i t y o f t h e c r o s s s e c t i o n r a t i o , R ( 9 ) , t o c n ( t h e F e r m i d i s t r i b u t i o n h a l f d e n s i t y r a d i u s p a r a m e t e r o f t h e n e u t r o n d e n s i t y d i s t r i b u t i o n ) o f t h e l a r g e r i s o t o p e , a t a n a n g l e , 0 , b y s ( 0 ) = = a*2 °R(e)  b K V J - ac OR(G) ac n n Now x 2 = ( R ( e ) - R ' ( 9 ) ) 2 / 0 2 D / , A N , w h e r e cr.,,-. i s t h e s t a n d a r d d e v i a t i o n o f KQtj j K ( 0 ) R ( 0 ) , a n d R ' ( 9 ) i s t h e c a l c u l a t e d c r o s s s e c t i o n r a t i o . T h e r e f o r e s ( e ) o o c ° R ( 0 ) n S i n c e o - 2 ^ ^ ^ i s r e l a t e d t o t h e n u m b e r s o f e v e n t s N^ a n d N 2 f r o m t h e two r u n s , b y fjm., - i + i .4- u H l - „ 2 R2(S) N l N2 N l 1 2 we h a v e R 2 ( 0 ) / a 2 , « N « t x XS , w h e r e t i s t h e t i m e t a k e n t o a c c u m u l a t e t h e d a t a a n d XS i s t h e c r o s s s e c t i o n a t a n g l e 0 . T h e r e f o r e s j e i a pmi x xs 5 t 9 c R n E f f ( 0 ) i s p l o t t e d i n F i g u r e 5 . I t c a n be s e e n t h a t t h e e f f i c i e n c y f o r m e a s u r i n g c n p e a k s a t a b o u t 1 1 0 " a n d t h e n f a l l s . T h e e f f i c i e n c y a l s o r i s e s a t s m a l l a n g l e s a s t h e c r o s s s e c t o n i s r i s i n g r a p i d l y . H o w e v e r , d R ( 0 ) / 9 c n a t t h e s e a n g l e s i s s m a l l a n d t h e e x t r a c t i o n o f c n w i l l be s e n s i t i v e t o a s m a l l s y s t e m a t i c e r r o r i n e x p e r i m e n t a l R ( 0 ) o r t o s m a l l m o d e l d e p e n d e n c i e s o f t h e t h e o r e t i c a l v a l u e . Two s e p a r a t e e x p e r i m e n t s w e r e p e r f o r m e d t o o b t a i n t h e d a t a p r e s e n t e d h e r e . T h e f i r s t e x p e r i m e n t , m e a s u r i n g t h e TC - 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 c r o s s s e c t i o n r a t i o o f 3 6 S t o 3 2 S , was d o n e w i t h t h e QD ( Q u a d r u p o l e / D i p o l e ) s p e c t r o m e t e r . T h i s s p e c t r o m e t e r h a d a s m a l l s o l i d a n g l e (4 m s r ) a n d a new s p e c t r o m e t e r s y s t e m , t h e QQD ( S o b 8 3 ) , wa s d e v e l o p e d t o p r o v i d e b o t h l a r g e r s o l i d a n g l e a n d i m p r o v e d r e s o l u t i o n . T h e o t h e r e x p e r i m e n t , p e r f o r m e d o n 2 6 M g , 2 4 M g a n d 3 H S , 3 2 S , was d o n e w i t h t h e QQD s p e c t r o m e t e r . B o t h e x p e r i m e n t s w e r e p e r f o r m e d o n t h e M13 beam l i n e ( 0 r a 8 1 ) ( F i g u r e 6 ) a t T R I U M F . T h e M13 beam l i n e i s a l o w e n e r g y ( 2 0 - 1 3 0 M e V / c ) p i o n a n d muon c h a n n e l . The c h a n n e l i s s y m m e t r i c , w i t h two 60° b e n d s , a n d i s 9 . 3 m l o n g . The beam l i n e m a g n e t i c e l e m e n t s , t h e s l i t s a n d t h e j a w s a r e c o n t r o l l e d b y c o m p u t e r t h r o u g h t h e REMCON s y s t e m . The e n e r g y o f t h e beam h a d b e e n c a l i b r a t e d w i t h a n oc-source ( 0 r a 8 1 ) a t 91 M e V / c a n d t h e d i p o l e m a g n e t i c f i e l d s w e r e m e a s u r e d w i t h N u c l e a r M a g n e t i c R e s o n a n c e p r o b e s t o s c a l e t o d i f f e r e n t m o m e n t a . T h e e n e r g i e s q u o t e d f o r t h e e x p e r i m e n t s a r e f o r t h e p i o n a t t h e c e n t r e o f t h e t a r g e t a f t e r c a l c u l a t i n g t h e e n e r g y l o s s i n a l l m a t e r i a l u p s t r e a m . T h e beam s p o t a t t h e t a r g e t p o s i t i o n was a p p r o x i m a t e l y 2 cm i n d i a m e t e r . The r a t i o o f n:\i:e i n t h e beam wa s a p p r o x i m a t e l y 9 2 : 3 : 5 ; m e a s u r e d a t 50 MeV p i o n e n e r g y b y t i m e o f f l i g h t ( s e e F i g u r e 1 0 ) . I n e a c h e x p e r i m e n t p a r t i c u l a r e m p h a s i s was p l a c e d o n m e a s u r i n g p r e c i s e l y t h e r a t i o o f c r o s s s e c t i o n s . The t a r g e t s w e r e c o m p a r e d b y p l a c i n g t h e m i n t h e beam i n s e q u e n c e f o r p e r i o d s o f a p p r o x i m a t e l y h a l f a n h o u r . The m e a s u r e m e n t s a t e a c h a n g l e w e r e d i v i d e d t h u s i n t o s e v e r a l s e t s o f r u n s . E f f e c t s d u e t o l o n g t e r m i n s t a b i l i t i e s ( w h e r e a s i g n i f i c a n t c h a n g e w o u l d be p r o d u c e d i n t i m e s g r e a t e r t h a n 1 h o u r ) o f beam m o n i t o r s , beam p r o p e r t i e s o r d e t e c t o r e f f i c i e n c i e s w e r e t h e r e f o r e m i n i m i s e d i n t a k i n g t h e c r o s s s e c t i o n r a t i o s . 42 FIGURE 6 M13 beamline and QQD spectrometer. BMl and BM2 are beam monitors, BI and B2, described i n the t e x t . 3.2 The 3 6 S , 3 2 S Experiment T h e 3 6 S / 3 2 S e x p e r i m e n t u s e d a Q u a d r u p o l e / M p o l e s p e c t r o m e t e r w i t h m u l t i - w i r e p r o p o r t i o n a l c h a m b e r s b e f o r e a n d a f t e r t h e s p e c t r o m e t e r m a g n e t s . The o v e r a l l l a y o u t o f t h i s s y s t e m i s s h o w n i n F i g u r e 7 . T h e c o o r d i n a t e s u s e d i n t h e f o l l o w i n g a r e d e f i n e d i n F i g u r e 8. 3.2.1 The QD spectrometer The QD s p e c t r o m e t e r u s e d i n t h i s e x p e r i m e n t c o m p r i s e d a v e r t i c a l b e n d d i p o l e w i t h a 4 cm p o l e f a c e g ap a n d a h o r i z o n t a l l y f o c u s s i n g q u a d r u p o l e . The s o l i d a n g l e o f t h e s p e c t r o m e t e r was 0 . 0 0 4 s r . T h e m a g n e t was m o u n t e d o n a c i r c u l a r t r a c k , a r o u n d w h i c h i t was p u s h e d t o t h e s c a t t e r i n g a n g l e ; t h e s p e c t r o m e t e r a n g l e c o u l d be r e a d f r o m a v e r n i e r s c a l e . T h e s c a t t e r e d p i o n s w e r e d e t e c t e d i n a n ' e n t r y ' w i r e c h a m b e r b e f o r e t h e d i p o l e a n d t h r e e ' e x i t ' w i r e c h a m b e r s a f t e r t h e d i p o l e . The p i o n s s c a t t e r e d f r o m t h e t a r g e t w e r e f o c u s s e d b y t h e d i p o l e m a g n e t b e c a u s e o f t h e s h a p e o f t h e d i p o l e p o l e f a c e s , a s w e l l a s b y t h e q u a d r u p o l e . T h e p o l e f a c e s w e r e o f a B r o w n e - B u e c h n e r ( B r o 5 6 ) d e s i g n . P a r t i c l e s o f h i g h e r m o m e n t a , p , w e r e t u r n e d t h r o u g h a s m a l l e r a n g l e , a s t h e r a d i u s o f c u r v a t u r e o f t h e p a r t i c l e t r a j e c t o r y i n t h e f i e l d , B , i s p r o p o r t i o n a l t o p / B . T h e y w e r e a l s o f o c u s s e d a t a d i s t a n c e f u r t h e r f r o m t h e d i p o l e c e n t r e , t h e r e f o r e t h e f o c a l p l a n e was a t a n a n g l e t o t h e w i r e c h a m b e r p l a n e s . The f o c a l p l a n e c r o s s e d t h e o p t i c a x i s a t a n a n g l e o f 2 8 " t o t h e h o r i z o n t a l b e t w e e n t h e f i r s t two e x i t w i r e c h a m b e r s . M o m e n t a w e r e r e s o l v e d b y p r o j e c t i n g t h e p a r t i c l e t r a j e c t o r y t o t h e f o c a l p l a n e u s i n g a l l t h r e e o r a n y p a i r o f e x i t w i r e c h a m b e r s a n d c o r r e c t i n g f o r m a g n i f i c a t i o n o f t h e f i n i t e t a r g e t s i z e a n d a b e r r a t i o n s i n t h e f o c u s s i n g . The h a n d l i n g o f t h e s e a b e r r a t i o n s i s d i s c u s s e d i n A p p e n d i x 1 . FIGURE 7 Schematic diagram of the QD spectrometer. FIGURE 8 Coordinates used i n discussion of the QD spectrometer. 46 3.2.2 Wire chambers A l l of the wire chambers had delay l i n e readout which gave no r a t e problem at the counting rates obtained (up to a few hundred h e r t z ) . The delay l i n e s f o r the entry wire chambers were of p r i n t e d c i r c u i t type w i t h both axes coming from g r i d s of cathode wires at ground p o t e n t i a l . The delay l i n e i n the X, d i s p e r s i v e , d i r e c t i o n of the e x i t wire chambers was the cathode wire i t s e l f . A continuous h e l i x of cathode wire was wound around the anode plane so that the pulses were delayed as they propagated down t h i s w i r e . Information i n the Y d i r e c t i o n was obtained from the anode wires which were connected to a p r i n t e d c i r c u i t delay l i n e . The wire spacing of the wire chambers was 1 mm f o r the cathodes and 2 mm f o r the anodes. The r e s o l u t i o n (Ope83) of the entry wire chambers was 0.6 mm. The r e s o l u t i o n of the e x i t chambers was not measured, but, since the d i s p e r s i o n i n the plane of these chambers was about lcm/%(6p/p), any reasonable r e s o l u t i o n (~1 mm) of the wire chambers would not have s i g n i f i c a n t l y a f f e c t e d the r e s o l u t i o n of the t o t a l system. A gas mixture of 69.7% argon, 0.3% freon and 30% isobutane was used w i t h a small amount of methylal added by bubbling the argon through l i q u i d m ethylal (dimethoxy methane : CH 2-(0CH 3) 2) at 0°C. 3.2.3 Beam monitors The primary beam monitor was a s c i n t i l l a t o r telescope comprised of: a 0.8 mm t h i c k s c i n t i l l a t o r , BI, at the e x i t of the beam pipe; a l a r g e 6.3 mm t h i c k s c i n t i l l a t o r , B2, upstream of the t a r g e t ; and a 1.6 mm t h i c k veto, V, s c i n t i l l a t o r at the target p o s i t i o n . A l l s c i n t i l l a t o r s were of NE110 w i t h RC8575R phototubes. The f l u x through the target was given by the coincidence r a t e B1»B2»V. The output of the B1»B2»V coincidence u n i t was t i m e d r e l a t i v e t o t h e B I s i g n a l t i m e . The t a r g e t s w e r e m o u n t e d b e h i n d a 9 mm d i a m e t e r h o l e i n t h e v e t o c o u n t e r . T h e v e t o c o u n t e r a l l o w e d t h e p i o n f l u x t h r o u g h t h e s m a l l ( 1 cm d i a m e t e r ) t a r g e t s t o be m e a s u r e d e v e n t h o u g h t h e b e a m s p o t was 2 cm FWHM. 9 t o 13% o f t h e beam p a s s e d t h r o u g h t h e h o l e , d e p e n d i n g o n t h e t a r g e t a n g l e . I t wa s i m p o r t a n t t h a t t h i s c o u n t e r was e f f i c i e n t a n d s t a b l e b e c a u s e i t s e f f i c i e n c y d i r e c t l y a f f e c t e d t h e f l u x m e a s u r e m e n t a n d a l s o b e c a u s e t h e s c i n t i l l a t o r was a p o s s i b l e s o u r c e o f b a c k g r o u n d . The e f f i c i e n c y wa s p l a t e a u e d ( i . e . , s e t t o be i n d e p e n d e n t o f s m a l l c h a n g e s i n h i g h v o l t a g e o r d i s c r i m i n a t o r t h r e s h o l d s e t t i n g s ) b y v a r y i n g t h e v o l t a g e t o t h e p h o t o t u b e u n t i l t h e r a t i o , B 1 » B 2 « V / B 1 « B 2 , o f c o i n c i d e n c e r a t e s , was m i n i m i s e d . T h i s r a t i o was t h e n m e a s u r e d w i t h t h e beam p a s s i n g t h r o u g h t h e v e t o c o u n t e r away f r o m t h e h o l e . The e f f i c i e n c y m e a s u r e d t h i s way was f o u n d t o be 97%, b u t p a r t o f t h e beam w o u l d s t i l l p a s s t h r o u g h t h e h o l e i n t h i s a r r a n g e m e n t , s o t h i s o n l y s e t s a l o w e r l i m i t o n t h e e f f i c i e n c y . The e f f i c i e n c y n e a r t h e h o l e s o n t h e s i d e o p p o s i t e t o t h e p h o t o t u b e w o u l d be e x p e c t e d t o b e l o w e r t h a n t h a t o f t h e r e s t o f t h e s c i n t i l l a t o r , s i n c e l i g h t f r o m t h i s p o s i t i o n i s s h i e l d e d f r o m t h e p h o t o t u b e . No e v i d e n c e o f t h i s was s e e n w h e n s t u d y i n g t h e t r a c e b a c k o f t h e p i o n t r a j e c t o r i e s t o t h e t a r g e t . A n y b a c k g r o u n d f r o m t h i s s o u r c e w a s p a r t l y e l i m i n a t e d b y m a k i n g c u t s o n t h e t r a c e b a c k p o s i t i o n . T h e r e m a i n i n g b a c k g r o u n d was m e a s u r e d b y p e r f o r m i n g r u n s w i t h a n e m p t y t a r g e t f r a m e b e h i n d t h e v e t o h o l e a n d m a k i n g t h e same c u t s . The v e t o c o u n t e r w a s m o s t l y l i g h t s e a l e d w i t h b l a c k p l a s t i c , b u t t h e a r e a a r o u n d a n d o v e r t h e h o l e was c o v e r e d i n 6 \m a l u m i n u m f o i l t o r e d u c e t h e b a c k g r o u n d . 3.2.4 Electronic logic The e l e c t r o n i c l o g i c d i a g r a m i s g i v e n i n F i g u r e 9 . A n e v e n t i n t h e s p e c t r o m e t e r wa s d e f i n e d b y t h e c o i n c i d e n c e B 1 « B 2 « V » S 1 « S 2 . T h i s c o i n c i d e n c e s t r o b e d t h e CAMAC c o i n c i d e n c e b u f f e r p a t t e r n u n i t w h i c h t h e n g e n e r a t e d a LAM ( a CAMAC i n t e r r u p t c a l l e d a L o o k A t M e ) , i n t e r r u p t i n g t h e c o m p u t e r . T h e e v e n t t r i g g e r a l s o s t a r t e d a l l t h e T D C ' s . T h e w i r e c h a m b e r d e l a y l i n e o u t p u t s p r o v i d e d TDC s t o p s . A l l o f t h e s c a l e r s a n d t h e e v e n t t r i g g e r w e r e g a t e d o f f f r o m t h e t i m e o f a n e v e n t s t r o b e u n t i l t h e c o m p u t e r h a d f i n i s h e d r e a d i n g a l l o f t h e CAMAC m o d u l e s a n d was r e a d y t o a c c e p t a n o t h e r e v e n t . T h e g a t e s o r v e t o s t o t h e v a r i o u s m o d u l e s came f r o m a l o g i c f a n - i n / o u t w h i c h a c t e d a s a l o g i c OR, g i v i n g a n o u t p u t N I M l e v e l a s l o n g a s a n y i n p u t h a d a N I M p u l s e a p p l i e d . T h e f i r s t i n p u t t o a r r i v e a t t h i s m o d u l e was t h e e v e n t s t r o b e . T h e o u t p u t t h e n s t a r t e d a g a t e g e n e r a t o r w h i c h was a d j u s t e d t o p r o v i d e a p u l s e l o n g e n o u g h t o o v e r l a p t h e b e g i n n i n g o f t h e ' c o m p u t e r b u s y ' s i g n a l f r o m a n o u t p u t r e g i s t e r . T he e v e n t s t r o b e p u l s e was l o n g e n o u g h t o o v e r l a p t h e s t a r t o f t h e p u l s e f r o m t h e g a t e g e n e r a t o r . I n t h i s w a y a c o n t i n u o u s ' b u s y ' g a t e was a c h i e v e d u n t i l t h e c o m p u t e r was r e a d y t o a c c e p t a n o t h e r e v e n t a n d r e m o v e d t h e s i g n a l f r o m t h e o u t p u t r e g i s t e r . A TDC s t o p wa s p r o v i d e d b y t h e s i g n a l f r o m t h e c a p a c i t i v e p r o b e i n t h e p r o t o n beam l i n e j u s t u p s t r e a m o f t h e p i o n p r o d u c t i o n t a r g e t T l . T h e t i m e b e t w e e n t h i s s i g n a l a n d t h e TDC s t a r t s i g n a l p r o v i d e d a m e a s u r e o f t h e d e t e c t e d p a r t i c l e ' s f l i g h t t i m e down t h e s e c o n d a r y beam l i n e , a s s e e n i n F i g u r e 1 0 . S i n c e a s t o p s i g n a l i s p r o v i d e d e v e r y 4 3 n s , t h e max imum t i m e b e t w e e n a s t a r t a n d a s t o p i s 4 3 n s . T h e d i f f e r e n c e i n t h e t i m e s t a k e n b y 128 M e V / c p i o n s a n d e l e c t r o n s t r a v e l l i n g t h e l e n g t h o f M13 ( 9 . 6 m) i s 1 5 n s a n d c o n s e q u e n t l y t h e p a r t i c l e p u l s e s c o u l d be f i t i n t o t h e 4 3 n s w i n d o w . E l e c t r o n i c l o g i c for the QD experiment. 50 FIGURE 10 Time-of-flight spectrum showing r e l a t i v e t r a v e l times and amounts of TT'S, U'S, and e's down the M13 beamline. This data i s from the QQD experiment and i s very s i m i l a r to that of the QD experiment. A n o t h e r e v e n t t r i g g e r wa s g e n e r a t e d b y t h e c i r c u i t l a b e l l e d ' b e a m s a m p l e r ' . T h i s c i r c u i t s a m p l e d t h e c o i n c i d e n c e s f r o m t h e B 1 » B 2 » V beam m o n i t o r . E a c h beam e v e n t s t a r t e d a 1 s e c o n d l o n g p u l s e f r o m a g a t e g e n e r a t o r w h i c h v e t o e d t h e beam e v e n t c o i n c i d e n c e u n i t , t h u s p r e v e n t i n g a n y m o r e beam e v e n t s f o r o n e s e c o n d . T h e f r a c t i o n o f p i o n s , muons a n d e l e c t r o n s i n t h e beam c o u l d t h e r e f o r e be c a l c u l a t e d f r o m t h e t i m e o f f l i g h t s p e c t r u m o f t h e s e e v e n t s ( F i g u r e 1 0 ) . 3 . 2 . 5 D a t a a c q u i s i t i o n The TRIUMF d a t a a c q u i s i t i o n p r o g r a m , DA , a n d t h e a n a l y s i s p r o g r a m , M U L T I ( F e r 7 9 ) , r u n n i n g u n d e r t h e o p e r a t i n g s y s t e m R S X 1 1 on a P D P 1 1 / 3 4 c o m p u t e r , was u s e d t o r e a d t h e CAMAC m o d u l e s , w r i t e d a t a o n t o m a g n e t i c t a p e a n d p e r f o r m o n l i n e a n a l y s i s . On i n t e r r u p t f r o m t h e CAMAC , t h e c o m p u t e r s u s p e n d e d a n y a n a l y s i s o f t h e p r e v i o u s e v e n t , w h i c h may h a v e b e e n i n p r o g r e s s , a n d r e a d a l l d e f i n e d CAMAC m o d u l e s , s a v i n g t h e v a l u e s i n a b u f f e r . When t h e r e was n o t e n o u g h r o o m l e f t i n t h e b u f f e r f o r a n e v e n t t h e s c a l e r s w e r e r e a d a n d t h e b u f f e r w r i t t e n t o m a g n e t i c t a p e a s o n e b l o c k . A u s e f u l f e a t u r e o f M U L T I i s t h a t t h e a n a l y s i s p r o g r a m may be m o d i f i e d a n d e n h a n c e d w h i l e s t i l l t a k i n g d a t a . 3 . 2 . 6 T a r g e t s B o t h t a r g e t s w e r e p r e p a r e d a t The M a x - P l a n c k I n s t i t u t f o r N u c l e a r P h y s i c s . T h e y c o n s i s t e d o f d i s c s o f p l a s t i c s u l f u r , 1 cm i n d i a m e t e r a n d a b o u t 1 mm t h i c k . T h e 3 6 S t a r g e t was 0 . 1 9 2 g / c m 2 t h i c k a n d h a d a n i s o t o p i c c o m p o s i t i o n : 8 1 . 1 % 3 6 S ; 18 .8% 3hS; a n d 0 . 13% 3 2 S . T h e 3 2 S t a r g e t was 0 . 1 9 3 g / c m 2 t h i c k a n d was n a t u r a l s u l f u r w i t h i s o t o p i c c o m p o s i t i o n : 9 5 . 0 2 % 3 2 S ; 0 . 75% 3 3 S ; 4 . 2 1 % 3kS a n d 0 . 0 2% 3 6 S . T h e t a r g e t s w e r e c o n t a i n e d i n p o l y e t h y l e n e b a g s o f 0 . 0 2 g / c m 2 t h i c k n e s s f o r some o f t h e m e a s u r e m e n t s . T h i s w a s f o u n d t o p r o d u c e t o o l a r g e a b a c k g r o u n d ( f r o m s c a t t e r i n g f r o m t h e c a r b o n ) a t t h e l a r g e r a n g l e s a n d wa s r e p l a c e d w i t h 6urn m y l a r . O n l y t h e d a t a a t 4 5 ° a n d 5 5 " w e r e t a k e n w i t h t h e t a r g e t s i n t h e p o l y e t h y l e n e b a g s a n d t h e b a c k g r o u n d was 25% h e r e . 3 .3 The 3 * S , 3 2 S and 2 6 M g , 2 H M g E x p e r i m e n t s T h e s e e x p e r i m e n t s w e r e p e r f o r m e d w i t h t h e QQD s p e c t r o m e t e r ( S o b 8 3 ) , b u t w e r e i n many r e s p e c t s s i m i l a r t o t h e 3 6 S , 3 2 S e x p e r i m e n t d e s c r i b e d a b o v e . 3 . 3 . 1 The QQD s p e c t r o m e t e r The QQD s p e c t r o m e t e r i s a l a r g e a c c e p t a n c e ( 0 . 0 1 8 s r ) s p e c t r o m e t e r w i t h two q u a d r u p o l e s a n d a h o r i z o n t a l l y b e n d i n g d i p o l e . T h e f i r s t q u a d r u p o l e f o c u s s e d i n t h e h o r i z o n t a l d i r e c t i o n a n d t h e s e c o n d q u a d r u p o l e i n t h e v e r t i c a l d i r e c t i o n . T h i s f o c u s s i n g s e r v e d t o i n c r e a s e t h e s o l i d a n g l e o f a c c e p t a n c e o f t h e s p e c t r o m e t e r . T he c h a r a c t e r i s t i c s o f t h e s p e c t r o m e t e r a r e g i v e n i n T a b l e I . T h e s p e c t r o m e t e r was a r r a n g e d t o b e n d i n t h e h o r i z o n t a l d i r e c t i o n t o a l l o w d i s p e r s i o n m a t c h i n g t o a d i s p e r s e d f o c u s a t t h e t a r g e t , b u t i n t h i s e x p e r i m e n t a n a c h r o m a t i c t u n e wa s u s e d . F i g u r e 6 s h o w s t h e s p e c t r o m e t e r a n d beam m o n i t o r i n g l a y o u t . The c o o r d i n a t e s u s e d i n t h e f o l l o w i n g d i s c u s s i o n a r e d e f i n e d i n F i g u r e 1 1 . Momentum r e s o l u t i o n was o b t a i n e d b y f i n d i n g t h e t r a n s f e r c o e f f i c i e n t s b e t w e e n t h e c o o r d i n a t e s o f t h e p a r t i c l e b e f o r e t h e d i p o l e a n d t h o s e a f t e r t h e d i p o l e . F o r i n s t a n c e , t h e X p o s i t i o n , X 3 , ( i n t h e d i s p e r s i v e d i r e c t i o n ) a t t h e e x i t w i r e c h a m b e r 3 c a n be e x p r e s s e d a s a p o l y n o m i a l F ( 6 p , X l , X 2 , Y l , Y 2 ) o f t h e X a n d Y p o s i t i o n s a t t h e t w o e n t r y TABLE I Specifications of QQD Spectrometer S o l i d a n g l e 0 . 0 1 8 s r Momentum a c c e p t a n c e ±20% Momen tum r a n g e <150 M e V / c F o c a l p l a n e : D i s p e r s i o n - 1 . 0 6 cm/% R a d i a l M a g n i f i c a t i o n - 0 . 5 4 T i l t a n g l e 7 2 ° P a t h l e n g t h 2 . 2 8 m A n g u l a r r a n g e 3 0 - 1 3 5 ° 54 dipole target WC1 WC2 FIGURE 11 Coordinates used i n discussion of QQD experiments. w i r e c h a m b e r s a n d 6p, t h e p e r c e n t a g e d e v i a t i o n f r o m t h e n o m i n a l m o m e n t u m . S i m i l a r l y , f o r c h a m b e r 4 , we h a v e a n e q u i v a l e n t f u n c t i o n G ( 6 p , X l , X 2 , Y 1 , Y 2 ) . T h e p o l y n o m i a l s w e r e f o u n d b y a n a l y s i n g d a t a i n w h i c h a l l f i v e c o o r d i n a t e s w e r e k n o w n . T h i s i s d i s c u s s e d i n A p p e n d i x 1 . O n l y t e r m s t o f i r s t o r d e r i n 6p w e r e u s e d f o r a n a l y s i s o f t h e s u l f u r d a t a ; a l l o f t h e s e c o e f f i c i e n t s w e r e o b t a i n e d f r o m d a t a t a k e n w i t h a c a r b o n t a r g e t a t 9 0 " , t h e d i s p e r s i v e t e r m s a r i s i n g f r o m t h e i n c l u s i o n o f t h e 1 s t e x c i t e d s t a t e a t 4 . 4 4 MeV i n t h e a n a l y s i s . T he d i s p e r s i o n i s n o t c o n s t a n t a c r o s s t h e w i d t h o f t h e e x i t w i r e c h a m b e r s ( s e e F i g u r e 1 2 ) a n d s o s m a l l t e r m s i n ( 6 p ) 2 s h o u l d be i n c l u d e d . S i n c e t h e r a n g e o f momen t a u s e d i n t h e s e e x p e r i m e n t s i s s m a l l (8%) t h e e f f e c t o f t h e s e t e r m s i s n e g l i g i b l e . H o w e v e r , new c o e f f i c i e n t s w e r e f o u n d f o r t h e a n a l y s i s o f t h e m a g n e s i u m d a t a . T h e s e c o e f f i c i e n t s w e r e o b t a i n e d f r o m a n a l y s i s o f d a t a t a k e n w i t h d i f f e r e n t f i e l d s e t t i n g s o n t h e m a g n e t i c e l e m e n t s . S i n c e t h e r a d i u s o f c u r v a t u r e o f t h e p a r t i c l e t r a j e c t o r y i n t h e m a g n e t i c f i e l d i s p r o p o r t i o n a l t o p / B ( m o m e n t u m / m a g n e t i c f i e l d s t r e n g t h ) , i n c r e a s i n g t h e m a g n e t i c f i e l d b y a g i v e n f a c t o r i s e q u i v a l e n t t o d i v i d i n g t h e momentum b y t h e same f a c t o r . T h e s p e c t r o m e t e r h a s a v a c u u m b o x e x t e n d i n g f r o m t h e e n t r a n c e o f t h e t a r g e t c h a m b e r t o a f t e r t h e l a s t w i r e c h a m b e r . T h i s b o x was n o t e v a c u a t e d i n t h i s e x p e r i m e n t b e c a u s e t h e l o w p r e s s u r e g a s s y s t e m w h i c h w o u l d h a v e b e e n n e c e s s a r y f o r t h e w i r e c h a m b e r s h a d n o t t h e n b e e n d e v e l o p e d . T h e m u l t i p l e s c a t t e r i n g f r o m a i r , i n t h e s p e c t r o m e t e r , a n d w i r e c h a m b e r w i n d o w s was c a l c u l a t e d ( u s i n g t h e p r o g r a m REVMOC ( K i t 7 1 ) ) t o c o n t r i b u t e 1 . 5 MeV t o t h e r e s o l u t i o n . F i l l i n g t h e v a c u u m c h a m b e r w i t h h e l i u m a t a t m o s p h e r i c p r e s s u r e r e d u c e s t h i s c o n t r i b u t i o n t o 7 6 0 k e V . T h e v a c u u m c h a m b e r was f i l l e d b y p u m p i n g down b o t h t h e v a c u u m c h a m b e r a n d w i r e 56 J 1 1 I I i tTp/p (V.) FIGURE 12 V a r i a t i o n i n p o s i t i o n ( X 3 ) o f p i o n s a t w i r e c h a m b e r 3 w i t h m o m e n t u m . S p / p i s t h e p e r c e n t a g e v a r i a t i o n o f t h e momentum f r o m t h e n o m i n a l v a l u e o f 128 M e V / c . The d i s p e r s i o n d X 3 / d ( 6 p / p ) , i s g i v e n b y t h e s l o p e o f t h e c u r v e a n d i s s e e n t o b e a l m o s t c o n s t a n t b e t w e e n 6 p / p = ~6% a n d 6 p / p = 4%. c h a m b e r s b e f o r e l e t t i n g h e l i u m i n t o t h e v a c u u m c h a m b e r a n d m a g i c g a s i n t o t h e w i r e c h a m b e r s . T h e r e s o l u t i o n o f t h e M13 beam l i n e was t h e m o s t s i g n i f i c a n t c o n t r i b u t i o n t o t h e o v e r a l l r e s o l u t i o n i n t h i s e x p e r i m e n t . The f o c a l p l a n e a t t h e momentum d e f i n i n g s l i t s S I a n d S2 ( F i g u r e 6 ) was i n c l i n e d a t a n a n g l e o f 81 ° t o t h e p l a n e o f t h e s l i t s . T h i s r e s u l t e d i n p a r t i c l e s , o f m o m e n t a w h i c h w e r e f o c u s s e d u p s t r e a m o r d o w n s t r e a m o f t h e s l i t s , d i v e r g i n g f r o m t h e f o c u s a n d p a s s i n g t h r o u g h t h e s l i t s . T he f i n i t e s i z e o f t h e p r o d u c t i o n t a r g e t a n d p r o t o n beam s p o t w i t h a m a g n i f i c a t i o n o f ~ 1 a t t h e d i s p e r s e d f o c u s , a n d a d i s p e r s i o n o f 1 . 2 6 c m / % ( 6 p / p ) , w o u l d h a v e g i v e n a n i n h e r e n t r e s o l u t i o n o f a b o u t . 5 % ( 6 p / p ) f o r a 1 cm t h i c k p r o d u c t i o n t a r g e t v i e w e d a t 4 5 ° , h a d t h e d i s p e r s i o n b e e n i n t h e p l a n e o f t h e s l i t s . T h e r e s o l u t i o n w h i c h was o b t a i n e d was 1 . 6 MeV FWHM i n t h e s u l f u r e x p e r i m e n t a n d 1 . 5 MeV i n t h e m a g n e s i u m e x p e r i m e n t , c o r r e s p o n d i n g t o 2% 6 p / p . 3 . 3 . 2 S c a t t e r i n g chamber T h e t a r g e t s w e r e c o n t a i n e d i n a n 8 " i n t e r n a l d i a m e t e r s c a t t e r i n g c h a m b e r w i t h a 127 um t h i c k K a p t o n w i n d o w . T h e w i n d o w a l l o w e d t h e s p e c t r o m e t e r t o be r o t a t e d b e t w e e n 4 5 ° a n d 135 ° w i t h o u t a n y o f t h e a l u m i n u m s t r u c t u r e c o m i n g w i t h i n 4 cm o f t h e beam c e n t r e . A 13 um t h i c k m y l a r w i n d o w b e t w e e n t h e s c a t t e r i n g c h a m b e r a n d t h e s p e c t r o m e t e r a l l o w e d t h e s c a t t e r i n g c h a m b e r t o be o p e n e d w i t h o u t l o s i n g h e l i u m f r o m t h e v a c u u m b o x . When t h e s p e c t r o m e t e r was e v a c u a t e d , t h e t a r g e t s w e r e p u l l e d o u t o f t h e s c a t t e r i n g c h a m b e r , w h i c h wa s t h e n s e a l e d w i t h a g a t e v a l v e a n d a l s o pumped down a n d f i l l e d w i t h h e l i u m . 3.3.3 Target ladder T h e t a r g e t s w e r e m o u n t e d o n a r e m o t e l y c o n t r o l l e d t a r g e t l a d d e r , a l l o w i n g t h e t a r g e t s t o be c h a n g e d q u i c k l y f r o m t h e e x p e r i m e n t c o n t r o l r o o m . T h e t o t a l d i s t a n c e o f t r a v e l , 16 c m , a l l o w e d u p t o f i v e t a r g e t s , e a c h 4 cm i n h e i g h t , t o be u s e d . T h e t a r g e t l a d d e r was d r i v e n b y a c h a i n , p o w e r e d b y a s y n c h r o n o u s m o t o r w i t h a c l u t c h a n d b r a k e . The l a d d e r i s d e s c r i b e d f u r t h e r i n A p p e n d i x 6 . 3.3.4 Wire chambers The w i r e c h a m b e r s w e r e a l l o f p r i n t e d c i r c u i t d e l a y l i n e t y p e . B o t h X a n d Y p l a n e s w e r e g r i d s o f c a t h o d e w i r e s a t g r o u n d p o t e n t i a l . T h e w i r e s p a c i n g s w e r e 1 mm i n t h e X d i r e c t i o n a n d 2 mm i n t h e Y d i r e c t i o n . The l a r g e e x i t w i r e c h a m b e r s w e r e d i v i d e d i n t h e X ( d i s p e r s i v e ) d i r e c t i o n i n t o t h r e e r e g i o n s , e a c h w i t h a s e p a r a t e d e l a y l i n e , t o r e d u c e a t t e n u a t i o n a n d p u l s e b r o a d e n i n g down t h e d e l a y l i n e . T h e e n d s o f e a c h d e l a y l i n e w e r e c o n n e c t e d t o f a s t ( 5 n s r i s e t i m e ) a m p l i f i e r s w i t h a g a i n o f 1 0 0 , g i v i n g o u t p u t p u l s e s o f a b o u t 3 0 0 mV . The g a s m i x t u r e u s e d i n t h e e x i t w i r e c h a m b e r s was t h e same a r g o n , f r e o n , i s o b u t a n e , m e t h y l a l m i x t u r e u s e d i n t h e p r e v i o u s e x p e r i m e n t . I n t h e e n t r y w i r e c h a m b e r s h o w e v e r a m i x t u r e o f 40% m e t h a n e , 25% i s o b u t a n e , 3 4 . 8 % a r g o n a n d 0 .2% f r e o n was u s e d t o r e d u c e t h e d e n s i t y a n d t h e a v e r a g e a t o m i c n u m b e r , a n d t h e r e f o r e t h e m u l t i p l e s c a t t e r i n g . 3.3.5 Beam monitors The beam f l u x was monitored w i t h s c i n t i l l a t o r s BI, B2, RRM, u. 1 and u2 (see Figure 6 ) . BI and B2 were i n coincidence and d i r e c t l y counted the p a r t i c l e s i n the beam. BI, upstream of the t a r g e t , was a l s o i n the spectrometer event d e f i n i n g coincidence. The Rate Reduction Monitor (RRM) was a la r g e (15 cm) counter comprised of a matrix of 1 cm diameter s c i n t i l l a t o r d i s c s placed 2 cm apart i n a p l e x i g l a s l i g h t g u i d e . This counter was downstream of the target where the beamspot was la r g e and so sampled the beam, reducing the counting r a t e . This i s of use at p a r t i c l e f l u x e s beyond the d i r e c t counting a b i l i t y of the s c i n t i l l a t o r s . u l and LI2 were i n coincidence and counted the muons from pion decays i n f l i g h t (Wad76). They were arranged to count muons coming through a reducing flange on the beam l i n e at an angle of 7°, where the Jacobian, dQ /dQ1 , i s f l a t ; t h i s ensures that small changes i n beam p o s i t i o n or cm lab extension do not a f f e c t the counting r a t e . The peak i n the Jacobian occurs at 18° at 50 MeV. The count rates of a l l the monitors scaled to w i t h i n 1% up to 8 x l 0 5 p a r t i c l e s / s , the highest f l u x encountered. A p l o t of ul»p,2 count ra t e v s . B1»B2 count ra t e i s shown i n Figure 13. The f l u x was v a r i e d by varying the width of the s l i t s , S2 i n the beam l i n e . One of the points shows the small e f f e c t of o f f s e t t i n g the s l i t s by 26 mm from the beam l i n e c e n t r e . This i s a s i g n i f i c a n t o f f s e t c o n s i d e r i n g that the s l i t s were only 18 mm wide during the experiment. These data were taken w i t h a i t + beam. The pion f l u x during the if experiments was about 4 x l 0 5 / s . J 1 I I I L FIGURE 13 Comparison of count rates of beam monitors Bl»B2 and yl«u2. The point circled shows the effect of offsetting the s l i t s by 26 mm from the beamline centre. 3.3.6 Electronic logic T h e l o g i c c i r c u i t , F i g u r e 1 4 , i s s i m i l a r t o t h a t o f t h e p r e v i o u s e x p e r i m e n t ( F i g u r e 9 ) . T h e s p e c t r o m e t e r s c i n t i l l a t o r p u l s e s o f E l a n d E 2 w e r e f e d i n t o ADCs t o h e l p i n p a r t i c l e i d e n t i f i c a t i o n . A l s o t h e l e f t a n d r i g h t s i g n a l s o f e a c h o f t h e s e s c i n t i l l a t o r s w e r e a d d e d i n l i n e a r f a n - i n m o d u l e s a n d t h e o u t p u t s o f t h e s e d i s c r i m i n a t e d o n p u l s e h e i g h t b e f o r e i n c l u s i o n i n t h e e v e n t c o i n c i d e n c e . T h i s summed s i g n a l was a l s o r e c o r d e d b y a n A D C . 3.3.7 Background A t some s p e c t r o m e t e r a n g l e s t h e r e was a s m a l l a m o u n t o f b a c k g r o u n d w h i c h d i d n o t s e em t o be a s s o c i a t e d w i t h s c a t t e r i n g f r o m t h e t a r g e t a r e a . I n t h e w o r s t c a s e t h i s p r o d u c e d e v e n t s a t a r a t e o f 20% o f t h a t o f t h e t r u e p i o n e v e n t s f r o m t h e t a r g e t b e f o r e r e j e c t i o n b y E2 p u l s e h e i g h t . The s o u r c e o f t h i s b a c k g r o u n d , w h i c h g a v e s m a l l o u t p u t p u l s e s f r o m t h e d e t e c t o r s E l a n d E 2 , wa s n o t d e t e r m i n e d . I t was f o u n d t o be l a r g e r o n o n e s i d e o f t h e beam l i n e t h a n t h e o t h e r . I t g a v e no p r o b l e m i n t h e f i n a l a n a l y s i s a s i t d i d n o t p r o d u c e s i g n a l s i n t h e w i r e c h a m b e r s . P u l s e h e i g h t d i s c r i m i n a t i o n o f t h e a d d e d l e f t + r i g h t s i g n a l s f r o m t h e d e t e c t o r E 2 wa s u s e d t o r e j e c t t h i s b a c k g r o u n d o n l i n e . T h e s p e c t r u m f r o m t h i s ADC wa s s t u d i e d w h i l e m a k i n g s o f t w a r e c u t s t o t h e d a t a o n l i n e . I t was f o u n d t h a t much b a c k g r o u n d c o u l d be r e m o v e d , w i t h o u t l o s i n g a n y o f t h e p i o n s , b y a d j u s t i n g t h e t h r e s h o l d o f t h e d i s c r i m i n a t o r o f t h i s summed s i g n a l . El left E l r igh t -L>i E2 l e E2 r igh t E3 B l • B2 • Ml U2 H i r e chamber outputs EVENT STROBE I EVENT STROBE Spectrometer event d e f i n i t i o n Bean Manual s t a r t / s t o p Latch D — TDC s t a r t a ADC gate __r~x I 1 o D 0 OR AND V To TDC atop O • D t s c r l a l n a t o r CF • Constant f r a c t i o n Cate generater Hean t l a e r Scalers (caaac * v i s u a l ) CAMAC output r e g i s t e r (computer busy) V i s u a l sca le r 3.3.8 Targets The s u l f u r t a r g e t s were made of s u l f u r powder compressed i n t o 2 mm t h i c k s t a i n l e s s s t e e l frames. The s u l f u r was compressed to about 200 kg/cm 2 pressure and became s e l f supporting. 12.5 ^ m t h i c k mylar windows were glued, under t e n s i o n , over the t a r g e t s . Although care was taken to add the powder evenly before compression, some i r r e g u l a r i t i e s i n d e n s i t y were i n e v i t a b l e . The v a r i a t i o n of d e n s i t y over the area of the targ e t s was, t h e r e f o r e , measured, as discussed i n Appendix 2. The average d e n s i t i e s of the ta r g e t s were 0.290 g/cm2 f o r the 3l*S target and 0.365 g/cm2 f o r the 3 2 S t a r g e t . The dimensions of the targ e t s and frames are shown i n Figure 15. S t e e l was chosen f o r the frames because the i r o n cross s e c t i o n f a l l s more r a p i d l y than the s u l f u r cross s e c t i o n at the l a r g e r angles s t u d i e d . The background from s c a t t e r i n g o f f the frame was reduced, t h e r e f o r e , at angles where the s u l f u r cross s e c t i o n was low and much of the frame was i n the beam because of the angle of the target to the beam d i r e c t i o n . The i s o t o p i c composition of the 3kS target was: 92.29% 3 4 S , 3.78% 3 3 S , 2.46% 3 2 S , 0.99% 3 6 S and 0.48% 1 2 C . The 3 2 S target was n a t u r a l s u l f u r of composition: 95.02% 3 2 S , 0.75% 3 3 S , 4.21% 3hS and 0.017% 3 6 S . The magnesium tar g e t s were of r o l l e d metal of uniform t h i c k n e s s . The d e n s i t i e s were ther e f o r e uniform and were given by the mass and the area. The t a r g e t s were r e c t a n g u l a r , each being 43 x 25.6 mm. The d e n s i t i e s were .300 g/cm2 f o r the 2 6Mg target and 0.296 g/cm2 f o r the 2 4Mg ta r g e t . The 2 6Mg tar g e t was 99.5% i s o t o p i c a l l y pure. The 2**Mg target was n a t u r a l magnesium w i t h composition: 79% 2 1 +Mg, 10% 2 5Mg and 11% 2 6Mg. FIGURE 15 Density p l o t of scattered pions from the empty s t e e l target frame used i n 3**S/ 3 2S experiment. The p o s i t i o n s at the target are c a l c u l a t e d from the measured p o s i t i o n s of the pions at wire chambers 1 and 2 . Superimposed i s a drawing of the target frame, to s c a l e . CHAPTER IV A n a l y s i s 4 .1 I n t r o d u c t i o n The d a t a w e r e a n a l y s e d o n t h e VAX 1 1 / 7 8 0 a t T R I I M F . H i s t o g r a m m i n g o f t h e d a t a was p e r f o r m e d w i t h t h e p r o g r a m M O L L I ( B e n 8 2 ) a n d t h e s u b s e q u e n t h i s t o g r a m s s t o r e d o n m a g n e t i c t a p e ; t h e s e h i s t o g r a m s w e r e t h e n d i s p l a y e d a n d a n a l y s e d w i t h t h e p r o g r a m R E P L A Y ( B e n 8 2 ) . A n a l y s i s o f t h e QD a n d QQD d a t a d i f f e r e d i n a n u m b e r o f r e s p e c t s a n d w i l l b e d i s c u s s e d s e p a r a t e l y . T h e t e r m ' c u t s ' i n t h i s d i s c u s s i o n i s u s e d t o s i g n i f y t h e r e j e c t i o n o f d a t a w h i c h h a v e m e a s u r e d p a r a m e t e r s o u t s i d e o f c e r t a i n l i m i t s ( t h e p o s i t i o n s o f t h e c u t s ) . 4 . 2 QD E x p e r i m e n t The c o o r d i n a t e s u s e d i n t h i s d i s c u s s i o n a r e d e f i n e d i n F i g u r e 8 . 4 . 2 . 1 C u t s The o p t i m u m p o s i t i o n s o f t h e c u t s w e r e f o u n d b y a n a l y s i n g t h e d a t a t a k e n a t a s m a l l a n g l e ( 3 0 ° ) w h e r e a l m o s t a l l o f t h e e v e n t s w e r e e l a s t i c a l l y s c a t t e r e d p i o n s . T h e p o s i t i o n s o f t h e p e a k s o n w h i c h t h e v a r i o u s c u t s w e r e made c o u l d t h e n be m o n i t o r e d f r o m r u n t o r u n b y h i s t o g r a m m i n g t h e r e l e v a n t c o o r d i n a t e w i t h a l l o t h e r c u t s a p p l i e d . C u t s w e r e made t o t h e d a t a t o r e m o v e e v e n t s w h i c h d i d n o t a r i s e f r o m p i o n s s c a t t e r e d f r o m t h e t a r g e t . T h i s i n v o l v e d r e m o v i n g e v e n t s w h e r e a p i o n s c a t t e r e d f r o m s o m e t h i n g o t h e r t h a n t h e t a r g e t , a n d a l s o e v e n t s w h e r e t h e s c a t t e r e d p a r t i c l e was n o t a p i o n . S i n c e t h e momentum o f t h e d e t e c t e d p a r t i c l e wa s m e a s u r e d b y t h e s p e c t r o m e t e r , t h e p a r t i c l e ' s v e l o c i t y t h e n f i x e d t h e m a s s . C u t s w e r e made o n t h e t i m e o f f l i g h t down t h e beam l i n e ( F i g u r e 1 0 ) a n d o n t h e f l i g h t t i m e b e t w e e n t h e beam m o n i t o r , B I , a n d t h e c o u n t e r , E l , a t t h e s p e c t r o m e t e r e x i t . T h e s e c u t s r e m o v e d muon s t h a t s c a t t e r e d f r o m t h e t a r g e t a n d some r a n d o m b a c k g r o u n d . A p r o j e c t i o n o f t h e p a r t i c l e t r a j e c t o r y b a c k t o t h e t a r g e t was made a n d a c u t t h e n made o n t h e r a d i a l d i s t a n c e o f t h i s p r o j e c t e d p o s i t i o n f r o m t h e t a r g e t c e n t r e . T h i s p r o j e c t i o n was made u s i n g : XT = f ( X l , 0 , . , X F P ) o u t a n d YT = Y I + g ( * o u t ) w h e r e t h e p o l y n o m i a l s f a n d g w e r e f o u n d u s i n g t h e m e t h o d d i s c u s s e d i n A p p e n d i x 1 . T h e t a r g e t c e n t r e was d e f i n e d a s t h e c e n t r e o f t h e p r o j e c t e d d i s t r i b u t i o n s i n b o t h h o r i z o n t a l a n d v e r t i c a l d i r e c t i o n s . C u t s a t d i f f e r e n t r a d i i w e r e t r i e d a n d a r a d i u s o f 8 mm was f o u n d t o g i v e t h e s m a l l e s t s t a t i s t i c a l e r r o r f o r t h e c r o s s s e c t i o n r a t i o s . O t h e r c u t s w e r e made t o r e m o v e a n y p i o n t h a t h a d s c a t t e r e d f r o m m a t e r i a l i n t h e s p e c t r o m e t e r , o r h a d d e c a y e d t o a muon i n s i d e t h e s p e c t r o m e t e r . A s t r a i g h t l i n e p r o j e c t i o n o f t h e p a r t i c l e t r a j e c t o r y b a c k t o t h e e n t r a n c e o f t h e d i p o l e i n t h e Y , n o n - d i s p e r s i v e , d i r e c t i o n was made u s i n g a p a i r o f - e x i t w i r e c h a m b e r s . E v e n t s w h i c h a p p e a r e d o u t s i d e o f t h e p e a k i n t h i s d i s t r i b u t i o n w e r e r e j e c t e d ; t h e s e c o n s t i t u t e d a b o u t 5% o f t h e e v e n t s w h i c h p a s s e d t h e o t h e r c u t s . T h e s e e v e n t s w e r e p i o n s t h a t e i t h e r h i t t h e p o l e f a c e o f t h e d i p o l e m a g n e t o r d e c a y e d i n f l i g h t t o m u o n s . A 50 MeV p i o n d e c a y i n g i n t o a muon w i l l a l t e r i t s f l i g h t p a t h b y a max imum o f 1 8 ° . T h i s c h a n g e i n t r a j e c t o r y may be i n a n y d i r e c t i o n , b u t w i l l h a v e t h e g r e a t e s t e f f e c t o n t h e c a l c u l a t e d momentum w h e n i t i s i n t h e same d i r e c t i o n a s t h e d i s p e r s i o n . A p r o j e c t i o n was made t o t h e f o c a l p l a n e u s i n g a p a i r o f e x i t w i r e c h a m b e r s , o r a s t r a i g h t l i n e l e a s t s q u a r e s f i t t o a l l t h r e e c h a m b e r s i f t h e y a l l g a v e s i g n a l s . T h e x 2 o f t h i s f i t was h i s t o g r a m m e d , s h o w i n g a p e a k , a s w o u l d b e e x p e c t e d , a n d a s m a l l f l a t b a c k g r o u n d o u t s i d e o f t h e p e a k . T h i s b a c k g r o u n d was p a r t l y d u e t o p i o n s t h a t d e c a y e d i n t o muons a f t e r t h e f i r s t e x i t w i r e c h a m b e r t h u s b r e a k i n g t h e s t r a i g h t l i n e t r a j e c t o r y . F o r e v e n t s o u t s i d e o f t h i s x 2 p e a k t h e t r a j e c t o r y was r e c a l c u l a t e d f r o m t h e f i r s t two w i r e c h a m b e r s . H a l f o f t h e e v e n t s w o u l d h a v e d e c a y e d b e t w e e n t h e s e c o n d two e x i t w i r e c h a m b e r s , t h e r e f o r e t h i s p r o j e c t i o n w o u l d now be c o r r e c t f o r t h e s e e v e n t s . P i o n s d e c a y i n g w i t h i n 7 . 5 cm o f e x i t w i r e c h a m b e r 2 w o u l d n o t p r o d u c e a n e r r o r o f g r e a t e r t h a n 10 mm i n t h e p r o j e c t i o n t o t h e f o c a l p l a n e . T h i s l e a v e s 30 cm i n w h i c h t h e p i o n c a n d e c a y a n d p r o d u c e a n e r r o r g r e a t e r t h a n 10 mm i n t h e p r o j e c t i o n t o t h e f o c a l p l a n e . The f r a c t i o n o f 5 0 MeV p i o n s d e c a y i n g i n 30 cm i s 4%; some o f t h e s e p i o n s w i l l c r o s s t h e f o c a l p l a n e o u t s i d e o f t h e g r o u n d s t a t e p e a k . T h i s f r a c t i o n i s c o n s t a n t a n d c o n t r i b u t e s t o t h e e f f i c i e n c y o f t h e s p e c t r o m e t e r a n d t h e l i n e s h a p e o f t h e e n e r g y s p e c t r u m . P i o n s t h a t d e c a y e d b e f o r e t h e f i r s t e x i t w i r e c h a m b e r c o u l d o n l y b e r e j e c t e d o n t h e b a s i s o f t h e p r o j e c t i o n b a c k t o t h e p o l e f a c e s s i n c e t h e t r a j e c t o r y o f t h e m u o n , t h r o u g h t h e e x i t w i r e c h a m b e r s , w o u l d f o r m a s t r a i g h t l i n e . T h i s d i d n o t r e j e c t p i o n s t h a t h a d d e c a y e d i n t o t h e d i s p e r s i v e d i r e c t i o n . H o w e v e r , t h e a c c e p t a n c e o f t h e s p e c t r o m e t e r w a s s m a l l a n d p i o n s f r o m t h e t a r g e t h a d a l i m i t e d r a n g e o f a n g l e s a t t h e e x i t o f t h e s p e c t r o m e t e r . T h i s d i s t r i b u t i o n o f a n g l e s e x t e n d e d f r o m - 1 1 ° t o +9° and some pions which decay produce muons at angles outside of th i s range. Cuts were therefore made on this e x i t angle. These cuts to remove pions which had decayed removed events that originated with pions scattering from the target and so reduced the o v e r a l l e f f i c i e n c y of the spectrometer. Since the cuts were made i d e n t i c a l l y with a l l three targets t h i s does not e f f e c t the cross section r a t i o s . 4 . 2 . 2 W i r e chamber e f f i c i e n c i e s The e f f i c i e n c y of a wire chamber i s defined as the p r o b a b i l i t y of the chamber f i r i n g when a pion passes through i t . The wire chamber e f f i c i e n c i e s were calculated for each run by comparing the number of events where a l l four wire chambers gave signals to that where one p a r t i c u l a r chamber only did not give a s i g n a l . For instance, the e f f i c i e n c y of wire chamber 2 i s given by the r a t i o of numbers of coincidences N(l»2»3»4)/N(l»3»4 ); where N(a»b) i s the number of events with both chambers a and b giving s i g n a l s . By counting only the events with a l l three other wire chambers giving signals, the background i n which p a r t i c l e s did not pass through a l l four wire chambers was removed. Also, only those events were counted which s a t i s f i e d a l l other cuts on the data and passed through the f o c a l plane within 4 cm of the ground state peak. These conditions were made to ensure that i t was the e f f i c i e n c y for pions only that was measured. The t o t a l p r o b a b i l i t y of the entry wire chamber and at least two ex i t wire chambers f i r i n g was then calculated since t h i s was a condition for accepting the event. Let P((a»b)+(a»c)) be the p r o b a b i l i t y o f b o t h c h a m b e r s a a n d b o r b o t h a a n d c f i r i n g ; we t h e n h a v e p ( ( 2 0 ) + ( 2 . 4 ) + ( 3 . 4 ) + ( 2 . 3 . 4 ) ) = 1 - P ( 2 ) . P ( 3 ) » P ( 4 ) ; w h e r e P ( a ) i s t h e p r o b a b i l i t y o f c h a m b e r a n o t f i r i n g . S i n c e t h e e x i t c h a m b e r s w e r e a l w a y s o v e r 90% e f f i c i e n t t h e p r o b a b i l i t y o f no p a i r o f e x i t w i r e c h a m b e r s f i r i n g was l e s s t h a n 10~~3 . I t was c h e c k e d t h a t t h i s e f f i c i e n c y d i d n o t v a r y d u r i n g e a c h s e t o f r u n s u s e d i n t h e c a l c u l a t i o n o f t h e c r o s s s e c t i o n r a t i o s ; i . e . t h a t t h e e r r o r s a n d e f f i c i e n c i e s w e r e c o n s i s t e n t w i t h a c o n s t a n t v a l u e . T h i s was a l w a y s t h e c a s e f o r t h e r u n s u s e d . 4 . 2 . 3 C a l c u l a t i n g t h e c r o s s s e c t i o n r a t i o s T h e momentum was c a l c u l a t e d f r o m t h e p o s i t i o n o n t h e f o c a l p l a n e ; c o r r e c t i n g f o r m a g n i f i c a t i o n o f t h e t a r g e t a n d s m a l l a b e r r a t i o n s i n t h e f o c u s s i n g . T h e s e c o r r e c t i o n s a r e d i s c u s s e d i n A p p e n d i x 1 . F r o m t h e momen tum, p , t h e p i o n k i n e t i c e n e r g y , T , was c a l c u l a t e d u s i n g it T = / p . p - m it + m<- it it w h e r e m^ i s t h e r e s t m a s s o f t h e p i o n . T h e numbe r o f c o u n t s i n t h e g r o u n d s t a t e p e a k o f t h e e n e r g y s p e c t r u m w e r e c o u n t e d f o r e a c h r u n . T h e e x c i t e d s t a t e s w e r e o n l y w e a k l y e x c i t e d a t t h e a n g l e s s t u d i e d a n d t h e r e s o l u t i o n wa s s u f f i c i e n t t o s e p a r a t e t h e 2 . 2 3 MeV s t a t e o f 3 2 S . T h i s s t a t e i s o n l y 4% o f t h e g r o u n d s t a t e c r o s s s e c t i o n a t 9 0 ° ( a s s e e n i n t h e QQD e x p e r i m e n t ) , s o t h a t a s m a l l f r a c t i o n o f t h i s p e a k b e i n g i n c l u d e d i n t h e i n t e g r a t i o n o f t h e g r o u n d s t a t e p e a k w o u l d n o t b e s i g n i f i c a n t . The r u n s w e r e d i v i d e d i n t o s e t s o f . t h r e e o r m o r e c o n s e c u t i v e r u n s , i n c l u d i n g a t l e a s t o n e r u n w i t h e a c h o f t h e t h r e e t a r g e t s ; 3 6 S , 3 2 S a n d e m p t y (MT ) t a r g e t . T h e r a t i o o f c r o s s s e c t i o n s , R, wa s c a l c u l a t e d f o r e a c h o f t h e s e s e t s a n d t h e w e i g h t e d mean o f t h e r e s u l t s c a l c u l a t e d f o r e a c h a n g l e . T h e c r o s s s e c t i o n r a t i o s w e r e c a l c u l a t e d f r o m t h e n u m b e r o f 36 32 MT c o u n t s , ' ' C , i n t h e g r o u n d s t a t e p e a k s a n d t h e n u m b e r s o f ___ 36 3 2 1*17 B 1 « B 2 « V c o u n t s , ' ' F , f o r t h e r e s p e c t i v e t a r g e t s u s i n g : E T , ( 3 6 C / 3 6 F -( 3 2 C / 3 2 F _ M T C / M T F ) w h e r e T i s t h e r a t i o o f t h e n u m b e r o f n u c l e i p e r u n i t a r e a i n t h e 3 2 S t a r g e t t o t h a t i n t h e 3 6 S t a r g e t . T h e i s o t o p i c c o m p o s i t i o n o f t h e t a r g e t ( s e e S e c t i o n 3 . 2 . 6 ) was t a k e n i n t o a c c o u n t when c a l c u l a t i n g t h e n u m b e r s o f n u c l e i / c m 2 . T he n u m b e r o f n u c l e i / c m 2 i n t h e 3 6 S t a r g e t i s 0 . 1 9 3 x N A / ( 0 . 8 1 1 x 3 5 . 9 7 + 0 . 1 8 8 x 3 3 . 9 8 + 0 . 0 0 1 3 x 3 1 . 9 7 ) = 0 . 0 0 5 4 2 x N A a n d i n t h e 3 2 S t a r g e t i s 0 . 1 9 2 x N A / ( 0 . 9 5 0 2 x 3 1 . 9 7 + 0 . 0 0 7 5 x 3 2 . 9 7 + 0 . 0 4 2 1 x 3 3 . 9 8 + 0 . 0 0 0 2 x 3 5 . 9 7 ) = 0 . 0 0 5 9 9 x N A . W h e r e N A i s A v a g a d r o s n u m b e r = 6 . 0 2 2 x 1 0 2 3 m o l e - 1 . T h e r e f o r e T = 1 . 1 0 4 The r e s u l t s f o r t h e c r o s s s e c t i o n r a t i o s a r e g i v e n i n T a b l e I I . TABLE II Cross Section Ratios of 3 6 S/ 3 2 S 0 cm d e g r e e s R ±6R ( m b / s r ) R ± 6R c c ( m b / s r ) 2 0 . 1 1 . 0 1 8 0 . 0 2 3 1 . 0 2 2 0 . 0 2 9 3 0 . 2 1 . 0 4 7 0 . 0 3 4 1 . 0 5 8 0 . 0 4 2 4 5 . 2 1 . 0 0 9 0 . 0 8 7 1 . 0 1 8 0 . 1 0 7 5 5 . 3 1 . 1 8 6 0 . 1 5 8 1 . 2 2 1 0 . 1 9 5 7 0 . 3 1 . 3 7 9 0 . 0 9 2 1 . 4 4 0 0 . 1 1 5 8 0 . 3 1 . 2 0 3 0 . 0 7 5 1 . 2 3 1 0 . 0 9 3 9 0 . 4 0 . 9 7 0 0 . 0 6 7 0 . 9 4 5 0 . 0 8 2 1 0 0 . 3 0 . 6 5 0 0 . 2 5 0 0 . 5 8 6 0 . 3 0 7 R i s c o r r e c t e d f o r i s o t o p i c c o m p o s i t i o n o f t h e t a r g e t s . 4 . 3 QQD E x p e r i m e n t The c o o r d i n a t e s u s e d i n t h i s d i s c u s s i o n a r e d e f i n e d i n F i g u r e 1 1 . 4 . 3 . 1 C u t s C u t s w e r e made o n t h e t i m e o f f l i g h t o f t h e p i o n down t h e beam l i n e ; t h i s r e m o v e d some r a n d o m b a c k g r o u n d a s w e l l a s a f e w muon s a n d e l e c t r o n s t h a t s c a t t e r e d f r o m t h e t a r g e t . C u t s w e r e made o n t h e p r o j e c t e d p o s i t i o n o f t h e p i o n a t t h e t a r g e t . T h e t r a n s f o r m a t i o n s o f t h e two e n t r y w i r e c h a m b e r c o o r d i n a t e s t o t h e c o o r d i n a t e s a t t h e t a r g e t w e r e g i v e n b y t h e p r o g r a m TRANSPORT ( B r o 8 0 ) . T h i s t r a n s f o r m a t i o n was f o u n d t o g i v e 2 mm r e s o l u t i o n b y c o m p a r i n g t h e p r o j e c t e d p o s i t i o n t o t h e p o s i t i o n g i v e n b y a w i r e c h a m b e r a t t h e t a r g e t . The s p e c t r o m e t e r was a t 0 ° f o r t h i s m e a s u r e m e n t . B e c a u s e t h e s l i t s a n d j a w s o f t h e M13 beam l i n e w e r e n e c e s s a r i l y v e r y n a r r o w , t o r e d u c e t h e f l u x t h r o u g h t h e w i r e c h a m b e r s , t h i s d a t a c o v e r e d a s m a l l e r p h a s e s p a c e t h a n t h e t r u e s c a t t e r i n g d a t a . H o w e v e r , i t c a n be s e e n f r o m t h e p r o j e c t e d d i s t r i b u t i o n o f p a r t i c l e s s c a t t e r e d a t 5 0 ° f r o m t h e e m p t y s t e e l t a r g e t f r a m e i n F i g u r e 1 5 , w h e r e t h e a c t u a l t a r g e t d i m e n s i o n s a r e s u p e r i m p o s e d , t h a t a r e a s o n a b l e p r o j e c t i o n was m a d e , a p a r t f r o m a s m a l l s c a l e f a c t o r i n t h e X d i r e c t i o n . L a c k o f p r e c i s i o n i n t h i s p r o j e c t i o n c o u l d a f f e c t t h e a m o u n t o f s c a t t e r i n g f r o m t h e t a r g e t f r a m e t h a t w o u l d be i n c l u d e d i n t h e c u t s . S i n c e t h e same c u t s w e r e made o n t h e e m p t y t a r g e t r u n s , h o w e v e r , s c a t t e r i n g f r o m t h e f r a m e w o u l d be s u b t r a c t e d w i t h t h e r e s t o f t h e b a c k g r o u n d . T h e c u t s w e r e made t o a c c e p t p i o n s w h i c h came f r o m t h e t a r g e t a t l e a s t 2 mm away f r o m t h e s t e e l f r a m e ; t h e y w e r e t h e n c h a n g e d a t e a c h a n g l e a s t h e a n g l e b e t w e e n t h e s p e c t r o m e t e r a n d t a r g e t was c h a n g e d . A s t r a i g h t l i n e p r o j e c t i o n o f t h e p a r t i c l e t r a j e c t o r y , i n t h e Y d i r e c t i o n , f r o m t h e e x i t w i r e c h a m b e r s b a c k t o t h e e n t r y o f t h e d i p o l e was m a d e , s i m i l a r t o t h a t i n t h e c a s e o f t h e QD. C u t s w e r e made t o r e m o v e p a r t i c l e s w h i c h h a d s c a t t e r e d f r o m t h e p o l e f a c e s . H a v i n g two w i r e c h a m b e r s i n f r o n t o f t h e d i p o l e a n d two a f t e r t h e d i p o l e , i t was p o s s i b l e t o c h e c k f o r c o n s i s t e n c y b e t w e e n t h e t r a j e c t o r i e s b e f o r e a n d a f t e r t h e d i p o l e . T o do t h i s , t h e t r a j e c t o r y a t t h e e x i t o f t h e d i p o l e was c a l c u l a t e d f r o m t h a t a t t h e e n t r y a n d c o m p a r e d t o t h e m e a s u r e d t r a j e c t o r y a t t h e d i p o l e e x i t . T h e t r a n s f o r m a t i o n s 0 Q u t = ( X 4 - X 3 ) / D 3 4 = f ( X l , Y l , X 2 , Y 2 , 6 p ) a n d • o u t = ( Y A - Y 3 > / D 3 4 = g ( X l , Y l , X 2 , Y 2 , 6 p ) , w h e r e D34 i s t h e d i s t a n c e b e t w e e n w i r e c h a m b e r s 3 a n d 4 , w e r e u s e d . T h e p o l y n o m i a l s f a n d g w e r e f o u n d b y f i t t i n g t o t h e d a t a a s d e s c r i b e d i n A p p e n d i x 1 . T h e p o l y n o m i a l f i s j u s t t h e d i f f e r e n c e b e t w e e n t h e p o l y n o m i a l s g i v i n g p o s i t i o n s X 3 a n d X4 d i v i d e d b y t h e d i s t a n c e b e t w e e n t h e c h a m b e r s . T h e s e same two p o l y n o m i a l s a r e u s e d t o f i n d t h e 6 p ' s f r o m c h a m b e r s 3 a n d 4 , s o s t r i c t l y i t i s n o t m e a n i n g f u l t o u s e t h e c a l c u l a t e d v a l u e s o f 6p t o c a l c u l a t e X3 a n d X4 a g a i n . D o i n g s o w i l l , o f c o u r s e , j u s t r e s u l t i n t h e o r i g i n a l v a l u e s o f X 3 a n d X4 u s e d t o c a l c u l a t e 6 p ; t h e r e f o r e , t h e mean o f t h e two v a l u e s o f 6 p c a l c u l a t e d f r o m t h e two w i r e c h a m b e r s was u s e d . T h i s i s e q u i v a l e n t t o n e g l e c t i n g t h e 6 p d e p e n d e n c e o f t h e a n g l e 9 o u t . T h i s i s a r e a s o n a b l e a p p r o x i m a t i o n f o r t h e r a n g e 74 o f 6 p u s e d a s a c h a n g e i n 6 p o f 1% w i l l o n l y c h a n g e t h e t r a j e c t o r y b y a p p r o x i m a t e l y 9 / 8 7 0 r a d i a n s ( m o v e m e n t a t w i r e c h a m b e r 4 d i v i d e d b y t h e d i s t a n c e b e t w e e n w i r e c h a m b e r 4 a n d t h e d i p o l e c e n t r e ) o r 0 . 6 d e g r e e s . C o n s i s t e n c y was c h e c k e d b y s u b t r a c t i n g t h e c a l c u l a t e d a n g l e <|> o r 8 Q u t f r o m t h e a n g l e m e a s u r e d w i t h c h a m b e r s 3 a n d 4. A l s o t h e p o l a r a n g l e , co, b e t w e e n t h e c a l c u l a t e d t r a j e c t o r y a f t e r t h e d i p o l e a n d t h e m e a s u r e d t r a j e c t o r y wa s c a l c u l a t e d ( F i g u r e 1 6 ) . N o t i c e t h a t t h i s d i s t r i b u t i o n d o e s n o t p e a k a t 0 ° . T h i s i s b e c a u s e t h e s o l i d a n g l e p e r a n g l e i n c r e m e n t d e c r e a s e s a s t h e a n g l e i s d e c r e a s e d . Q = 2TC ( 1 -COS(U)) dQ/dco = 2TC S i n ( c o ) D i v i d i n g t h e n u m b e r o f c o u n t s p e r b i n i n F i g u r e 16 b y t h e a n g l e , co, r e s u l t s i n t h e d i s t r i b u t i o n a l s o s h o w n i n F i g u r e 1 6 . A c u t was made i n t h i s p o l a r a n g l e t o a c c e p t o n l y e v e n t s w h e r e t h e a n g l e wa s l e s s t h a n 3 . 5 ° . T h i s c u t wa s e f f e c t i v e i n r e m o v i n g e v e n t s w h e r e t h e p i o n d e c a y e d i n t h e s p e c t r o m e t e r , s i n c e f o r t h e s e e v e n t s t h e t r a j e c t o r y w o u l d c h a n g e b y a n a n g l e c l o s e t o 1 8 ° . P i o n s h i t t i n g a w i r e i n a w i r e c h a m b e r ( c a l c u l a t e d t o be 20% o f t h e p i o n s ) a n d s c a t t e r i n g a t a n a n g l e g r e a t e r t h a n 3 . 5 ° w o u l d a l s o be r e m o v e d . M u l t i p l e s c a t t e r i n g i n t h e w i r e c h a m b e r w i n d o w s a n d g a s i n s i d e t h e s p e c t r o m e t e r c o u l d a l s o c h a n g e t h e d i r e c t i o n o f t h e p i o n t r a j e c t o r y b y m o r e t h a n 3 . 5 ° . T h e mean m u l t i p l e s c a t t e r i n g a n g l e ( i g n o r i n g t h e e f f e c t s o f t h e m a g n e t i c f i e l d s ) f o r t h e a m o u n t o f m a t e r i a l b e t w e e n w i r e c h a m b e r 1 a n d w i r e c h a m b e r 4 i s c a l c u l a t e d t o be 1 . 3 ° w h i c h a g r e e s r e a s o n a b l y w i t h t h e w i d t h o f t h e p e a k i n F i g u r e 1 6 . FIGURE 16 Polar angle (u>) between the measured and calculated t r a j e c t o r i e s of the p a r t i c l e at the ex i t wire chambers of the QQD spectrometer. The dashed l i n e r e s u l t s from d i v i d i n g by co. The p o s i t i o n of the cut made i s shown. T h e c u t o n t h i s p o l a r a n g l e made t h e s h a p e o f t h e p e a k s i n t h e c a l c u l a t e d e n e r g y d i s t r i b u t i o n c l o s e r t o G a u s s i a n , b y r e m o v i n g e v e n t s f a r f r o m t h e m e a n . T h i s made t h e f i t t i n g a n d s u b t r a c t i o n o f t h e e x c i t e d s t a t e c o n t r i b u t i o n s m o r e p r e c i s e b y r e m o v i n g a ' b a c k g r o u n d t a i l ' o f t h e g r o u n d s t a t e p e a k f r o m u n d e r t h e s m a l l e x c i t e d s t a t e p e a k s . 4 . 3 . 2 Peak f i t t i n g As t h e e n e r g y r e s o l u t i o n was n o t s u f f i c i e n t t o c o m p l e t e l y s e p a r a t e t h e g r o u n d s t a t e s f r o m t h e f i r s t e x c i t e d s t a t e s , p a r t i c u l a r l y f o r t h e m a g n e s i u m i s o t o p e s , p e a k f i t t i n g was u s e d t o i n t e g r a t e t h e c o u n t s u n d e r t h e g r o u n d s t a t e s . T h i s i s d i s c u s s e d i n A p p e n d i x 4 . Some o f t h e f i t s o b t a i n e d a r e s h o w n i n F i g u r e s 17 a n d 1 8 . 4 . 3 . 3 C r o s s s e c t i o n r a t i o s The c r o s s s e c t i o n r a t i o s , R, a t e a c h a n g l e w e r e c a l c u l a t e d u s i n g 1 . , 1 1 . MT / / T n MT R = C / ( E x F ) - C / ( E x F ) " 2 C / ( 2 E x 2 F ) - ^ C / C E x ^ F ) w h e r e 1 2 MT . ^ ±s t h e n u m b e r o f p i o n s d e t e c t e d w i t h t a r g e t 1 , 2 , e m p t y r e s p e c t i v e l y ; 1 2 MT » » F i s t h e n u m b e r o f p i o n s w h i c h p a s s e d t h r o u g h t a r g e t 1 , 2 , e m p t y ; 1 2 ' E i s t h e e f f i c i e n c y f o r a l l f o u r w i r e c h a m b e r s f i r i n g d u r i n g t h e r u n w i t h t a r g e t 1 , 2 ; E i s t h e mean o f 1 E a n d 2 E ; a n d T i s t h e r a t i o o f t h e n u m b e r o f t a r g e t n u c l e i p e r u n i t a r e a i n t a r g e t 2 t o t h a t i n t a r g e t 1 . 77 FIGURE 17 E n e r g y s p e c t r u m f r o m 3kS a t 1 00 ° w i t h f i t t e d p e a k s a t t h e g r o u n d s t a t e , a n d e x c i t e d s t a t e s a t 2 . 1 3 MeV a n d 4 . 5 M e V . FIGURE 18 Energy spectrum from 2 5Mg at 100° showing fitted ground state and 1.81 MeV state peaks. 79 OR was c a l c u l a t e d from: In the case of the magnesium i s o t o p e s , the e r r o r s of *C and 2C are complicated by the peak f i t t i n g procedure, but the f i n a l r e s u l t i s very l i t t l e d i f f e r e n t to that which would be obtained w i t h simple Poisson s t a t i s t i c s on C. This i s discussed i n Appendix 4. There were too few events i n the empty target runs to give a good measure of the wire chamber e f f i c i e n c i e s , so the mean of the e f f i c i e n c i e s measured with the two other t a r g e t s was used. Since the background and the v a r i a t i o n s i n the e f f i c i e n c i e s were s m a l l , the cross s e c t i o n r a t i o s were not s e n s i t i v e to t h i s approximation. The e f f i c i e n c y of a l l four wire chambers f i r i n g was t y p i c a l l y 95% with a s t a t i s t i c a l e r r o r i n each run of 0.5%. The numbers of atoms per u n i t area i n the tar g e t s were c a l c u l a t e d from the mass per u n i t area and the known i s o t o p i c compositions and atomic weights. In the case of the 3t*S and 3 2 S t a r g e t s , the non-uniform t a r g e t d e n s i t i e s were averaged, weighted by the beam p r o f i l e , as described i n Appendix 2. The ul»u2 monitor was used f o r the measure of the numbers of pions passing through the t a r g e t s . Since only r a t i o s are i n v o l v e d no c a l i b r a t i o n was needed at t h i s p o i n t . The r a t i o , R, was c a l c u l a t e d f o r each set of three consecutive runs w i t h each isotope and empty t a r g e t . The mean value of these r e s u l t s was then c a l c u l a t e d at each angle. 4.4 C o r r e c t i o n f o r I s o t o p i c Composition Because the t a r g e t s were not 100% i s o t o p i c a l l y pure, the cross s e c t i o n r a t i o s were not the r a t i o s of cross s e c t i o n s of i s o t o p e s . The r a t i o s were corrected f o r the i s o t o p i c composition of the t a r g e t s . For the magnesium experiment we have R ^ 0.79 2ha + 0.10 2 5 a + 0.11 2 6 a where N a i s the cross s e c t i o n f o r the isotope of mass N. Therefore, 2 6 q R(0.79 + 0.1 2 5 0 / 2 t t q  2 h a = (1 - 0.11 R) To c o r r e c t f o r the 10% 2 5Mg contamination of the n a t u r a l magnesium we must make some assumption about the r a t i o 2 5 ( j / 2 4 a . For the purpose of the f o l l o w i n g a n a l y s i s the c o r r e c t i o n was made by assuming that 2 5 a = 2 4 o + 2 6 q 2 A 20% e r r o r i n t h i s assumption however would produce only about a 2% change i n the c a l c u l a t e d cross s e c t i o n r a t i o 2 6 a / 2 4 a . The s u l f u r cross s e c t i o n r a t i o s i n v o l v e a more complex mixture of isotopes but the treatment i s s i m i l a r to the magnesium-case. For the 3 4 S , 3 2 S experiment we have _ _ 0.9229 3 V + 0.0246 32a + 0.0378 3 3 a + 0.0099 3 6 a + 0.0048 1 2 a K — .. - , ,— .9502 3 2 a + .0075 3 3 a + 0.0421 3 a + 0.0002 3 6 a where ^ 2 a i s the 1 2C cross s e c t i o n . Therefore 33 36 33 36 RfO.9502 + 0.0075 — - + 0.0002 — - ) - 0.0246 — - - 0.0099 — -3^ a 3 2 0 3 2 a 3 2 o 3 2 a 32^ ( 0.9229 - 0.0421R) 1 2 o - 0.0048 - 0.0378 + U* (0.9229 - 0.0421R) The corrections for ^ 2 a / 3 2 a and 3 3 c r / 3 2 a were made with the measured r a t i o s . Since the carbon measurement was done i n a separate experiment (with the magnesium) a 30% error was a l l o t t e d to the r a t i o 1 2 o / 3 2 o . The 3 3 0 / 3 2 o correction was made by assuming that 3 3 0 = ( 3 4 0 + 3 2o)/2 Therefore 3 6rt 36-, 12, Rf0.9450 + 0.0002 —-) - 0.0099 — - - 0.0048 — - -0.0501 34 0 32 a 32 a 32 a 32 a ( 0 . 9 3 5 2 - 0 . 0 4 5 9 R ) Corrections to the r a t i o 3 6 o / 3 2 o were made s i m i l a r l y . In the case of the su l f u r data the process was i t e r a t e d , using the corrected 3 6 o / 3 2 0 re s u l t s to correct the 3 4 0 / 3 2 0 data and vice-versa. The cross section r a t i o s before and a f t e r these corrections are given i n Tables I I , III and IV. The errors of the corrected r a t i o s include the e f f e c t s of the errors i n the cross sections of the contaminants. TABLE III Cross Section Ratios of 3 I*S/ 3 2S ®cm degrees cm degrees R ± 6R (mb/sr) R ± 6R c c (mb/sr) 39.47 39.2 0.998 0.027 1.002 0.031 49.49 48.2 0.966 0.017 0.965 0.019 59.55 59.5 1.098 0.032 1.111 0.037 69.6 69.6 1.143 0.027 1.165 0.031 79.6 79.6 1.076 0.030 1.087 0.034 84.6 84.6 1.059 0.033 1.069 0.038 89.6 89.6 1.035 0.031 1.042 0.035 96.1 96.1 0.993 0.037 0.994 0.042 101.1 100.9 0.884 0.034 0.871 0.038 0' i s co r r e c t e d f o r cross s e c t i o n v a r i a t i o n across the spectrometer cm acceptance, as described i n s e c t i o n 4.5 R i s co r r e c t e d f o r isotope composition of the t a r g e t s . TABLE IV Cross Section Ratios of 26Mg/2**Mg 0 cm 0' cm R i b 6R R i c b 6R c 38.8 38.5 0.973 0.037 0.968 0.044 48.9 48.6 0.982 0.038 0.978 0.044 59.3 59.25 1.217 0.057 1.269 0.074 69.3 69.3 1.242 0.047 1.301 0.061 79.3 79.3 1.293 0.053 1.369 0.071 89.4 89.4 1.179 0.052 1.220 0.066 96.6 96.6 1.092 0.047 1.111 0.058 101.6 101.6 1.076 0.051 1.091 0.063 106.6 106.6 1.058 0.065 1.069 0.079 0' is corrected for cross section variation across spectrometer cm acceptance, as described in section 4.5 R is corrected for isotope composition of the targets. 4 . 5 Absolute Cross Sections The absolute cross sections were normalised to the •JI+(P,P)TI;+ cross section. This cross section was measured at 48.3 MeV and 91.6° (102.8° i n the centre of mass frame) using a CH2 target. The spectrometer was tuned for the momentum of the e l a s t i c a l l y scattered pions from the hydrogen. The analysis was i d e n t i c a l to that described above. The d i f f e r e n t i a l cross sections were calculated from where: C = the number of pions scattered from the hydrogen and detected; 9^ = the angle between the normal to the target and the beam d i r e c t i o n ; F E the number of pions incident on the target = the t o t a l e f f i c i e n c y for a l l four wire chambers; E^ E the f r a c t i o n of pions accepted i n the cuts on consistency between entry and exit t r a j e c t o r i e s ; Q = the s o l i d angle of the spectrometer (0.018 sr) T E the number of hydrogen atoms/cm2 i n the target. The r e s u l t i n g laboratory cross section was The corresponding cm. cross section i s (-£•) = 0.765 ± 0.028 mb/sr. v d0'cm This was compared with the cross section calculated from phase s h i f t s r e s u l t i n g from a f i t to the "world TC+P data". The c a l c u l a t i o n and the f i t t i n g of the phase s h i f t s were performed with the program SAID (Arn82). da dQ c x cos(eT) fi x F x 1 x E I T x E = 0.739 + 0.027 mb/sr T h e r e s u l t i n g c r o s s s e c t i o n a t 1 0 2 . 8 ° was = 1 . 0 0 2 ± 0 . 0 5 m b / s r . ° v d f r cm T h e c r o s s s e c t i o n s f r o m t h i s e x p e r i m e n t m u s t be i n c r e a s e d b y a f a c t o r o f 1 . 3 1 t o a g r e e w i t h t h i s . T h i s d i s c r e p a n c y i s d u e t o p i o n s d e c a y i n g i n t h e s p e c t r o m e t e r . A s m a l l f r a c t i o n o f t h e r e s u l t i n g muons r e a c h t h e e x i t s c i n t i l l a t o r s t o c r e a t e a n e v e n t . T h e f r a c t i o n o f 35 MeV p i o n s ( s c a t t e r e d o f f h y d r o g e n ) d e c a y i n g i n t h e 2 . 8m f l i g h t p a t h b e t w e e n t h e t a r g e t a n d WC4 was e s t i m a t e d f r o m t h e v e l o c i t y a n d t h e l i f e t i m e o f t h e p i o n t o be 40%. I n a d d i t i o n , some o f t h e p a r t i c l e s c o u n t e d i n t h e beam f l u x a s p i o n s w i l l b e muons f r o m p i o n s t h a t d e c a y e d b e f o r e t h e t a r g e t . F o r a s h o r t f l i g h t p a t h t h e y w i l l b e i n d i s t i n g u i s h a b l e f r o m p i o n s b y t h e i r t i m e o f f l i g h t . T h i s r e s u l t s i n a n o v e r e s t i m a t e o f t h e p i o n f l u x a n d a c o r r e s p o n d i n g l y l o w e s t i m a t e o f t h e c r o s s s e c t i o n . T h e p i o n s s c a t t e r e d f r o m t h e h y d r o g e n h a d a n e n e r g y o f 35 M e V . T h e h i g h e r e n e r g y o f t h e p i o n s s c a t t e r e d f r o m s u l f u r a n d m a g n e s i u m m e a n s t h a t f e w e r p i o n s w i l l d e c a y i n t h e s e c a s e s a n d t h e n o r m a l i s i n g f a c t o r s h o u l d b e l e s s t h a n 1 . 3 1 . To c o r r e c t f o r t h i s k i n e m a t i c e f f e c t , t h e d i s t a n c e o v e r w h i c h 24% ( 1 - 1 / 1 . 3 1 ) o f t h e p i o n s d e c a y was c a l c u l a t e d . T h i s r e s u l t e d i n n o r m a l i s a t i o n f a c t o r s o f 1 . 2 7 f o r s u l f u r a n d 1 . 2 9 f o r m a g n e s i u m . C h a n g e s i n t h e b e a m s p o t a t t h e t a r g e t w i l l c h a n g e t h e a v e r a g e s o l i d a n g l e s u b t e n d e d a t t h e s p e c t r o m e t e r , s i n c e t h e s o l i d a n g l e d e c r e a s e s a t p o i n t s away f r o m t h e t a r g e t c e n t r e . B e c a u s e t h e p o l a r i t i e s o f a l l m a g n e t s i n t h e beam l i n e a n d s p e c t r o m e t e r w e r e n e c e s s a r i l y r e v e r s e d b e t w e e n i t + a n d TZ~ e x p e r i m e n t s t h e b e a m s p o t c o u l d c h a n g e . T h e a v e r a g e s o l i d a n g l e o v e r t h e b e a m s p o t d i s t r i b u t i o n was c a l c u l a t e d f o r t h e TC + ( n o r m a l i s a t i o n ) a n d %~ ( r a t i o ) e x p e r i m e n t s . T h i s p r o c e d u r e i s d i s c u s s e d i n A p p e n d i x 3 . T h e r e s u l t s w e r e 1 5 . 5 ± 2 . 3 m s r f o r t h e TC+ e x p e r i m e n t a n d 1 7 . 0 ± 2 . 1 m s r f o r t h e TC- e x p e r i m e n t . T h e d i f f e r e n c e i s c o n s i s t e n t w i t h z e r o a n d no s c a l i n g o f t h e i t " c r o s s s e c t i o n s was m a d e . T h e a b s o l u t e v a l u e o f t h i s s o l i d a n g l e c a l c u l a t i o n i s n o t r e l i a b l e , h o w e v e r , s i n c e o n l y f i r s t o r d e r c o e f f i c i e n t s w e r e u s e d t o c a l c u l a t e t h e a n g l e s u s e d . To make a n a b s o l u t e c r o s s s e c t i o n m e a s u r e m e n t t h i s c a l c u l a t i o n w o u l d n e e d t o be d o n e m o r e p r e c i s e l y a n d a c c u r a t e c a l c u l a t i o n s made o f t h e e f f e c t s o f p i o n d e c a y s . I t i s m o r e p r e c i s e t o n o r m a l i s e t o a c r o s s s e c t i o n , w h i c h i s k n o w n w e l l f r o m a m e a s u r e m e n t w h e r e t h e s o l i d a n g l e a n d d e c a y e f f e c t s a r e m o r e s t r a i g h t f o r w a r d t o c a l c u l a t e . T h e i x + p c r o s s s e c t i o n i s n o t i d e a l f o r t h i s b e c a u s e o f : t h e r a p i d v a r i a t i o n o f t h e e n e r g y o f t h e s c a t t e r e d p i o n w i t h a n g l e ; t h e r a p i d v a r i a t i o n o f t h e c r o s s s e c t i o n w i t h b o t h a n g l e a n d e n e r g y ; t h e n e c e s s i t y t o r e t u n e t h e b e a m l i n e f o r n~ e x p e r i m e n t s a n d p o s s i b i l i t y t h e r e i n o f c h a n g i n g t h e n o r m a l i s a t i o n . I n f u t u r e , n o r m a l i s a t i o n t o d e u t e r i u m c r o s s s e c t i o n s w i l l b e u s e d . The 3 2 S c r o s s s e c t i o n s w e r e c a l c u l a t e d f r o m d a t a t a k e n w i t h a l a r g e ( 1 0 b y 5 . 5 cm) t a r g e t . The c a l c u l a t i o n was a s d e s c r i b e d a b o v e a n d t h e f a c t o r 1 . 2 7 was a p p l i e d . Due t o t h e s m a l l s i z e o f t h e m a g n e s i u m t a r g e t s , o n l y a b o u t 70% o f t h e p i o n beam p a s s e d t h r o u g h t h e t a r g e t a n d t h e r e f o r e t h e beam f l u x m o n i t o r s d i d n o t d i r e c t l y m e a s u r e t h e f l u x t h r o u g h t h e t a r g e t . D u r i n g t h e m a g n e s i u m e x p e r i m e n t , h o w e v e r , d a t a was t a k e n w i t h a l a r g e c a r b o n t a r g e t a t e v e r y a n g l e . T h e c r o s s s e c t i o n f o r c a r b o n w a s c a l c u l a t e d w i t h o u t a n y c u t s a p p l i e d t o t h e p o s i t i o n o n t h e t a r g e t . T h e c r o s s s e c t i o n wa s t h e n c a l c u l a t e d w i t h t h e same c u t s , o n t h e p o s i t i o n a t t h e t a r g e t , a s w e r e a p p l i e d t o t h e m a g n e s i u m d a t a . T h e r a t i o o f t h e s e t w o c r o s s s e c t i o n r e s u l t s g i v e s a s c a l i n g f a c t o r f o r t h e m a g n e s i u m r e s u l t s t o a c c o u n t f o r t h e s m a l l t a r g e t s i z e . The c r o s s s e c t i o n s w e r e a l l c o r r e c t e d f o r i s o t o p i c c o m p o s i t i o n o f t h e t a r g e t i n a m a n n e r s i m i l a r t o t h a t u s e d o n t h e c r o s s s e c t i o n r a t i o s . T h e c r o s s s e c t i o n s a r e g i v e n i n T a b l e s V a n d V I . A t s m a l l a n g l e s , t h e d i f f e r e n t i a l c r o s s s e c t i o n s v a r y n o n - l i n e a r l y a c r o s s t h e a c c e p t a n c e o f t h e s p e c t r o m e t e r . T h e mean c r o s s s e c t i o n s m e a s u r e d w e r e t h e r e f o r e n o t t h e c r o s s s e c t i o n s a t t h e c e n t r a l a n g l e o f t h e s p e c t r o m e t e r a c c e p t e n c e . T h i s e f f e c t was c o m p e n s a t e d f o r b y c a l c u l a t i n g t h e a n g l e a t w h i c h t h e same c r o s s s e c t i o n w o u l d be m e a s u r e d w i t h a n e g l i g i b l y s m a l l s p e c t r o m e t e r a c c e p t a n c e . F o r t h i s c a l c u l a t i o n t h e c r o s s s e c t i o n v a r i a t i o n wa s a p p r o x i m a t e d b y a n e x p o n e n t i a l , f i t t e d t o t h e c a l c u l a t e d c r o s s s e c t i o n , a n d t h e s p e c t r o m e t e r a c c e p t a n c e i n 0 w a s a p p r o x i m a t e d b y a G a u s s i a n , f i t t e d t o t h e m e a s u r e d 0 d i s t r i b u t i o n a t 8 0 ° . The e f f e c t was v e r y s m a l l f o r t h e QD s p e c t r o m e t e r , b e c a u s e o f t h e s m a l l a c c e p t a n c e . T h e a n g l e s 0' i n T a b l e s I I , I I I , I V , V a n d V I i n c l u d e t h i s c o m p e n s a t i o n . T h e e r r o r o f t h e c a l c u l a t e d T t + p c r o s s s e c t i o n u s i n g S A I D was e s t i m a t e d t o be 5%. A d d i t i o n a l e r r o r s e n t e r i n g t h e n o r m a l i s a t i o n o f t h i s e x p e r i m e n t d u e t o c h a n g i n g t h e p o l a r i t y o f t h e beam l i n e a n d t h e t u n e o f t h e s p e c t r o m e t e r w e r e e s t i m a t e d t o be l e s s t h a n 5%. A d d i n g t h e s e t w o c o n t r i b u t i o n s l i n e a r l y r e s u l t s i n a n o r m a l i s a t i o n e r r o r o f a b o u t 10%. TABLE V Absolute Differential Cross Sections for 48.4 MeV n~ Scattering on 3 2S do dQ ± Error (mb/sr) do c ± Error (mb/sr) 0 cm degrees 28.6 38.6 48.7 58.75 66.6 81.9 92.0 102.2 0' cm degrees 28.3 38.4 48.4 58.7 66.6 81.9 92.0 102.0 212.2 75.5 26.4 11.81 10.03 10.38 8.54 4.07 5.9 2.6 1.04 0.60 0.47 0.55 0.42 0.24 212.2 75.5 26.5 11.75 9.97 10.31 8.52 4.09 6.1 2.7 1.10 0.63 0.50 0.58 0.44 0.25 6 ' i s corrected for cross section v a r i a t i o n across spectrometer cm r acceptance. da -T7T- i s corrected for i s o t o p i c composition of the targets. TABLE VI Absolute Differential Cross Sections for 43.9 MeV vT Scattering on 2l*Mg 0 cm degrees 0' cm degrees 4s + dQ -(mb/j E r r o r da _ c H dQ , " (mbi b E r r o r 'sr) 39.0 38.8 47.21 1.87 46.35 2.37 49.1 48.9 16.17 0.65 15.83 0.82 59.3 59.3 7.02 0.40 6.39 0.51 69.3 69.3 7.48 0.36 6.75 0.46 79.3 79.3 9.89 0.44 8.77 0.56 89.4 89.4 11.37 0.45 10.49 0.57 96.6 96.6 11.00 0.43 10.42 0.55 101.6 101.6 10.43 0.64 9.93 0.73 106.6 106.6 9.62 0.50 9.19 0.58 0' i s co r r e c t e d f o r cross s e c t i o n v a r i a t i o n across spectrometer cm acceptance. da c -j - r — i s correct e d f o r i s o t o p i c composition of the t a r g e t . CHAPTER V Density Distribution Analysis 5.1 Introduction T h e d a t a w e r e f i t t e d w i t h c a l c u l a t i o n s u s i n g a n o p t i c a l p o t e n t i a l . T he n e u t r o n d i s t r i b u t i o n o f t h e l a r g e r i s o t o p e o f e a c h p a i r wa s v a r i e d t o p r o d u c e a f i t t o t h e c r o s s s e c t i o n r a t i o s . T h e p r o t o n d i s t r i b u t i o n s o f b o t h i s o t o p e s w e r e k e p t f i x e d a t F e r m i d i s t r i b u t i o n s w h i c h a p p r o x i m a t e d t h e r e s u l t s f r o m e l e c t r o n s c a t t e r i n g . T h e n e u t r o n d i s t r i b u t i o n o f t h e s m a l l e r i s o t o p e was s e t e q u a l t o t h a t o f t h e r e s p e c t i v e p r o t o n d i s t r i b u t i o n . S e n s i t i v i t y o f t h e e x t r a c t e d n e u t r o n d i s t r i b u t i o n s t o v a r i a t i o n s i n t h e s e a s s u m e d n e u t r o n a n d p r o t o n d i s t r i b u t i o n s was i n v e s t i g a t e d . A l s o , t h e s e n s i t i v i t y t o v a r i a t i o n s i n t h e f o r m o r p a r a m e t e r s o f t h e o p t i c a l p o t e n t i a l was s t u d i e d . 5.2 Optical Potential The o p t i c a l p o t e n t i a l u s e d was t h a t o f S t r i e k e r , McManus a n d C a r r ( S t r 7 9 , S t r 8 0 ) , w h i c h i s s i m i l a r i n f o r m t o t h e E r i c s o n - E r i c s o n p o t e n t i a l ( E r i 6 6 ) . T he o r i g i n a l f o r m ( S M C 7 9 ) o f t h i s p o t e n t i a l i s ( P l _ i ) c 0 ( p 2 _ i ) 2SU o p t=-47t (b ( r ) + p 2 B 0 p 2 ( r ) + — V 2 c ( r ) + j — V 2 p 2 ( r ) ) c0 + 4it( V » L ( r ) c ( r ) V + V » p 2 ( r ) v ) + 2wV ( r ) P 2 c w h e r e L ( r ) = ( i + * 2 ^ c ( r ) £ ± ) ) " ! , b ( r ) = p 1 ( b 0 p ( r ) - b 1 e i t [ 6 p ( r ) ] ) , 6 p ( r ) = p n ( r ) - P p ( r ) , c ( r ) = ( c Q p ( r ) - c , e [ 6 p ( r ) ] ) , V c ( r ) i s t h e C o u l o m b p o t e n t i a l d u e t o t h e p r o t o n c h a r g e s , E i s t h e p i o n c h a r g e , ± 1 , a n d TC co i s t h e r e d u c e d e n e r g y , r e l a t e d t o t h e t o t a l e n e r g y o f t h e p i o n i n t h e p i o n n u c l e u s c e n t r e o f m a s s s y s t e m , co, a n d t h e n u c l e a r m a s s , M , b y co = co / ( 1 + co/M ) T he k i n e m a t i c f a c t o r s p^ a n d p 2 a r e d i s c u s s e d i n r e f e r e n c e ( S t r 7 9 ) . T h e t e r m s ( T h i 7 6 ) i n V 2 p a n d V 2 p 2 a r i s e f r o m t h e t r a n s f o r m a t i o n o f t h e k ' » k f a c t o r i n t h e p - w a v e t e r m f r o m t h e n - n u c l e o n c e n t r e o f m a s s s y s t e m t o t h e T t - n u c l e u s c e n t r e o f ma s s s y s t e m . T h i s i s r e f e r r e d t o a s t h e a n g l e t r a n s f o r m a t i o n . T h e c o m p l e x p a r a m e t e r s b Q , b^, c Q a n d c^ a r e r e f e r r e d t o a s t h e s i n g l e n u c l e o n p a r a m e t e r s . T h e i r r e a l p a r t s may be c a l c u l a t e d f r o m p i o n - n u c l e o n p h a s e s h i f t s , g i v i n g r e a s o n a b l e a g r e e m e n t w i t h t h e e m p i r i c a l v a l u e s . T h e ' a b s o r p t i o n p a r a m e t e r s ' , B Q a n d C Q , a r e a l s o c o m p l e x . The p o t e n t i a l was a l t e r e d i n a l a t e r p u b l i c a t i o n ( S t r 8 0 ) t o i n c l u d e t h e L o r e n t z - L o r e n z e f f e c t a l s o i n t h e a b s o r p t i o n , c h a n g i n g L ( r ) f r o m ( l + ^ \ c ( r ) ) - l t o ( l + X [ c ( r ) + C ( r ) ] ) ~ l a n d 4 T c { " V » L ( r ) c ( r ) V + V » C ( r ) V } t o 4 n V » L ( r ) { c ( r ) + C ( r ) } V , C 0 P 2 ( r ) w h e r e C ( r ) = P 2 a n d C Q i s t h e p - w a v e a b s o r p t i o n p a r a m e t e r . We r e f e r t o t h e s e two v e r s i o n s o f t h e p o t e n t i a l a s SMC79 a n d SMC81 r e s p e c t i v e l y . F i t s o f t h e p o t e n t i a l p a r a m e t e r s t o p i o n s c a t t e r i n g d a t a o v e r a w i d e r a n g e o f n u c l e i , u s i n g t h e s e p o t e n t i a l s , w e r e p u b l i s h e d . We u s e a s t h e b a s i c s e t s o f p a r a m e t e r s , S e t 1 ( S t r 7 9 ) a n d S e t E ( C a r 8 2 ) , a s g i v e n i n T a b l e V I I , f o r p o t e n t i a l s SMC79 a n d SMC81 r e s p e c t i v e l y . TABLE V I I Optical Potential Parameter Sets SET 1 SET E PARAMETER REAL IMAGINARY REAL IMAGINARY UNITS b o -0.028 +0.004 -0.043 +0.0042 b l -0 .08 -0.0015 -0.092 -0.0014 B0 -0 .04 +0.04 -0.005 +0.028 n-* c0 +0.27 +0.01 +0.248 +0.0099 n " 3 c l +0.22 +0.005 +0.163 +0.0046 H~3 c o -0 .10 +0.10 +0.045 +0.068 X 1.0 1.4 Set 1 and Set E are from References (Str79, Car82) m c 2 \i i s the inverse pion mass, — r - — = 1.4 fm 5.2.1 Fits to absolute cross sections V a r i o u s f i t s o f t h e o p t i c a l p o t e n t i a l p a r a m e t e r s t o t h e a b s o l u t e 3 2 S a n d 2 1 + M g c r o s s s e c t i o n s w e r e m a d e . T h e s e f i t s a r e i l l u s t r a t e d i n F i g u r e s 19 a n d 2 0 . T h e r e s u l t s a r e t a b u l a t e d i n T a b l e V I I I . I t was f o u n d t h a t t h e c r o s s s e c t i o n s c o u l d b e f i t w e l l b y v a r y i n g e i t h e r b Q a n d c Q , o r B Q a n d C Q . T h i s i s t o be e x p e c t e d f r o m t h e s i m i l a r i t y i n t h e t e r m s c o n t a i n i n g t h e p a r a m e t e r s b Q a n d B Q , o r c Q a n d C Q . T h e SMC81 p o t e n t i a l w i t h p a r a m e t e r S e t E d i d n o t g i v e g o o d a g r e e m e n t w i t h t h e a b s o l u t e 2 i + M g o r 3 2 S c r o s s s e c t i o n s . T h e s e p a r a m e t e r s w e r e o b t a i n e d f r o m f i t s t o TC + e l a s t i c s c a t t e r i n g a n d r e a c t i o n c r o s s s e c t i o n d a t a . I t s h o u l d b e n o t e d , h o w e v e r , t h a t t h e f i t s o b t a i n e d ( C a r 8 2 ) t o t h e TC+ e l a s t i c s c a t t e r i n g d a t a a r e n o t g o o d e i t h e r , e s p e c i a l l y i n t h e 1 + 0 C a r e g i o n . T h e a b s o r p t i o n c r o s s s e c t i o n s u s e d i n t h e s e f i t s w e r e f r o m N a k a i ( N a k 8 0 ) . W h i l e i t i s i m p o r t a n t t o i n s i s t t h a t t h e p o t e n t i a l p r o d u c e s t h e c o r r e c t r e a c t i o n c r o s s s e c t i o n s , t h e r e b y f i x i n g t h e b a l a n c e b e t w e e n t h e i m a g i n a r y b Q , c Q a n d i m a g i n a r y B Q , C Q p a r a m e t e r s , i t i s i m p e r a t i v e t h a t t h e e r r o r s a s s i g n e d t o t h e s e c r o s s s e c t i o n s a r e r e a l i s t i c . I t i s p o s s i b l e t h a t , w i t h S e t E , t h e p u b l i s h e d a b s o r p t i o n c r o s s s e c t i o n s h a v e b e e n r e p r o d u c e d a t t h e e x p e n s e o f t h e e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s . 5.3 Fits to Ratios of Cross Sections I n a n a l y s i n g t h e c r o s s s e c t i o n r a t i o s we a r e q u i t e i n s e n s i t i v e t o t h e p a r a m e t e r s e t u s e d , a s l o n g a s t h e a b s o l u t e c r o s s s e c t i o n i s r e p r o d u c e d ( s e e S e c t i o n 5 . 3 . 1 . 2 ) . A l l o f t h e p a r a m e t e r s a r e , i n t h e e n d , c o m b i n e d i n t h e p o t e n t i a l t o g i v e two c o m p l e x p o t e n t i a l s , t h e l o c a l a n d n o n - l o c a l p o t e n t i a l s . T h e c o r r e c t r a t i o o f l o c a l t o n o n - l o c a l p o t e n t i a l i s e s t a b l i s h e d q u i t e s e n s i t i v e l y b y f i t t i n g t h e a b s o l u t e c r o s s s e c t i o n s . TABLE VIII Parameter Sets used i n Fitting the Data Parameter Set Cross Section f i t t e d Parameters f i t t e d F i t t e d Values of X f f i t 1 NONE l f l 32S Re Re b o c o -.0165 .248 + + .0025 .0039 71 l f 2 X-0 32 S Re Re b o c o -.0205 .205 + + .0034 .0038 30 l f 3 3 2 S x 1.1 Re Re b0 c o -.0158 .263 + + .0026 .004 61 E NONE E f l 32 S Re Re b0 c o -.0368 .183 + + .0017 .0024 44 Ef2 3 2 S Re Re B o c o .0102 -.114 + + .0037 .0048 34 Ef3 3 2 S Im Im B o c o .00152 ±.0064 .421 ±.020 40 EflO 2 4Mg Re Re b0 c0 -.0448 .225 + + .001 .0027 8.1 l f l O 2hMg Re Re b0 c o -.0296 .271 + + .0032 .0096 56 The ba s i c parameter sets are Set 1 and Set E (Table V I I ) . Other sets have the i n d i c a t e d parameters f i t t e d to absolute cross s e c t i o n s . 2 0 4 5 7 0 9 5 1 2 0 ANGLE (COM.) FIGURE 1 9 48.4 MeV TT- on 3 2 S absolute d i f f e r e n t i a l cross s e c t i o n s , compared to c a l c u l a t i o n s w i t h the two p o t e n t i a l s SMC79 and SMC81. The e r r o r bars are smaller than the data p o i n t s . The p o t e n t i a l parameter sets are defined i n Tables VII and V I I I . 10 : ' 1 1 I I I I I I L 10 2. \ 10 " SMC 79 *« i SMC 81 s « m o SMC 81 S « E * • • c; l l 0 ° d 10" —I 1 1— AO 60 —i 1 1 1 1 — 80 100 120 20 c m . ANGLE (degrees) FIGURE 20 43.9 MeV TT~ on 2 uMg absolute d i f f e r e n t i a l cross s e c t i o n s , compared to c a l c u l a t i o n s w i t h the two p o t e n t i a l s SMC79 and SMC81. The p o t e n t i a l parameter s e t s : Set 1, Set E, and Set EflO are defined i n Tables VII and V I I I . However, the r e l a t i v e sizes of single nucleon parameters b Q and c Q compared to absorption parameters B Q and C Q are not well established i n thi s way. In the f i r s t Born approximation, the form f a c t o r , F(q), i s the Fourier transform of the po t e n t i a l ; CO F(q) = / V(r) e l q * r dr . 0 Taking the inverse ; V(r) = / F(q) e ± q ' r dq . 0 However, since only a l i m i t e d range of q i s covered i n the experiment, i t i s not possible to extract a unique V(r) this way. This i s equivalent to the statement that there i s a range of po t e n t i a l s , V ( r ) , which w i l l produce the same cross s e c t i o n . The c o r r e l a t i o n of the parameters b Q and B Q or c 0 and C Q i s p a r t l y due to th i s i n s e n s i t i v i t y to the d e t a i l e d form of V ( r ) . If we write, for instance, the l o c a l p o t e n t i a l , V^Cr), as V x ( r ) = bp(r).+ Bp 2(r) i t can be seen that v a r i a t i o n s i n b and B w i l l produce d i f f e r e n t forms f o r V * ( r ) . However, the shapes of p(r) and p 2 ( r ) are not very d i f f e r e n t , apart from an o v e r a l l m u l t i p l i c a t i o n f a c t o r , therefore changes i n b can be compensated for by changes i n B, within the s e n s i t i v i t y to the form of V ( r ) . The cross section r a t i o s , R, are s e n s i t i v e to the change i n V(r) between one isotope and another. We have 0-0(0) o,(0) + 6a(0) . R ( 0 ) a x(0) a x(0) ffl ' and 6a(0) i s i n some way related to the change i n the p o t e n t i a l V(r) b e t w e e n o ne i s o t o p e a n d t h e o t h e r . T h e c h a n g e i n V ( r ) , 6 V ( r ) , i s r e l a t e d t o a c h a n g e i n p ( r ) t h r o u g h t h e f o r m o f t h e o p t i c a l p o t e n t i a l . A g a i n u s i n g t h e l o c a l p a r t o f t h e p o t e n t i a l a s a n e x a m p l e we h a v e V ( r ) - p ^ V b ^ p ^ r ) + ( b Q + b ^ p ^ r ) ) + p 2 B Q ( p n ( r ) + P p ( r ) )2 , a n d f o r a c h a n g e i n p n ( r ) o f 6 p n ( r ) , t h e p o t e n t i a l V ( r ) w i l l c h a n g e b y 6 V ( r ) cc P j x ( b g + b ^ f i p ^ r ) + 2 p 2 B Q ( p p ( r ) + p n ( r ) )&p(r) . F o r t h e p o t e n t i a l p a r a m e t e r s o f S e t 1 ( T a b l e V I I ) a n d f o r a t o t a l d e n s i t y o f 0 . 2 n u c l e o n s / f m 3 ( c o r r e s p o n d i n g t o t h e n u c l e a r c e n t r e ) we h a v e P l < b 0 + b l > _ i . U x . 1 6 _ . 1 8 2 p 2 B 0 ( p n + p p ) 2 x 1 . 0 7 x . 1 7 x . 2 . 0 7 2 ' a n d s i m i l a r l y f o r S e t E we f i n d a r a t i o o f 5 4 . A s i m i l a r t r e a t m e n t o f t h e n o n - l o c a l p o t e n t i a l r e s u l t s i n t h e r a t i o s , P 2 ( c 0 + C l ) - z — — - — r - = 8 . 7 f o r S e t 1 a n d 1 6 . 1 f o r S e t E . 2 P 1 C 0 ( p n + p p ) S i n c e t h e a b o v e r a t i o s a r e m u c h g r e a t e r t h a n 1 ( e x c e p t f o r t h e S e t 1 , l o c a l p o t e n t i a l ) , t h e u n c e r t a i n t y i n t h e s i n g l e n u c l e o n a n d a b s o r p t i o n p a r a m e t e r s , c a u s e d b y t h e c o r r e l a t i o n s b e t w e e n t h e m , s h o u l d n o t a p p r e c i a b l y a f f e c t t h e m o d e l d e p e n d e n c y o f t h e e x t r a c t i o n o f p ( r ) , o r < r 2 > . T h i s i s i n d e e d f o u n d i n f i t t i n g t h e r a t i o d a t a u s i n g t h e p o t e n t i a l s c o n t a i n i n g v a r i e d a m o u n t s o f s i n g l e n u c l e o n a n d a b s o r p t i o n p a r a m e t e r s , a s l o n g a s t h e p o t e n t i a l i s c o n s t r a i n e d t o p r o v i d e a f i t t o t h e a b s o l u t e c r o s s s e c t i o n . T h e r a t i o d a t a w e r e f i t t e d w i t h c a l c u l a t e d c r o s s s e c t i o n r a t i o s . T h i s a n a l y s i s i s d i v i d e d h e r e i n t o two p a r t s . I n S e c t i o n 5 . 3 . 1 t h e n e u t r o n d i s t r i b u t i o n o f t h e l a r g e r i s o t o p e i s r e p r e s e n t e d w i t h a two p a r a m e t e r F e r m i f u n c t i o n . I n S e c t i o n 5 . 3 . 2 F o u r i e r B e s s e l t e r m s a r e a d d e d t o t h i s F e r m i f u n c t i o n d e s c r i p t i o n . T h e p r o t o n d i s t r i b u t i o n s w e r e d e s c r i b e d b y F e r m i f u n c t i o n s h a v i n g rms r a d i i g i v e n b y s u b t r a c t i n g t h e p r o t o n f o r m f a c t o r f r o m t h e e l e c t r o n s c a t t e r i n g c h a r g e r a d i i ( R y c 8 3 , L e e 7 4 ) . 5.3.1 Fermi function f i t s T h e two p a r a m e t e r s c n a n d t n ' o f a F e r m i f u n c t i o n d e s c r i p t i o n o f t h e n e u t r o n d i s t r i b u t i o n o f t h e l a r g e r i s o t o p e w e r e v a r i e d t o m i n i m i s e t h e x 2 o f t h e f i t . T h e d e p e n d e n c e o f t h e r e s u l t s o n t h e n u c l e o n d i s t r i b u t i o n s u s e d f o r t h e s m a l l e r i s o t o p e a n d t h e p r o t o n d i s t r i b u t i o n u s e d f o r t h e l a r g e r i s o t o p e i s d i s c u s s e d i n S e c t i o n 5 . 3 . 1 . 1 . I n S e c t i o n 5 . 3 . 1 . 2 t h e d e p e n d e n c e o f t h e r e s u l t s o n t h e p a r a m e t e r s a n d f o r m o f t h e o p t i c a l p o t e n t i a l i s d i s c u s s e d . The f i t t i n g was p e r f o r m e d w i t h b o t h p o t e n t i a l s SMC79 a n d SMC81 a n d w i t h v a r i o u s p a r a m e t e r s e t s . The r e s u l t s a r e t a b u l a t e d i n T a b l e s X , X I a n d X I I , a n d X I I I , f o r r a t i o s 3 6 S / 3 2 S , 3hS/32S, 2 6 M g / 2 4 M g r e s p e c t i v e l y . 5*3.1.1 Sensitivity to assumed nucleon distributions T h e d e p e n d e n c e o f t h e e x t r a c t e d rms n e u t r o n r a d i u s o n t h e a s s u m e d n e u t r o n d i s t r i b u t i o n o f t h e s m a l l e r i s o t o p e a n d t h e a s s u m e d p r o t o n d i s t r i b u t i o n o f b o t h i s o t o p e s was s t u d i e d u s i n g t h e 3 I + S / 3 2 S d a t a . T h e s e r e s u l t s a r e i n c l u d e d i n T a b l e X I . I t c a n be s e e n t h a t i n c r e a s i n g t h e rms n e u t r o n r a d i u s o f 3 2 S b y 0 . 0 6 0 fm ( F i t 6 ) r e s u l t s i n a n i n c r e a s e o f t h e e x t r a c t e d rms n e u t r o n r a d i u s o f 3 4 S b y 0 . 0 5 2 f m . T h e r m s n e u t r o n r a d i u s d i f f e r e n c e ( o r ^ ) b e t w e e n t h e t w o i s o t o p e s , t h e r e f o r e , i s m e a s u r e d , n o t t h e a b s o l u t e rms n e u t r o n r a d i u s o f 3 I + S . C h a n g i n g t h e p r o t o n d i s t r i b u t i o n s h a s a l e s s e r e f f e c t o n t h e e x t r a c t e d n e u t r o n r a d i u s . A c h a n g e i n t h e p r o t o n r a d i u s o f 3 1 *S b y 0 . 0 6 fm ( F i t 7 ) r e s u l t s o n l y i n a c h a n g e i n t h e e x t r a c t e d n e u t r o n r a d i u s o f 3 I + S b y 0 . 0 1 8 f m , o r 1 /3 o f t h e a m o u n t . T h i s means t h a t t h e n e u t r o n r a d i i r e s u l t s a r e l a r g e l y i n d e p e n d e n t o f t h e m e a s u r e m e n t s o f t h e p r o t o n r a d i i u s e d , e s p e c i a l l y s i n c e t h e s e a r e t y p i c a l l y m e a s u r e d t o h i g h p r e c i s i o n . C h a n g i n g t h e s h a p e s o f t h e p r o t o n d i s t r i b u t i o n s ^ o f b o t h i s o t o p e s ( F i t 8 ) b y ' d e c r e a s i n g b o t h 3**Cp a n d 3 2Cp b y 0 . 1 0 fm r e s u l t s i n n o c h a n g e i n t h e e x t r a c t e d n e u t r o n r a d i u s . The e x t r a c t e d n e u t r o n r a d i u s d i f f e r e n c e , 6 r n , i s , t h e r e f o r e , l a r g e l y i n d e p e n d e n t o f t h e a s s u m e d m a t t e r d i s t r i b u t i o n s . H o w e v e r , i t c a n be s e e n f r o m F i g u r e 21 t h a t t h e c r o s s s e c t i o n r a t i o s a r e s e n s i t i v e t o t h e d i f f e r e n c e i n s h a p e s o f t h e 3 2 S a n d 3 1 *S n e u t r o n d e n s i t y d i s t r i b u t i o n s . 5.3.1.2 Sensitivity to optical potential parameters T h e 3 l * S / 3 2 S d a t a was a g a i n u s e d f o r t h e s e t e s t s . T he f i t s w e r e p e r f o r m e d w i t h p o t e n t i a l p a r a m e t e r s S e t E , w h e r e o n e p a r a m e t e r h a d b e e n c h a n g e d b y +10%. The r e s u l t s a r e i n T a b l e I X . The s e n s i t i v i t y t o t h e p a r a m e t e r c h a n g e i s v e r y s m a l l e x c e p t f o r t h e c a s e o f c Q . T h i s i s a s w o u l d b e e x p e c t e d f r o m s t u d y i n g F i g u r e A . A n u n a m b i g u o u s s e t o f p a r a m e t e r s c a n n o t b e f i t t e d t o a n e l a s t i c s c a t t e r i n g c r o s s s e c t i o n a s t h e r e i s n o t e n o u g h i n f o r m a t i o n t o s e p a r a t e c o r r e l a t i o n s b e t w e e n t h e e f f e c t s o f t h e p a r a m e t e r s . F i t t i n g t o a w i d e r a n g e o f n u c l e i h e l p s f i x t h e p a r a m e t e r s b u t i t d o e s n o t s e em p o s s i b l e t o f i t c r o s s s e c t i o n s w e l l f r o m 1 6 0 t o 2 0 8 P b w i t h p a r a m e t e r S e t E , w i t h 50 MeV T t + . I t i s s i g n i f i c a n t t h a t t h e d i s c r e p a n c y b e t w e e n t h e d a t a a n d 101 FIGURE 2 1 The e f f e c t of changing the shape of the 3 4 S neutron d i s t r i b u t i o n while keeping the rms radius constant at 3.259 fm. The p o t e n t i a l used was SMC81 with parameter Set E f l . c a l c u l a t i o n ( C a r 8 2 ) f o r TC+ o n 4 0 C a i s v e r y s i m i l a r t o t h a t f o r 50 MeV n~ o n 3 2 S w i t h S e t E ( F i g u r e 1 9 ) . T h e p a r a m e t e r s w e r e f i t t e d , t h e r e f o r e , t o t h e a b s o l u t e c r o s s s e c t i o n o f 3 2 S f o r t h e s u l f u r i s o t o p e s d a t a , a n d 2 4 M g f o r t h e m a g n e s i u m i s o t o p e s d a t a . A s p r e v i o u s l y m e n t i o n e d , a n u n a m b i g u o u s s e t o f p a r a m e t e r s c a n n o t b e o b t a i n e d t h i s w a y , b u t i t was f o u n d t h a t t h e e x t r a c t e d n e u t r o n r a d i u s was i n d e p e n d e n t o f t h e p a r a m e t e r s a s l o n g a s t h e a b s o l u t e c r o s s s e c t i o n was f i t t e d . T h i s i s s e e n i n f i t s 2 , 4 , 1 0 a n d 11 i n T a b l e s X I a n d X I I . I n p a r t i c u l a r , i t s h o u l d b e n o t e d t h a t s e t t i n g t h e L o r e n t z - L o r e n z p a r a m e t e r , \ , t o 0 ( t u r n i n g o f f t h e L o r e n t z - L o r e n z e f f e c t a l t o g e t h e r ) b e f o r e f i t t i n g b Q a n d c Q t o t h e a b s o l u t e c r o s s s e c t i o n s , r e s u l t s i n a c h a n g e o f o n l y 0 . 0 1 0 fm i n 6 r n . I t i s s e e n f r o m F i g u r e 23 t h a t t h e f i t t o t h e r a t i o d a t a , w i t h \ = 0 , i s v e r y p o o r a t s m a l l a n g l e s , e v e n t h o u g h t h e f i t t o t h e a b s o l u t e c r o s s s e c t i o n was o p t i m i s e d b y v a r y i n g b Q a n d c Q . When X i s s e t t o 1 , t h e f o r w a r d a n g l e r a t i o d a t a a r e r e p r o d u c e d . T h e SMC79 a n d SMC81 p o t e n t i a l s d i f f e r i n t h e L o r e n t z - L o r e n z e f f e c t a n d i n d e e d t h e f i t s u s i n g t h e s e two p o t e n t i a l s d i f f e r a t s m a l l a n g l e s ( s e e F i g u r e s 23 a n d 2 4 ) . F i t 1 2 , w h e r e t h e p a r a m e t e r s w e r e S e t E w i t h Im B Q a n d Im C Q f i t t e d t o t h e 3 2 S c r o s s s e c t i o n , p r o d u c e s a v a l u e o f 6 r n w h i c h i s s i g n i f i c a n t l y d i f f e r e n t t o t h e o t h e r r e s u l t s . H o w e v e r , t h e s e i m a g i n a r y p a r a m e t e r s a r e n o t c o n s t r a i n e d v e r y w e l l b y t h e a b s o l u t e c r o s s s e c t i o n . 5.3.1.3 3 6 S/ 3 2 S ratios The r e s u l t s o f t h e 3 6 S / 3 2 S f i t s a r e g i v e n i n T a b l e X . Some d i f f e r e n c e was f o u n d h e r e b e t w e e n t h e r e s u l t s w i t h t h e SMC79 a n d SMC81 p o t e n t i a l s . T h i s i s p a r t l y d u e t o t h e h i g h w e i g h t i n g o f t h e 2 0 ° p o i n t b e c a u s e o f i t s s m a l l e r r o r . A t 2 0 ° t h e r a t i o i s n o t v e r y s e n s i t i v e t o 6 r n , t h e d i f f e r e n c e i n t h e rms r a d i i f o r 3 6 S - 3 2 S , ( F i g u r e 4 ) b u t i s s e n s i t i v e t o t h e p o t e n t i a l p a r a m e t e r s . T he SMC79 p o t e n t i a l p r o d u c e s a f i t w h i c h i s w e l l o u t s i d e o f t h e e r r o r b a r o f t h e 20° p o i n t , a n d h e n c e t h e s l o p e o f t h e x 2 f r o m t h i s p o i n t OX2 = 9 l i OR o ( 6 r ) oR d ( 6 r ) n n i s l a r g e e v e n t h o u g h dR/o ( 6 r n ) i s s m a l l . R e m o v i n g t h e 20° p o i n t f r o m t h e f i t p r o d u c e s a v a l u e o f 6 r n w h i c h i s m o r e i n a g r e e m e n t w i t h t h a t o b t a i n e d f r o m SMC81 . T he SMC81 r e s u l t s a r e n o t s e n s i t i v e t o t h e i n c l u s i o n o f t h i s p o i n t a s t h e f i t g o e s t h r o u g h t h i s p o i n t e i t h e r way a n d ox 2/dR i s t h e r e f o r e s m a l l . T h e d i f f e r e n c e b e t w e e n t h e SMC79 a n d SMC81 p o t e n t i a l s i s i n t h e m a n n e r i n w h i c h t h e L o r e n t z - L o r e n z e f f e c t i s i n c l u d e d . I t h a s b e e n s e e n a b o v e ( i n S e c t i o n 5 . 3 . 1 . 2 ) t h a t t h i s t e r m a f f e c t s t h e c r o s s s e c t i o n r a t i o s a t s m a l l a n g l e s . The x 2 v a l u e s f o r t h e f i t s t o t h i s d a t a a r e v e r y s m a l l . T h e e r r o r s o f t h e d a t a p o i n t s a r e p u r e l y s t a t i s t i c a l , h o w e v e r . I g n o r i n g t h e r e s u l t f o r t h e SMC79 f i t w i t h t h e 2 0 ° p o i n t b e c a u s e o f t h e m o d e l d e p e n d e n c y , we h a v e 6 r n = 0 . 1 5 2 ± 0 . 0 2 7 fm f r o m t h e SMC81 p o t e n t i a l a n d 6 r n = 0 . 1 2 6 ± 0 . 0 3 5 fm f r o m t h e SMC79 p o t e n t i a l . T h e r e s u l t s e x t e n d f r o m 0 . 0 9 1 fm t o 0 . 1 7 9 fm w i t h i n a x 2 c h a n g e o f 1 a n d we may e x p r e s s t h e c o m b i n e d r e s u l t s a s 6 r n = 0 . 1 3 5 ± 0 . 0 4 4 f m . A d d i n g 0 . 0 1 3 fm e r r o r f o r t h e c o n t r i b u t i o n f r o m t h e a b s o l u t e c r o s s s e c t i o n n o r m a l i s a t i o n u n c e r t a i n t y ( s e e T a b l e X I , F i t 5 ) we h a v e 6 r n = 0 . 1 3 5 ± 0 . 0 5 7 f m . 5.3.1.4 3 I »S/ 3 2 S ratios The r e s u l t s f r o m t h e s e f i t s a r e i n T a b l e s X I a n d X I I . A s d i s c u s s e d e a r l i e r , 6 r n i s i n s e n s i t i v e t o v a r i a t i o n s i n t h e r e a l p o t e n t i a l p a r a m e t e r s a s l o n g a s t h e a b s o l u t e c r o s s s e c t i o n i s f i t . T h e p a r a m e t e r s e t E f 3 , w h e r e I m B Q a n d I m C 0 w e r e f i t t e d t o t h e a b s o l u t e c r o s s s e c t i o n , p r o d u c e s a v a l u e o f 6 r n w h i c h i s 0 . 0 3 fm l e s s t h a n t h a t p r o d u c e d w i t h p a r a m e t e r s e t s E f l a n d E f 2 . I t i s n o t r e a s o n a b l e t o f i t t h e i m a g i n a r y p a r t s o f BQ a n d C Q t o e l a s t i c s c a t t e r i n g d a t a , a s i s e v i d e n c e d b y t h e c o m p a r a t i v e l y l a r g e u n c e r t a i n t i e s . We w i l l , t h e r e f o r e , r e l y o n t h e a b s o r p t i o n p a r a m e t e r s f r o m t h e f i t t i n g t o t h e r e a c t i o n c r o s s s e c t i o n s ( C a r 8 2 ) . T h e v a l u e s f o r a l l o f t h e r e m a i n i n g f i t s t o t h e c r o s s s e c t i o n r a t i o s may be w r i t t e n 6 r n = 0 . 1 0 3 ± 0 . 0 3 2 fm w h e r e t h e e r r o r e n c o m p a s s e s a l l r e s u l t s w i t h t h e i r s t a n d a r d d e v i a t i o n s a n d 0 . 0 1 3 fm f r o m t h e n o r m a l i s a t i o n u n c e r t a i n t y . 5.3.1.5 26Mg/2 l»Mg ratios The r e s u l t s f r o m t h e s e f i t s a r e g i v e n i n T a b l e X I I I a n d t h e f i t s o b t a i n e d a r e i l l u s t r a t e d i n F i g u r e s 25 a n d 2 6 . T h e r e s u l t s f r o m t h e f i t s w i t h t h e v a r i o u s p o t e n t i a l s a n d p a r a m e t e r s e t s a r e v e r y s i m i l a r . T he s e t E p a r a m e t e r s a r e now n o t a l t e r e d v e r y much i n f i t t i n g t h e a b s o l u t e c r o s s 105 TABLE IX Change i n 6rn from 3 HS/ 3 2S Data with Variations of the Optical Potential Parameters Parameter varied 3 4 c n (fm) 24 t n (fm) l i n Change i n 6r n NONE 3.243 2.394 0.61 -Re b Q 3.227 2.425 0.72 +.007 Re b x 3.252 2.383 0.59 -.001 Re B Q 3.244 2.394 0.62 .000 Re c Q 3.262 2.331 0.58 -.022 Re Cj 3.235 2.412 0.63 + .004 Re C 0 3.246 2.387 0.62 -.002 \(to 1.6) 3.243 2.414 0.64 + .010 The parameters were increased by 10% i n each case. The values of c n and t n are the r e s u l t s for 3t*S from the f i t . T A B L E X Fits to 3 6S/ 3 2S Data with Fermi Function Distribution Densities P o t e n t i a l P a r a m e t e r S e t 2 0 ° p o i n t i n c l u d e d 36 r  L n ( f m ) 36 t n ( f m ) X 2/* (fm) SMC 7 9 l f l Y ES 3 . 4 5 8 ± 0 . 0 6 5 2 . 1 9 4 ± 0 . 1 1 2 0 . 8 0 . 0 9 6 ± 0 . 0 2 8 SMC 7 9 l f l NO 3 . 4 6 9 ± 0 . 0 8 9 2 . 2 4 1 ± 0 . 1 5 5 0 . 3 0 . 1 2 6 ± 0 . 0 3 5 SMC81 E f l Y E S 3 . 5 4 4 ± 0 . 0 7 7 2 . 1 9 7 ± 0 . 1 3 0 0 . 2 0 . 1 5 2 ± 0 . 0 2 7 SMC 81 E f l NO 2 . 2 0 3 ± 0 . 0 7 1 2 . 2 0 3 ± 0 . 1 2 1 0 . 2 0 . 1 5 6 ± 0 . 0 2 8 F o r a l l f i t s : 3 2 c n = 3 . 1 5 3 f m ; 3 2 t „ = 2 . 3 7 6 fm p , n p , n 3kc = 3 . 2 1 8 fm ; 3ht = 2 . 3 7 6 f m . p p TABLE XI F i ts to 3 I*S/ 3 2S Data using Optical Potential SMC79 F I T R E S U L T S F I T P A R A -METER SET 3 2 S D E N S -I T I E S PROTON D E N S I T Y 3 V n ( f m ) 3 4 t n ( f m ) x2/ n 6 r n ( f m ) 1 1 A M 3 . 1 1 9 ± 0 . 0 2 7 2 . 5 9 2 ± 0 . 0 4 3 2 . 2 3 0 . 1 0 0 ± 0 . 0 1 4 2 l f l A M 3 . 1 5 9 ± 0 . 0 2 4 2 . 5 5 1 ± 0 . 0 4 0 1 . 5 6 0 . 1 0 0 ± 0 . 0 1 2 3 l\ A M 3 . 2 3 9 ± 0 . 0 2 6 2 . 3 4 7 ± 0 . 0 4 3 7 . 2 7 0 . 0 3 3 ± 0 . 0 1 4 4 U2 A M 3 . 1 5 3 ± 0 . 0 5 1 2 . 5 7 6 ± 0 . 0 7 8 4 . 3 0 . 1 1 0 ± 0 . 0 1 2 5 l f 3 A M 3 . 1 7 4 ± 0 . 0 2 5 2 . 5 1 2 ± 0 . 0 4 2 1 . 1 6 0 . 0 8 7 ± 0 . 0 1 3 6 1 B M 3 . 2 3 4 ± 0 . 0 2 2 2 . 5 6 7 ± 0 . 0 3 9 1 . 6 0 . 0 9 2 ± 0 . 0 1 2 7 1 A N 3 . 1 5 0 ± 0 . 0 3 9 2 . 5 9 4 ± 0 . 0 6 6 2 . 9 0 . 1 1 8 ± 0 . 0 1 3 8 1 C P 3 . 1 1 4 ± 0 . 0 3 6 2 . 5 9 8 ± 0 . 0 5 8 2 . 1 0 . 1 0 0 ± 0 . 0 1 3 P a r a m e t e r s e t s a r e d e f i n e d i n T a b l e V I I I . S e t 1 \ i s S e t l f l w i t h \=0 A 3 2 c „ = 3 . 1 5 3 fm , 3 2 t „ = 2 . 3 7 6 f m p , n p , n B 3 2 c ~ « - 3 . 2 5 3 fm , 3 2 t n = 2 . 3 7 6 f m p , n p , n C 3 2 c „ _ = 3 . 0 5 3 fm , 3 2 t = 2 . 3 7 6 f m M 3 l t c „ = 3 . 2 1 5 fm , 3 1 t t = 2 . 3 7 6 fm P ' P N 3 1 + c = 3 . 3 1 5 fm , 3 1 t t = 2 . 3 7 6 f m P P P 3 1 + c = 3 . 1 1 5 fm , 3kt = 2 . 3 7 6 fm TABLE XII Fits to 3 I»S/ 3 2S Data using Optical Potential SMC81 F I T # P A R A -METER SET 3 2 S D E N S I T I E S 3 4 s PROTON D E N S I T Y 3 ^ n ( f m ) 3 1 + t n ( f m ) *i n * r n ( f m ) 9 E A M 3 . 2 4 3 ± 0 . 0 2 3 2 . 3 9 4 ± 0 . 0 4 2 0 . 6 1 0 . 0 6 4 ± 0 . 0 1 3 10 E f l A M 3 . 2 3 9 ± 0 . 0 1 8 2 . 4 5 9 ± 0 . 0 4 0 . 9 9 0 . 0 9 6 ± 0 . 0 1 2 11 E f 2 A M 3 . 2 2 0 ± 0 . 0 1 8 2 . 4 7 7 ± 0 . 0 3 0 . 6 8 0 . 0 9 5 ± 0 . 0 1 0 12 E f 3 A M 3 . 2 1 3 ± 0 . 0 2 7 2 . 4 3 3 ± 0 . 0 4 1 . 9 0 . 0 6 7 ± 0 . 0 1 2 13 E X A M 3 . 2 4 3 ± 0 . 0 2 1 2 . 4 1 4 ± 0 . 0 4 0 . 6 4 0 . 0 7 4 ± 0 . 0 1 1 EX i s S e t E w i t h X = 1 .6 A a n d M a s i n T a b l e X I TABLE XIII Fits to 26Mg/21*Mg Data with Fermi Function Densities P o t e n t i a l P a r a m e t e r S e t 26*n (fS) 2 6*n ( f m ) xi n * rn ( f m ) SMC 79 1 2 . 8 5 5 ± 0 . 0 7 4 2 . 5 0 5 ± 0 . 0 9 6 1 . 0 0 . 0 7 2 ± 0 . 0 2 6 SMC 7 9 l f l O 2 . 8 2 5 ± 0 . 1 1 0 2 . 5 4 4 ± 0 . 1 4 0 1 . 1 0 . 0 7 8 ± 0 . 0 2 6 SMC 81 E " 2 . 967 ± 0 . 0 3 6 2 . 4 1 3 ± 0 . 0 5 5 0 . 5 6 0 . 0 8 2 ± 0 . 0 1 6 SMC 81 E f l O 2 . 9 3 1 ± 0 . 0 3 3 2 . 4 6 2 ± 0 . 0 5 1 0 . 5 6 0 . 0 9 0 ± 0 . 0 1 6 T h e p a r a m e t e r s e t s a r e d e f i n e d i n T a b l e s V I I a n d V I I I . F o r a l l f i t s : 2 i * c p , n = 2 . 8 2 8 fm ; 2ktPin = 2 ' 4 0 8 fm-2 6 c p = 2 . 8 9 6 f m ; 2 6 t p = 2 . 3 0 2 f m . 110 FIGURE 22 3 6 S / 3 2 S cross s e c t i o n r a t i o s . The SMC79 p o t e n t i a l curves used Set 1 parameters, defined i n Table V I I . The SMC81 p o t e n t i a l curve used Set E f l parameters. The curve l a b e l l e d NO FIT used equal neutron d i s t r i b u t i o n shapes f o r 3 6 S and 3 2 S . FIGURE 23 3 l tS/ 3 2S cross section ratios fitted with the SMC79 potential. The parameter sets indicated are defined in Tables VII and VIII. Set If2 had X=0. The curve labelled NO FIT used equal neutron distribution shapes for 3 1 +S and 3 2 S. 112 c.o.m. ANGLE FIGURE 24 3 4 S / 3 2 S cross section r a t i o s f i t t e d with the SMC81 p o t e n t i a l . The parameter sets indicated are defined i n Tables VII and VIII. The curve l a b e l l e d NO FIT used equal neutron d i s t r i b u t i o n shapes for 3**S and 3 2 S . 113 13 to 12 rsi rs M rO 09 Set 1 NO FIT Set 1 FIT Set I f FIT -i l_ 20 40 60 80 c.o.m. A N G L E J 1 1 1 i » • 100 FIGURE 25 2 6Mg/ 2 1 +Mg cross section r a t i o s f i t t e d with the SMC79 p o t e n t i a l . The parameter sets are defined i n Tables VII and VIII. The curve l a b e l l e d NO FIT used equal neutron d i s t r i b u t i o n shapes for 2 6Mg and 2l+Mg. T 1 r 1 1 1 1 f i r J 1 1 1 1 1 1 i • • « 20 40 60 80 100 c o m ANGLE (degrees) FIGURE 26 2 6Mg/ 2 l tMg cross section r a t i o s f i t t e d with the SMC81 p o t e n t i a l . The parameter sets are defined i n Tables VII and VIII. The curve labeled NO FIT used equal neutron d i s t r i b u t i o n shapes for 2 6Mg and 2 t fMg. s e c t i o n and consequently the d i f f e r e n c e i n 6 r n obtained from Sets E and EflO i s only 0.008 fm. The r e s u l t s may be w r i t t e n 6 r n = 0.076 ± 0.043 fm i n c l u d i n g a l l e r r o r s as above. 5 . 3 . 2 F o u r i e r B e s s e l a n a l y s i s I t i s now usual to analyse e l e c t r o n s c a t t e r i n g data (Neg79a) by representing the charge d i s t r i b u t i o n as a sum of orthogonal f u n c t i o n s , f n ( r ) , or as p ( r ) " P , ( r ) + Ia f ( r ) , ^c r c l '•' n n n where p ^ C r ) i s an approximation to the charge d e n s i t y , such as a Fermi d i s t r i b u t i o n , and the a n are c o e f f i c i e n t s to be found from f i t t i n g to the cross s e c t i o n data. The f n ( r ) a r e o f t e n represented by s p h e r i c a l Bessel f u n c t i o n s , i n ( \ r) = s i n ( \ r ) / \ r , where X = mt/R and R i s a u n n n n c c c u t o f f radius beyond which the f n ( r ) are set to zero. The j ( X n r ) are orthogonal i n the sense / r 2 j (\ r) j (X r) dr = 0 ; n * m  J o n o m This procedure has a l s o been a p p l i e d to the a n a l y s i s of proton s c a t t e r i n g (Ray81) and alpha s c a t t e r i n g ( G i l 8 0 ) . A l s o Friedman et a l . ( F r i 8 2 ) have studied the r a d i a l s e n s i t i v i t y of p, a, and % probes to the neutron d e n s i t y using t h i s technique w i t h pseudodata. The cross s e c t i o n data presented here were analysed by r e p r e s e n t i n g the neutron d i s t r i b u t i o n , p n ( r ) , of the l a r g e r isotope by s i n ( \ r ) P n ( r ) = P l ( r ) + I <xn F - 2 — = P l ( r ) + P ] ? B ( r ) n w h e r e a n a r e t h e F o u r i e r B e s s e l ( F B ) c o e f f i c i e n t s , wa s n o r m a l i s e d s u c h t h a t 4TC / p x ( r ) r 2 d r = N , w h e r e N i s t h e n u m b e r o f n e u t r o n s . T h e r e f o r e we h a v e t h e c o n s t r a i n t , s i n ( \ r ) I % I ' 2 — - r - -° • n w h i c h r e s u l t s i n ( s e e S e c t i o n 5 . 3 . 2 . 1 ) a, = I a n . ( 5 . 1 ) n=2 T h i s c o n s t r a i n t was a c c o m p l i s h e d b y f r e e l y v a r y i n g o n l y t h e F B c o e f f i c i e n t s f o r n > 1 , a n d a d j u s t i n g a c c o r d i n g t o ( 5 . 1 ) a t e a c h s t e p i n t h e s e a r c h . F e r m i d i s t r i b u t i o n s w e r e u s e d f o r p^. Two f o r m s f o r p^  w e r e u s e d : 1 ) P i ( r ) = ~ * 1 P ( r ) > w h e r e t h e s u p e r s c r i p t 1 , 2 s i g n i f i e s t h e l i g h t e r , 1 N n h e a v i e r i s o t o p e a n d 1 > 2 N i s t h e r e s p e c t i v e n umbe r o f n e u t r o n s ; 2 ) p^( r ) i s t h e r e s u l t o f a two p a r a m e t e r F e r m i f u n c t i o n f i t t o t h e d a t a . T h e d e n s i t y wa s c o n s t r a i n e d t o be n o n - n e g a t i v e e v e r y w h e r e b y i n c r e a s i n g t h e x2 f o r t h o s e t r i a l s i n t h e f i t t i n g w h e r e t h e d e n s i t y d i d b e c o m e n e g a t i v e a t some r a d i i . T h e x2 was i n c r e a s e d i n a s m o o t h b u t r a p i d l y i n c r e a s i n g q u a d r a t i c m a n n e r i n o r d e r n o t t o a d v e r s e l y i n f l u e n c e t h e f i t t i n g p r o c e d u r e , w h i c h c o n v e r g e s m o s t r a p i d l y o n a q u a d r a t i c x2 s u r f a c e ; P n ( r ) 2 X = X x ( 1 + 9 x f r - N — 1 x 1 00 I w h e r e A * L ^ p ^ ( r ) -'max J 117 0 = 0 f o r p o s i t i v e d e n s i t y , 0 = 1 i f the d e n s i t y becomes negative anywhere, and f 1 s i g n i f i e s the maximum value over a l l r a d i i . v 'max This c o n s t r a i n t only became a c t i v e f o r f i t s w i t h large R c, when the FB terms were added at r a d i i where was very s m a l l . The number of FB terms used was increased u n t i l the x2 P e r degree of freedom showed no improvement. For the s u l f u r experiments t h i s required 3 f r e e FB c o e f f i c i e n t s and f o r the magnesium i t required only two. The x2 m i n i m i s a t i o n was performed by a r o u t i n e which c a l c u l a t e d the curvature matrix, M, where nm 2 9a 9a n m Care was taken that the e v a l u a t i o n of the elements of the curvature matrix d i d not i n c l u d e x2 values which were adjusted f o r negative d e n s i t y . The e r r o r matrix a i s then given by a = (M . The e r r o r a, of a nm nm f f u n c t i o n f ( p n ( r ) ) w a s then c a l c u l a t e d i n the usual way: n=2 n m>n n m The i n c l u s i o n of the o f f - d i a g o n a l terms of the e r r o r matrix reduced the c a l c u l a t e d e r r o r s s i g n i f i c a n t l y because of the strong c o r r e l a t i o n s between the FB terms. 5.3.2.1 Radial moments The r a d i a l moments of the f i t t e d d e n s i t y d i s t r i b u t i o n s were c a l c u l a t e d f rom 4 * / f p ( r ) + p F f i ( r ) ) T"*2 d r <r > = 4TC / ( p ( r ) + p F B ( r ) ) r 2 d r But 4TC / ( p ( r ) + Ppg ( r ) ) r 2 = N, the number of n e u t r o n s , <r m > <r > = <r > : + 4ic — , where : < r m > 1 i s the m*"*1 r a d i a l moment o f the Fe rmi d i s t r i b u t i o n , p ^ ( r ) , and R c m+1 = / T a s i n ( X r ) r dr , n i n n 0 n=l U s i n g r-i L 2 J , m-2v J r s i n ( a r ) d r = c o s ( a r ) £ ( - 1 ) + v=0 a We have r—i L 2 J , m - 2 v - l s i n ( a r ) y ( - l ) v ^ 1 ^ , v=0 a J c r m 8 i n ( X n r ) d r - c o S ( n „ ) J ( - l ) v _ ^ - f ^ 0 v=0 ' X and [f] R»-2v m-1 r. n r. . , x v + l m! c n=l v=0 \ n [ 2 ^ ] m-2v +l FB . , _ „ n r , . .n r. . v+i (m+i;! c Since <r°> = 0, we ob t a i n the c o n d i t i o n FB a, a a -~ ^ <_1) r 5al " l n~ 1 n=2 n n=2 Imposing t h i s c o n d i t i o n r e s u l t s i2 FB L "n v E v (m-2v+iyr "c n=2 v=0 < r % « = I «„ ("I)" ! ( " 1 ) V + 1 - T ^ M i r r R m " 2 V + 1 \ 2 v + l .2v+l J n 1 - T a A n=2 n m ' n The e r r o r of t h i s q u a n t i t y i s then given by a 2 = I ( a 2 A + 2 Y a 2 A . A m ^_ 2 V nn m,n n i m,i m,n n -^ ^  nul/m r y m v . 4 it m -il/m We want <r > = I <r >, + -£ <r J> 1 J- N FB ' m ' = <rm>j/m ( 1 + % ! L ! l L . )!/» 1 N ^m^ m - < r m > } / m ( 1 + £ — — — ) Nm . m. <r > 1 ^ .^m^l/m The corresponding error, E = a { ) v e ' m m 1 „ m ; FB m Nm ^ In p a r t i c u l a r for the rms radius < r 2 > l / 2 = < r 2 > l / 2 + *ZL < r > F B 1 N m < r 2 > l / 2 and E 2  2 N < r 2 > l / 2 1 5.3.2.2 Dependence on the cut off radius, R£ I t was found that the r e s u l t s for 6 r n depended on the value of the cut off radius used. This i s i l l u s t r a t e d i n Figures 27 and 28. If p ^ C r ) i s set to be proportional to the neutron d i s t r i b u t i o n of the smaller isotope, then 6 r n increases as RQ i s increased. However, i f p i ( r ) i s set to be equal to the f i t t e d Fermi d i s t r i b u t i o n of the larger isotope, then 6 r n has a minimum at some value of R c and increases for values of Rc above and below t h i s . In t h i s case the decrease of 6 r n at large values of R c i s caused by the constraint keeping the density p o s i t i v e everywhere; larger values of 6 r n require the density to become negative, 121 •15 h (fm) •10 h •05 t FIGURE 27 V a r i a t i o n of x 2 of the Fourier Bessel f i t to 3 t f S / 3 2 S data as the cutoff radius, Rc, i s changed. The s t a r t i n g density for the 3l*S neutron d i s t r i b u t i o n was proportional to that of 3 2 S. 09 H •08 H (fm) •07 H •06 H Rc (fm) FIGURE 28 Variation of x 2 of the Fourier Bessel f i t to 3 I + S / 3 2 S data as the cutoff radius, R„ the distribution to the data. . -c, is changed. The starting density for 3hS neutron distribution was the best f i t Fermi 1 2 3 at some r a d i i , when Rc is large. The cut off radius was therefore incorporated into the f i t as an additional fitted parameter. Extra terms in the error matrix, from the variation of Rc, were included in the evaluations of the errors of the density distributions and the radial moments. The SMC81 potential, with parameter sets E f l and EflO, was used for the f i t s since this gave better f i t s to the absolute cross sections and to the ratios. The imaginary parameters have also been obtained from analysis of reaction cross sections and hence should be more r e a l i s t i c than those of Set 1. The 6r n results of the f i t s are given in Tables XIV, XV and XVI for the 3 6S/ 3 2S, 3 4S/ 3 2S, and 26Mg/21*Mg data respectively. The values of 6r n differ from those obtained with Fermi functions although the two methods are consistent within the errors. The errors for the Fourier Bessel f i t s are larger, as should be expected. Two different starting distributions were used: p^ proportional to the neutron distribution of the smaller isotope and p^ equal to the Fermi function f i t . The value of 6r n obtained was found to be only slightly dependent on p^. The results may be summarised: 3 6S - 3 2S : 6r n = 0.14 ± 0.07 fm 3hS - 3 2S : 6r n = 0.061 ± 0.035 fm 2 6 ^ _ 2 4 M g . 5 r n = 0.077 ± 0.056 fm where 0.013 fm error has been added for the contribution from the absolute cross section uncertainty as in Section 5.2. 124 TABLE XIV Results of Fits to 3 6S/ 3 2S Data with Fourier Bessel Series for 3 6S Neutron Distribution Pl x2 n 6r n (fm) Rc (fm) A .25 0.15±0.04 6.5 B .25 0.12±0.04 5.3 The Potential used was SMC81. Rc is the cutoff radius in fm. is the starting density to which are added the Fourier Bessel terms. A: 3 6 c n = 3.544 fm.; 3 6 t n = 2.197 fm. B: 3 6 c n = 3.153 fm.; 3 6 t n = 2.376 fm. For a l l f i t s : 3 6 c = 3.218 fm, 3 6 t = 2.376 fm, P P 3 2 c = 3.153 fm, 3 2 t = 2.376 fm P P 125 TABLE XV Results of F i ts to 3**S/32S Data using Fourier Bessel Series for 3 t tS Neutron Distribution Rc(fm) Pl x 2 n 6r (fm) n ' 6.3 A 0.54 0.061±0.022 6.8 B 0.54 0.062±0.019 The potential used was SMC81. Rc was free in the f i t s . R c is the cutoff radius in fm. is the starting density distribution to which the Fourier Bessel terms are added. A: 3 k c n = 3.153 fm; 3 2 t n = 2.376 fm 3 4Pn(r) « 3 2 p ( r ) B: 3 4 c n = 3.239 fm; 3 l + t n = 2.459 fm Best f i t Fermi For both f i t s : 3 2 c = 3.153 fm; 3 2 t = 2.376 fm p,n p,n 3 l + c p = 3.215 fm; 3 l t t p = 2.376 fm 126 TABLE XVI Results of Fits to 26Mg/2l*Mg Data using a Fourier Bessel Series for 2 6Mg Neutron Distribution R c Pl x2 n 6 r n (fm) (fm) 5.6 A 0.54 0.068±0.034 7.6 B 0.50 0.090±0.030 The potential used was SMC81. Rc was free in the f i t s . Rc is the cutoff radius in fm. is the starting density distribution to which the Fourier Bessel terms are added. A: 2 6 c n - 2.828 fm; 2 6 t n = 2.408 fm. B: 2 6 c n = 2.931 fm: 2 6 t n = 2.463 fm. For both f i t s : 2 1 + cp,n " 2.828 fm; 2htn = 2.408 fm. 2 6 c p = 2.896 fm; 2 6 t n = 2.302 fm. 1 2 7 5.3.2.3 Neutron density distribution differences The neutron density distributions of the larger isotopes of the pairs were obtained from the f i t s . The relevant quantities here, however, are the differences in densities, 6pn, between the two isotopes of each pair. These are plotted as 6p nr 2 in Figures 29, 30 and 31 for each of 3 6S - 3 2S, 3 4S - 3 2S, and 26Mg - 24Mg. The multiplication by r 2 means that the height of the curve is proportional to the amount of extra neutron density at that radius. The s t a t i s t i c a l error band resulting from the f i t is shown. However, since only a limited number of Fourier Bessel terms was used, the analysis is not truly model independent as not a l l distributions could be generated. The resulting error of the distribution is referred to as the completeness error. This error w i l l decrease, and the s t a t i s t i c a l error w i l l decrease, as the number of Fourier Bessel terms is increased. However, the spatial resolution of the experiment is limited by the momentum transfer. Bessel function terms with structure smaller than this inherent resolution w i l l not be determined and can cause large oscillations in the density without altering the x2 of the f i t . The completeness error was estimated in a manner similar to that sometimes used in the analysis of electron scattering experiments (Neg79b). In the electron scattering experiments, however, the completeness error is much smaller than the s t a t i s t i c a l error, as the momentum transfer covered can be large. A theoretical model for the density (Shell Model or Hartree-Fock calculation) is f i t with the same functional form and Fourier-Bessel sum as used in the analysis of the experimental data. The deviation of the f i t from the model density is then the completeness error. In this case, however, density distribution differences are being extracted in the analysis. Therefore, the c o m p l e t e n e s s e r r o r was e v a l u a t e d b y r e p r e s e n t i n g t h e S i n g l e P a r t i c l e P o t e n t i a l ( S P P ) ( S t r 8 2 , H o d 8 3 ) p r e d i c t i o n f o r t h e d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s w i t h t h e same f u n c t i o n a l f o r m u s e d i n t h e f i t t i n g , i . e . , 6 p s p p ( r ) was f i t w i t h s i n ( \ r ) P x ( r ) + I <xn ^ p 2 ( r ) n = l w h e r e p 2 ( r ) i s t h e F e r m i f u n c t i o n u s e d t o r e p r e s e n t t h e n e u t r o n d e n s i t y o f t h e s m a l l e r i s o t o p e , a n d t h e <xn w e r e v a r i e d t o p r o d u c e t h e f i t . 6 p ( r ) was w e i g h t e d a c c o r d i n g t o a c o n s t a n t p e r c e n t a g e u n c e r t a i n t y a t a l l v a l u e s o f r . T h e d i f f e r e n c e b e t w e e n t h e f i t a n d t h e m o d e l d e n s i t y d i s t r i b u t i o n was t h e n a d d e d t o e a c h s i d e o f t h e e r r o r e n v e l o p e i n F i g u r e s 2 9 , 30 a n d 3 1 . 1 2 3 4 5 6 7 8 RADIUS ( fm ) FIGURE 29 Results of Fourier Bessel analysis of 3 6S/ 3 2S cross section ratios. The dashed curves are the s t a t i s t i c a l error envelope and the solid line has the completeness error added. The SMC81 potential was used with parameter set E f l . The solid dots are the results of SPP (Hod83) calculations, which were evaluated every 0.5 fm. 0 2 4 6 8 R A D I U S ( f e r m i s ) FIGURE 30 Results of Fourier Bessel analysis of 3 1 +S/ 3 2S cross section ratios. The dashed line is the s t a t i s t i c a l error envelope and the solid line has the completeness error added. The SMC81 potential was used with parameter set E f l . The solid dots are the results of SPP (Hod83) calculations, which were evaluated every 0.5 fm. .03 .02 .01 - .01 1 1 • 1 1 T • 1 -j 9 * ^ X t \ % \ -_ / ' \#\ w I / V I \\\ - \ * V 1 -/ i 1 i i \ > \ \ -1 ll • / 1 — / / J if v \ /'I  # /in // ' I/// • \\\\ -\ \ \ \ X \ • • • • i i i i i • 1 2 3 4 5 6 7 R A D I U S ( f e r m i s ) FIGURE 31 R e s u l t s o f F o u r i e r B e s s e l a n a l y s i s o f 2 6 M g / 2 1 + M g r a t i o s . The d a s h e d l i n e i s t h e s t a t i s t i c a l e r r o r e n v e l o p e a n d t h e s o l i d l i n e h a s t h e c o m p l e t e n e s s e r r o r a d d e d . The SMC81 p o t e n t i a l was u s e d w i t h p a r a m e t e r S e t E f l O . The s o l i d d o t s a r e t h e r e s u l t s o f SPP ( H o d 8 3 ) c a l c u l a t i o n s , w h i c h w e r e e v a l u a t e d e v e r y 0 . 5 f m . CHAPTER V I C o n c l u s i o n s The l a c k o f u n d e r s t a n d i n g o f t h e p i o n - n u c l e u s i n t e r a c t i o n l i m i t s t h e a p p l i c a b i l i t y o f p i o n s f o r s t u d y i n g n u c l e a r s t r u c t u r e . T h e p i o n o t h e r w i s e h a s c h a r a c t e r i s t i c s w h i c h make i t a v e r y a t t r a c t i v e n u c l e a r p r o b e : l o n g mean f r e e p a t h , z e r o s p i n a n d a v a i l a b i l i t y i n beams o f two c h a r g e d s t a t e s w i t h v e r y d i f f e r e n t n e u t r o n a n d p r o t o n i n t e r a c t i o n . T h e s e e x p e r i m e n t s w e r e d e s i g n e d t o t a k e a d v a n t a g e o f t h e s e u s e f u l p r o p e r t i e s , w h i l e m i n i m i s i n g t h e d e p e n d e n c e o f t h e a n a l y s i s o n t h e d e t a i l s o f t h e p i o n -n u c l e u s i n t e r a c t i o n . T h i s wa s a c h i e v e d b y m e a s u r i n g c r o s s s e c t i o n r a t i o s ( a t t h e same t i m e r e m o v i n g many e x p e r i m e n t a l s y s t e m a t i c e r r o r s ) a n d l i m i t i n g t h e a n g u l a r r a n g e t o b e c l e a r o f t h e d i f f r a c t i v e r e g i o n o f t h e c r o s s s e c t i o n . The d a t a w e r e a n a l y s e d f o r d i f f e r e n c e s i n t h e n e u t r o n d i s t r i b u t i o n s o f t h e two i s o t o p e s o f e a c h p a i r : 3 6 S / 3 2 S , 3 4 S / a ; i S a n d 2 b M g / 2 H M g . D e s c r i b i n g t h e n u c l e a r d i s t r i b u t i o n s b y F e r m i f u n c t i o n s g a v e p r e c i s e r e s u l t s f o r t h e d i f f e r e n c e s i n rms n e u t r o n r a d i i . T he r e s u l t s w e r e f o u n d t o be i n d e p e n d e n t o f c h a n g e s i n t h e p a r a m e t e r s o f t h e o p t i c a l p o t e n t i a l u s e d f o r t h e a n a l y s i s . T he i n s e n s i t i v i t y o f t h e r e s u l t s t o t h e L o r e n t z -L o r e n z e f f e c t i s p a r t i c u l a r l y r e a s s u r i n g , a s t h i s p a r t o f t h e p o t e n t i a l i s a f f e c t e d ( L a n 8 0 ) b y t h e r a n g e o f t h e TC-N i n t e r a c t i o n , P a u l i c o r r e l a t i o n s a n d t h e % c o u p l i n g t o t h e p m e s o n ; t h e b e s t v a l u e t o u s e i s n o t c l e a r . T h e r e s u l t s f o r t h e rms n e u t r o n r a d i i d i f f e r e n c e s , 5r n, w i t h i n a F e r m i f u n c t i o n d e s c r i p t i o n w e r e : 3 6 S - 3 2 S : 6 r n = 0 . 1 3 5 ± 0 . 0 5 7 f m 3hS - 3 2 S : 6 r n = 0 . 1 0 3 ± 0 . 0 3 2 fm 133 2 6Mg- 2 4Mg : 6 r n = 0.076 ± 0.043 fm The data were a l s o analysed by representing the neutron d i s t r i b u t i o n of the l a r g e r nucleus with a Fermi f u n c t i o n modified by a sum of s p h e r i c a l Bessel f u n c t i o n s . The r e s u l t s f o r the rms r a d i i d i f f e r e n c e s were c l o s e , but not equal, to those from the Fermi d i s t r i b u t i o n f i t s , except f o r the case of 3 i +S - 3 2 S . The r e s u l t s were: 3 6 S - 3 2 S : 6 r n = 0.14 ± 0.07 fm 3t*S - 3 2 S : 6 r n = 0.061 ± 0.035 fm 2 6 M g _ 2 4 ^ . 6 r n = 0 > 0 7 7 + 0.056 fm The r e s u l t s of Single P a r t i c l e P o t e n t i a l ( S P P ) c a l c u l a t i o n s by Hodgson(Str82,Hod83) are: 3 6 S - 3 2 S : 6 r n = 0.171 fm 3kS - 3 2 S : 6 r n = 0.091 fm 2 6Mg - 21+Mg : 6 r n = 0.121 fm For a nucleus of incompressible nuclear matter we have A l / 3 , 6r 1 6A r A and — = -r- - — r 3 A This model r e s u l t s i n r a d i i d i f f e r e n c e s of : 3 6 S - 3 2 S : 6 r n = 0.13 fm 3hS - 3 2 S : 6 r n = 0.065 fm 2 6 ^ _ 2*+Mg . 6 r n = 0.093 fm, i n remarkably good agreement with the r e s u l t s of t h i s experiment and the SPP c a l c u l a t i o n s . The r e s u l t s f o r the Fo u r i e r Bessel f i t of 3kS - 3 2 S and both Fermi f u n c t i o n and F o u r i e r Bessel f i t s of 2 6Mg - 2l*Mg agree with the SPP c a l c u l a t i o n s w i t h i n a standard d e v i a t i o n . The rms radius i s very s e n s i t i v e to d e n s i t i e s at large r a d i i ; the f a c t o r p ( r ) r ' t i s i n t e g r a t e d to evaluate the rms radius. The Fourier Bessel expansions are unstable at large r a d i i for small numbers of c o e f f i c i e n t s ; o s c i l l a t i o n s , rather than a slowly varying density, are introduced around zero density. For a low momentum transfer probe with a l o c a l i n t e r a c t i o n the form f a c t o r would s t i l l be d i r e c t l y related to the rms radius, and not be s e n s i t i v e to the p a r t i c u l a r behaviour of the density at large r a d i i . The non-local nature of the pion i n t e r a c t i o n w i l l change this since the Fourier transform i s of the p o t e n t i a l , not the density. The Fermi d i s t r i b u t i o n constrains the density at large r a d i i to be smooth, and although this introduces t h e o r e t i c a l bias to the a n a l y s i s , the r e s u l t s for 6 r n may s t i l l be more r e l i a b l e than those of the Fourier Bessel f i t s . Indeed, the r e s u l t s for the density d i s t r i b u t i o n difference extracted from a Fermi function f i t are very close to the SPP values at r a d i i above about 4 fm for the n u c l e i studied here. This suggests that the Fermi model i s quite r e l i a b l e for analysis of rms r a d i i d i f f e r e n c e s . This i s not necessarily so for the rms r a d i i themselves. For a low momentum transfer probe, only a small number of Fourier Bessel terms may be f i t t e d and therefore the analysis i s not s t r i c t l y model independent, although the bias introduced should be manifested i n the completeness e r r o r s . It would be preferable i n t h i s case to choose a d i f f e r e n t set of functions which behave more reasonably at large r a d i i . The squares of harmonic o s c i l l a t o r wavefunctions (Laguerre polynomials m u l t i p l i e d by an exponential factor) might be appropriate, although t h e i r behaviour at large r a d i i i s not i d e a l . The analysis of the data for neutron density d i s t r i b u t i o n differences shows a good s e n s i t i v i t y from r a d i i around 7 fm down to about 2 fm. Although only a few Fourier Bessel terms can be f i t , the shape of t h e d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s a r e o f a s i m p l e f o r m a n d may be f i t q u i t e w e l l w i t h a f e w t e r m s ; t h i s i s d e m o n s t r a t e d b y t h e f a i r l y s m a l l s i z e s o f t h e c o m p l e t e n e s s e r r o r s . T h e c o m p l e t e n e s s e r r o r s d o , h o w e v e r , s e t a l i m i t o n t h e a t t a i n a b l e p r e c i s i o n o f t h e r e s u l t s w i t h l o w e n e r g y p i o n s . T h e s e e r r o r s a r e a r e f l e c t i o n o f t h e l o w momentum t r a n s f e r o f t h e s c a t t e r i n g . I n c r e a s i n g t h e momentum t r a n s f e r o f t h e p i o n , h o w e v e r , b r i n g s t h e s c a t t e r i n g i n t o t h e d i f f r a c t i v e r e g i o n , w h e r e t h e a n a l y s i s i s m o d e l d e p e n d e n t ( s e e R e f e r e n c e ( S t e 7 8 ) a n d F i g u r e 4 ) . I t i s i n t e r e s t i n g t h a t t h e d i s c r e p a n c y b e t w e e n t h e SPP c a l c u l a t i o n a n d e x p e r i m e n t a l r e s u l t s i s r e p r o d u c e d i n c o m p a r i n g t h e n e u t r o n d e n s i t y d i f f e r e n c e s o f t h e C a i s o t o p e s ( S t r 8 2 ) . T h e C a e x p e r i m e n t a l r e s u l t s w e r e f r o m a m o d e l i n d e p e n d e n t a n a l y s i s o f 8 0 0 MeV p r o t o n e l a s t i c s c a t t e r i n g b y R a y e t a l . ( R a y 8 1 ) T h e e x p e r i m e n t a l r e s u l t s a g a i n do n o t h a v e t h e n e g a t i v e d e n s i t y d i f f e r e n c e a t s m a l l r a d i i , a s p r e d i c t e d b y t h e t h e o r y . The e x p e r i m e n t a l r e s u l t s f o r t h e rms r a d i i a n d d e n s i t y d i s t r i b u t i o n d i f f e r e n c e s f o r t h e s u l f u r i s o t o p e s a g r e e q u i t e w e l l w i t h SPP c a l c u l a t i o n s ; t h e m a g n e s i u m r a d i i m e a s u r e d h e r e a r e o n e s t a n d a r d d e v i a t i o n l o w e r t h a n t h e SPP c a l c u l a t i o n s . T h e s e n s i t i v i t y o f t h e l o w e n e r g y p i o n s t o n e u t r o n d i s t r i b u t i o n s h a s b e e n d e m o n s t r a t e d . R e a s o n a b l e a g r e e m e n t w i t h SPP c a l c u l a t i o n s h a s b e e n f o u n d , e x c e p t i n t h e n u c l e a r i n t e r i o r . A u n i q u e a d v a n t a g e o f p i o n s i s t h a t t h e m e t h o d may be t e s t e d b y u s i n g TC+ t o m e a s u r e k n o w n ( f r o m e l e c t r o n s c a t t e r i n g ) p r o t o n d i s t r i b u t i o n s with the same s e n s i t i v i t y . 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C23, 533 (1981) APPENDIX 1 Finding Transfer Coefficients A program was written to find the polynomials which would express one coordinate of a trajectory as a function of the others. The program was organised so that the maximum order for each coordinate in the polynomial and the total maximum order for any term could be defined. For example, consider the polynomial, f, where x = f(y,z) By setting the maximum order of y to be 2, that of z to be 1 and the total maximum order to be 3 the polynomial is C Q + C xy + C 2y 2 + C 3z + C^yz + C 5y 2z where the C n are coefficients to be found. Data for which a l l of the coordinates (x,y,z in this case) were known were used to find the th coefficients. For the i event we have the measured values of x^, y^, zi» A multiple linear regression routine, RMUL (IMS82), was then used to find the C n, defined above, that exactly minimise the quantity I (x - f C y ^ z , ) ) 2 i About 1000 events were used to find fewer than 60 coefficients i n each case. A partial F test was made on each coefficient, allowing the rejection of s t a t i s t i c a l l y insignificant terms in the polynomial. The deviations ux - f(y ,z )) were then histogrammed and data rejected which gave very large deviations. The regression f i t t i n g was then repeated. When using the carbon data f o r events w i t h two known values of the momentum d e v i a t i o n , 6p/p, (from the ground and f i r s t e x c i t e d s t a t e s ) the c o e f f i c i e n t s from TRANSPORT (Bro80) were f i r s t used to f i n d the approximate values of 6p/p. This gave good enough r e s o l u t i o n to i d e n t i f y the ground and f i r s t e x c i t e d s t a t e s and the corresponding values of 6p/p were assigned to each event. C o e f f i c i e n t s only up to f i r s t order i n 6p/p could then be found, since only two values of 6p/p were f i t . Higher order terms were found by using the data at va r i o u s magnetic f i e l d (B) s e t t i n g s . The radius of curvature, p, of the pion t r a j e c t o r y i n the magnetic f i e l d i s p r o p o r t i o n a l to p/B. We want to know what change i n B w i l l produce the same change i n p as a change i n momentum, 6p. We have p = K x p/B where K i s a constant. 6p = K x 6p/B = K x p x (-6B/B2 + ( 8 B ) 2 / B 3 ) 6p/p = (6B/B) 2 - 6B/B A value of dp/p equal to -6B/B + (6B/B) 2 i s therefore assigned f o r each magnet s e t t i n g . I n f i n d i n g c o e f f i c i e n t s to trace back to the target w i t h the QD spectrometer, data at two p o s i t i o n s of the veto counter separated by 7 mm i n the v e r t i c a l , x, d i r e c t i o n were used. The x p o s i t i o n at the ta r g e t was taken to be that of the centre of the hole i n the veto counter. Only one p o s i t i o n of the target i n the h o r i z o n t a l , y, d i r e c t i o n e x i s t e d . F i t t i n g the c o e f f i c i e n t s of the polynomial YT = f(Xl,Yl,<t>) where YT = 0, and XI, YI are the coordinates at wire chamber 1, i s meaningless, unless part of the polynomial i s f i x e d . The c o e f f i c i e n t f o r Y l was ther e f o r e f i x e d at u n i t y , which i s reasonable since there were no magnetic elements between the target and wire chamber 1. The e f f i c a c y of t h i s procedure was demonstrated by f i t t i n g the x p o s i t i o n i n the same manner, using only one veto p o s i t i o n . The p o s i t i o n of the projected d i s t r i b u t i o n was found to move by the correct amount i n the data where the counter was moved. The procedure for f i n d i n g the c o e f f i c i e n t s was performed i t e r a t i v e l y . The difference between the known coordinate and the calculated coordinate was histogrammed and cuts made to the data to remove events far from the peak before r e f i t t i n g the c o e f f i c i e n t s . It was always this difference which was a c t u a l l y f i t and the values of the c o e f f i c i e n t s determined were added to the c o e f f i c i e n t s used i n the c a l c u l a t i o n . Let X = f(Y,Z), and l e t f'(Y,Z) be the approximate polynomial found from f i t t i n g or from TRANSPORT. Then f i t t i n g the d i f f e r e n c e , X-f'(Y.Z), r e s u l t s i n a polynomial g(Y,Z) where g(Y,Z) = X - f ( Y . Z ) and g(Y,Z) + f'(Y,Z) = f( Y , Z ) . This procedure allows the addition of new terms i n the polynomial without f i t t i n g a l l the e x i s t i n g c o e f f i c i e n t s again. It also allows s t a r t i n g values for the c o e f f i c i e n t s to be used (taken, for instance, from TRANSPORT). APPENDIX 2 Sulfur Target Density Measurement Because the s u l f u r targets were made from compressed powder, they were not completely uniform i n density. The average target density was known from the t o t a l mass of s u l f u r and the area. The beamspot, however, did not cover the targets uniformly, sampling the target centres more than the edges. The e f f e c t of t h i s on the number of scattered pions detected can be seen by considering the scattering from a small square target c e l l at p o s i t i o n x,y with dimensions dx and dy. We have for the d i f f e r e n t i a l cross section da = K x N(x,y)  dQ T(x,y) x Q(x,y) x F(x,y) where K i s a constant; N(x,y) i s the number of pions per unit area detected from the target element at x,y; T(x,y) i s the density of the target element (g/cm 2); Q(x,y) i s the s o l i d angle subtended by the spectrometer from the p o s i t i o n of the target element; and F(x,y) i s the number of pions per unit area which passed through the target element. Therefore f § / F(x,y) Q(x,y) dx dy _ K J dx dy J dx dy / dx dy But 148 f F(x,y) Q(x,y) dx dy = j? x Q / dx dy " a V e r a g e Let T(x,y) = T a v e r a g e x C-r(x,y) . Therefore we have f dx dy F Q = x average T r , , average J dx dy T T average To calculate the average, KT, we need to know the distribution C^,(x,y). The target density was measured at points forming a grid with a separation of 3.5 mm in the y vertical direction and 4.0 mm in the horizontal direction. The degradation of a collimated beam of electrons from a l°6Ru source was used to measure the density. The target was mounted on a transit table in front of the source and the electrons detected with a s c i n t i l l a t i o n counter. The pulses from the detector were discriminated on pulse height and counted with a scaler. The scaler was gated on and off with a gate generator which simultaneously started or stopped a 100 Hz clock. The electrons were degraded in energy as they passed through the target so that, as the target thickness increased, fewer electrons gave pulse heights over the discriminator threshold level. The system was calibrated with small sulfur targets of known average densities. These calibration targets were sampled at 9 points which were spaced to be at the centres of 9 equal areas, symmetrically arranged. Calibration targets that showed greater than 5% variation in density between any two sampled points were rejected. The average count rate of the 9 values was used as the calibration point for the average density of that target. The calibration curve thus obtained is shown in Figure 32. The average count rates for the sampled points of the real targets are also shown. The 3t*S target density was scaled to that of a n a t S target having the same number of atoms per unit area. Points were repeated throughout the measurement every few hours to check for any d r i f t s in count rates due to gain changes in the phototube or discriminator threshold changes. For each point about 20,000 counts were taken. The slopes of the calibration curve at the positions of the average target densities were used to calculate the matrix of density factors C T(x,y). These factors were constrained such that the mean was equal to 1, so that the numerical integrals then performed over the targets were correctly normalised. The resulting distributions are shown in Figure 33. F I G U R E 3 2 C a l i b r a t i o n curve for measuring s u l f u r target thickness by attenuation of electrons from a collimated 1 0 6 R u source. The point for the 3 4 S target has been adjusted to account for the higher atomic weight of 3 4 S. 151 2mm. FIGURE 33 Variation of 3 2S and 3 4S target densities as measured from the attenuation of electrons from a 1 0 6Ru source. The densities were measured at the points of intersection of the lines. The mean target densities were 365 mg/cm2 for 3 2 S and 290 mg/cm2 for 3 1 +S. APPENDIX 3 Spectrometer Acceptance It i s usual to calculate a detector s o l i d angle by Monte-Carlo simulation. The f r a c t i o n of p a r t i c l e s leaving the target i n a s p h e r i c a l l y symmetric d i s t r i b u t i o n (or d i s t r i b u t e d according to the d i f f e r e n t i a l cross section) which passes through the detectors i s ca l c u l a t e d . Because of the complex nature of the r e s t r i c t i n g apertures of the spectrometer and the ef f e c t s of the magnetic f i e l d s , this i s not a simple procedure. The s o l i d angle was calculated from the d i s t r i b u t i o n of angles of the p a r t i c l e s as they l e f t the target. The angles were calculated from the positions of the pions at wire chambers 1 and 2 using transformations given by TRANSPORT. The target was divided into 1 cm wide v e r t i c a l s t r i p s and the s o l i d angle found for each of these s t r i p s . The s o l i d angle, Q ' , subtended by the base of a cone from the apex of the cone i s given by Q ' = 2 i t(l - Cos (9 ) ) , where 9 i s the hal f angle of the cone. Therefore the s o l i d angle contained within a polar angle, 9, of a reference l i n e i s also given by the same equation. Consider this reference l i n e as some i n i t i a l t r a j e c t o r y from the target to the spectrometer. The numbers of p a r t i c l e s with t r a j e c t o r i e s having d i r e c t i o n s within a polar angle, 9, of th i s reference l i n e w i l l be proportional to Q ' , as defined above, i f there are no r e s t r i c t i n g apertures l i m i t i n g the t r a j e c t o r i e s i n th i s range. If the reference l i n e i s the mean d i r e c t i o n of p a r t i c l e s from a p a r t i c u l a r p o s i t i o n on the target, then, up to a l i m i t i n g value of 9, the s o l i d angle subtended by the spectrometer from this p o s i t i o n w i l l be equal to 0.'. For each target strip the mean direction for pions leaving the target was found. The mean direction was defined as mean (0), mean (<)>); where 9 = Tan _ 1(6X/6Z), cj> = Tan _ 1(6Y/6Z), and X, Y, Z are defined in Figure 11. For each event the polar angle between the mean direction and the actual direction was then calculated. The solid angle, corresponding to this polar angle was then calculated. Assuming that for .004 sr around the mean direction there are no restrictions on the direction, the true solid angle, Q , for the target strip is given by Q = .004 x N/Nin where N is the total number of events from the target strip and N^n is the number of events with Q" less than or equal to .004 sr. The distribution of from the whole target is shown in Figure 34. Notice that the distribution is fl a t to well beyond .004 sr. To compute the average solid angle i t is necessary to weight the solid angle from each target strip with the flux through that strip. The flux through the strip is proportional to the number of pions, N_^ , detected from the strip divided by the solid angle, Q ^ , at the position of the strip. Therefore 154 J L J L J L J I L 4 0 - o o CD) o lO J 30-1 E 20-10-II o 0 o (I (> o o <> o O 11 o o o OHO O O o o o o) o O O (i To o 1 o o, 4> O " \ ' t°o°o ° ^ Q P'Q Q, n , a 1 3 10 12 i i 16 18 20 22 24 26 28 30 32 Q' (.001 sr) FIGURE 34 D i s t r i b u t i o n of 0' (defined i n the text) from the whole carbon target. 155 Horizontal target position (cm) 0 FIGURE 3 5 Acceptance of the spectrometer, horizontally across the target position. This was calculated at each position on the target from the distribution of trajectories of the detected particles. 156 APPENDIX 4 Peak Fitting The resolution in the experiments was not sufficient to cleanly separate the ground and f i r s t excited states, particularly for the magnesium data. The relative heights of the peaks in the energy spectrum were found by f i t t i n g with a x2 minimisation routine. This routine followed, in principle, the algorithm used by CURFIT, as described in Reference (Bev69). The lineshape of the peaks was found by f i t t i n g the carbon data, where the f i r s t excited state did not overlap the ground state, and by f i t t i n g the sulfur and magnesium data at 40°, where the f i r s t excited state was very small compared to the ground state. A4.1 Sulfur Data It was found that, for the sulfur data, the lineshape was well described by the addition of two normal distributions at the same mean position and with standard deviations of 0.7 MeV and 1.7 MeV. The height of the wide peak was kept at 1/10 the height of the narrow peak. The width of the wide peak was kept fixed but the narrow peak was allowed to vary to account for the changes due to changing target thickness at different angles. The difference in positions between the ground state and the f i r s t excited state was kept fixed at the known value. A further excited state at 4.5 MeV was included in the f i t to represent the states seen at around this energy. A4.2 Magnesium Data A s l i g h t l y d i f f e r e n t lineshape was used for the 21+Mg and 2 6Mg spectra. The procedure used was f i r s t to f i t the spectrum of 1 2 C obtained at the same angle with a ground state and a peak at an e x c i t a t i o n energy of 1.81 MeV for 2 6Mg, or 1.34 MeV for 2i+Mg. The r a t i o of the heights of ground to excited states was recorded i n each case. This r a t i o was then subtracted from the r a t i o obtained i n f i t t i n g the 2 4Mg or 2 6Mg. This i s equivalent to approximating the lineshape of 2 6Mg, for instance, with a sum of two normal d i s t r i b u t i o n s separated by 1.81 MeV. In evaluating a cross section r a t i o the r a t i o of the number of events i n the ground state peaks i s needed. This i s given by 26 C f x _ ( 2 6 R _ C 1 R ) x 2 6 ^ _ 26 c ^  2t*C (1 - ( 2 4R - ° 2R) x 2^F) = 2 4 C * where 24»26Q I s t n e n u m b e r Q f events, counted with the 2k,26^g target, having a pion energy between the l i m i t s set to include the whole ground state peak (some of the f i r s t excited state i s therefore also included); 26,24j> i s the f r a c t i o n of f i r s t excited state between the same l i m i t s ; 26,24 R i s t n e r a t i o of f i r s t excited state to ground state peak heights i n f i t t i n g 2 6 » 2 1 + M g ; c 1 AR i s the r a t i o 1.81 MeV state to ground state peak heights i n f i t t i n g 1 2C; and C 2 R i s the r a t i o 1.34 MeV state to ground state peak heights i n f i t t i n g 1 2C. We wish to evaluate the contribution (a ) to the error of G a r i s i n g from the f i t s to the carbon data. Since C l R and C 2 R are evaluated from the same data (the peaks at 1.81 MeV and 1.34 MeV largely overlap since the standard deviation of the peaks is 0.7 MeV), the errors a and a C l R C 2 R are correlated and w i l l cancel to some extent in the ratio. cl c2 Let a = R and B = R. We wish to evaluate \ - °l (I'2 + i f '2+ © tf) 2 The problem is to find a „, the covariance, since a and 6 are the results a,8 r of separate f i t s to the same data. The value of a from the f i t is the same as would be obtained by taking the weighted mean of the values, a^, obtained by f i t t i n g each histogram bin separately ( a l l other parameters being fixed at their f i t values). That i s , both methods minimise the total x2. Therefore we may write a. i cr a. 1 + F 2 6 x ^ - F 2 6 x R 2 6 ? T -i az a G - F (1) 1 + F24 i _ F24 x R24 a 2 o. c x i i r i M i i " i I K i To find the terms o 2 , o 2 , a 2 0 , imagine N (many) measurements of the a i ^ a i P i histogram bin i . Each measurement w i l l give a value of a^, 6^  distributed about mean values of a. , 8.. Let the i measurement of a. be 1 ' K i J l a„ . = a. + 6a. . , and similarly for 8.; then we have i , J i i,3 ' ' H i 6 G _ ... ,&G , , .„ r5G c. X > J X X Using 5a. 58. 6a, . = 6y, . (x—) and 68. . = 6y, . — ) i , J i , J ^oy ; h i , j i , J ^ 5y ; tti where 6y. . = deviation of the i measurement of the number of counts in i,3 bin i from the mean, we have % = ° y. " y i > 56. p. y i ^5y J and a a 6 = y i ( ' <• ^ V p i 1 5y 5y assuming Poisson s t a t i s t i c s . Therefore a 2 „ = a a„ . a ±,6. a. 8 ± For the lineshapes used we have, in f i t t i n g to the carbon data, y ± = Y ± + a ,B x N x normal (x ±,u a p,co) , where is the number of counts at bin i from the f i t with a = 0; is the distance in MeV of bin i from the centre of the ground state peak (i.e., ground state = 0 MeV); u = -1.81, -1.34 MeV respectively; <x,B 1 1 X i ~ ^ 8 normal (x^ n »o>' , J ) = — exp {- j ( )} x b i n width; P co/2n N is the number of counts under the ground state peak; ct,B x N are the numbers of counts under the 1.81, 1.43 MeV peaks; and co is the standard deviation of the peaks. Therefore ba Joy = 1/(N x normal(x^,u ,co))« From (1) and (2) we have a2 ( 2 6 F / a 2 )2 0 2 ( 2 4 F / ( J 2 ) ( 1 + 2 6 F ( ~ _ 2 6 R ) ) 2 2 = y g i g i + B i P i  ~ i il — a + 2MF - 2 , +R))) 2 (I — a + 2hnv - 2 ^ R ) ) 2 ) 2 3 H J ( ( 2 6 F - 2 1 t F ) ( l + a - 2 6 F 2*R) _L_ !_) a 2 a 2  a i P i 2 a a x , , i L a i P i £ — £ — ( l - p " - 2 ^ 2 4 R ) 3 j a 2 k a 2 a. 6, After some manipulation and using J a a . J = a 2 a a . J we f i n d 4 • 4 (I'2 + i (If)+ 2 and by comparison we have V 1 /pG^ /-5G-. i % \ [*«' lw a,6 i 3 < k This procedure was tested by evaluating two known cases: 1. Where the two excited states do not overlap. Separating the two excited states by 4.5 MeV should give o a > f 3 = ^» s :*- n c e there i s no c o r r e l a t i o n and 4c - 4 (I) 2 + i ( f ) 2 Agreement here was within 0.1%. 2. The two excited states overlap completely. Putting the two excited states i n the same p o s i t i o n ( i . e . , performing the same f i t twice) should give a G c= 0, i f 2 6R =  2kR and 2 6 F = 2 H F . We a c t u a l l y obtained or = 0.0005: with o = 0.0137 and rr = 0.0142, G a 8 c K since the two f i t s were not identical because different starting points were chosen. In the real case of magnesium isotopes at 50° we get a = 0.0107, cr„ = 0.016; o„ = .0065, a 8 G r c whereas adding a - r — and aa -r^— in quadrature gives 'cr' = 0.0192. We CC Off p Op find, therefore, that the error of the cross section ratio introduced from the f i t s to the carbon data is negligible because of the correlations between the two f i t s . We have therefore a 'model independent' lineshape for the extraction of excited state contributions. APPENDIX 5 Reprint of Reference (Joh79) Neutron Radii Determinations from the Ratio of w Elastic Scattering from 1 J 1 3 C and , 6 l«0 R. R. Johnson, T. Masterson, B. Bassalleck, W. Gyles, T. Marks, and K. L. Erdman Physics Department. University of British Columbia, Vancouver. British Columbia V6TIW5, Canada and A. W. Thomas and D. R. Gill TRTUMF. Vancouver, British Columbia V6T1W5, Canada and E. Rost and J. J. Krauahaar'*' University of Colorado, Boulder, Colorado 80309 and J..Alster<" Physics Department, Tel-Aviv University, Ramat Aviv, Israel and C. Sabev*" CERN, 1211 Geneva, Switzerland and . J. Arvieux1*' Institut des Sciences Nucliaires, Institut National de Physique NuclSaire et de Physique des Particules, F3S044 Grenoble, France and M. Krell ( , ) Universiti de Sherbrooke, Sherbrooke. Qufbec J1K 2R1, Canada (Received 7 May 1979) Differentia] elastic cross-section ratios and absolute cross sections nave been measur-ed for 1]C at 29.2-and 49.5-MeV average »" energy and for '*• "o at 29.2 MeV. Range telescopes detected the scattered pions. The ratio data were compared with different op-tical-potential calculations to extract neutron radii of 2.35* 0.03 tm for 1JC and 2.81* 0.03 fm for " O , relative to the neutron radii of l ! C (2.31) and i e O (2.60), respectively. Our studies Indicate little sensitivity to the optical model used. A long-standing question of nuclear structure ly scattered IT" at low energy from a pair of iso-concerns the neutron density distribution which topes, one of which has reasonably well-estab-is not nearly as well known as the proton distri- lished neutron and proton density distributions, button. Various methods have been applied but We measured the ratio of the differential cross there are still major discrepancies between the sections for the pair since both systematic er-results obtained by different techniques.1 Here rors in the data and uncertainties In the theory we consider a new method that Involves the meas- cancel to a large extent. In the low-energy re-tirement of the angular distributions of elastical- gion the v'n elastic-scattering amplitude is much 844 © 1979 The American Physical Society V O L U M E 4 3 , N U M B E R 1 2 PHYSICAL REVIEW LETTERS 1 7 S E P T E M B E R 1 9 7 9 larger than that for ir'p mainly because cf a can-cellation in the i-nucleon p wave. This feature occurs in any reasonable low-energy pion-nucle-us optical model and we shall see that this makes the results insensitive to the precise form at the model employed. A similar experiment has been carried out on '^Li and U , U C using 50-MeV if V However, the results showed little sensitivity to the neutron density distribution. At low energies, where the v-N interaction is relatively weak, one can develop an optical poten-tial in terms of the density of nucleons and medi-um-corrected IT -N scattering amplitudes.'"7 Two different n -nucleus optical potentials were used in our analysis, that of Strieker, McManus, and Carr (SMC)3 and that of DiGiacomo, Rosenthal, Host, and Sparrow (DRRS).8 The SMC form is closely related to that derived by Ericson and Ericson"'9 for pionic atoms. It contains, in addition to the usual impulse (or Kisslinger potential) term, corrections due to Fermi motion, Pauli blocking, true pion S- and P-wave absorption, and a P-wave Lorentz-Lor-enz effect. Details are described in the SMC paper.3 We have used the SMC parameter set 1 unless otherwise stated. The DRRS form3 is also taken from the theory of pionic atoms extended to positive energy. The most significant differences are the inclusion of i-N phase shifts rather than scattering lengths, a different relativistic reduction (angle transfor-mation), and a phenomenological extraction10 of the S-wave absorption term using an analysis of 50-MeV IT* scattering from many target nuclei. We have taken the optical parameter set from Ref. 5 in which there is no P-wave absorption and the Lorentz-Lorenz parameter is zero (DRRS-A). Each of the ir-nucleus potentials, SMC set 1 and DRRS-A, fits ir* elastic-scattering data quite well for nuclei from "C to a MPb. This strongly suggests that variations in parameters between neighboring nuclei will be negligible. Neither code includes spin flip, but an impulse approxi-mation gave a maximum (incoherent) contribu-tion for l3C of less than 0.02 mb/sr. For the mat-ter density we used a modified harmonic oscilla-tor form fitted to electron scattering." For the tf -Z nuclei l2C and "O we assumed equal neutron and proton density distributions as is reasonable from their closed-shell character. It should be noted that low-energy pions determine only low-? features of the neutron distribution and thus a smooth density form should be adequate. As an additional test of model dependency, we have allowed the S- and P-wave absorption pa-rameters, the Lorentz-Lorenz term, and (for SMC only) the v-N phase parameters to vary freely to obtain a best fit to the 12C and "O rr" an-gular distributions. Then, using these sets of pa-rameters, the ratio data were fitted by varying the neutron and proton radii of 13C and iaO. The experiments were performed at TRIUMF on the MS and Af9 pion channels. The details of the range telescopes used in this experiment, as well as the beam-monitoring devices, have been reported elsewhere."113 The major emphasis in this experiment was to determine the ratio of the differential cross section for the two isotopes of interest as a function of angle. For this purpose data were accumulated on identically shaped tar-gets of natural U C (or water) immediately fol-lowed by runs with a 99.7% U C target (or 99.5% H20") and then with an empty target frame. The ratios hence involved essentially none of the un-certainties associated with the absolute normal-ization of the data and removed most systematic errors as well. Corrections for scattering to excited states were based on the measured efficiency for detec-tion of lower-energy pions (« 10%), and used dif-ferential cross sections determined with a dis-torted-wave Born-approximation calculation. These corrections were smaller than 1.5%. Cor-rections for tr" elastic scattering from hydrogen 8cm (dag) 8cm (dag) FIG. 1. Angular distributions for the elastic scat-tering of *" bom " C and " C (on the left) at 29.2 MeV. The ratio of the " C to the " C cross sections are shown on the right. The curves are the best-fit calculations described In the text with the DRRS code (solid line) and SMC code (dashed line). The dot-dashed curve Is for SMC with r .("c) = r , ( » 0 . 84S V O L U M E 4 3 , N U M B E R 1 2 PHYSICAL REVIEW LETTERS 1 7 S E P T E M B E R 1979 20 60 100 140 0cm (deg) 20 140 60 100 0cm (deg) F I G . 2. A n g u l a r dis t r ibut ions and rat io for > 3 C at 49.5 M e V . T h e c u r v e s a r e as d e s c r i b e d In F i g . 1. in the water target were found to be negligible, as were the corrections for scattering to the 6.6-MeV 1 60 state. Corrections for scattering to the 1.98-MeV ,0O state reached a maximum of 3.5% at 150°. The measured differential cross sections and their ratios are shown in Figs. 1-3. The absolute normalization uncertainties in the differential cross sections are ± 15%. Relative errors in-clude statistics and a T% systematic error. Our previously published iaC results" have been fold-ed in with the new results to give the U C differen-tial cross section. The errors in the ratios are statistical only. •"20 60 100 WO a cm (deg) The optical-potential parameters DRRS-A and SMC set 1 were each used to calculate differen-tial cross sections. The cross-section ratios were calculated by varying the neutron radii for 13C and laO until a best fit was obtained (typically X2/N * 2). The neutron radii found to give the best fit are given in Table I. The absolute differential cross sections pre-dicted by these parameters were as much as 40% low at back angles for carbon at 50 MeV and the UC and "O absolute differential cross sections were fitted by allowing the absorption and Lorentz-Lorenz parameters (and the v-N phases for the SMC code) to vary freely until good fits to the an-gular distributions were obtained. S- and P-wave absorption parameters were varied about ± 30% for both codes. The Lorentz-Lorenz parameter varied from 0.5 to 1.0 for SMC and from 0 to 0.6 for DRRS. The analysis of the ratio data was repeated and statistically very well-defined values for the neu-tron radii of the neutron-rich isotopes were ob-tained. The radii determined from DRRS-A and SMC set 1 as well as from the parameter sets found with the best fits at each energy are listed in Table I. For carbon, radii found by combining both the 30- and the 50-MeV data are also listed. All radii determinations assume the neutron ra-dius of the comparison isotope f c or 10O) equal to its proton radius. We feel that a comparison of the neutron radii extracted from an analysis in these two models should give a fair indication T A B L E I. Neutron r m s r a d i i (in f e r m l s ) for l 3 C and " O obatalned for v a r i o u s o p t i c a l - m o d e l calculations— a s s u m i n g r„ (-r>) - 2.31 f m for 1 J C , and r , ( = r „ ) » 2.60 fa for ' « 0 . P a r a m e t e r set " 0 D R R S - A 2 .32*0.02 2.82*0.02 D R R S , 3 0 - M e V fit 2.32 * 0.02* 2.76*0.02 D R R S , 5 0 - M e V a t 2 .35*0 .01 D R R S , c o m b i n e d 2 .34*0 .01 D R R S , m e a n 2 .34*0 .01 S M C (set 1) 2 .33*0 .02 2.84*0.01 S M C , 3 0 - M e V fit 2 .41*0 .02 2.83*0.01 S M C , 5 0 - M e V fit 2 .35*0 .01 S M C , c o m b i n e d 2 .36*0 .01 S M C , m e a n 2 .36*0 .03 O v e r a l l m e a n 2 .35*0 .03 2.81*0.03 F I G . 3. A n g u l a r dis t r ibut ions and rat io for the e l a s -tic scat ter ing of » ' f r o m " O and " O at 29.2 M e V . T h e c u r v e s are as d e s c r i b e d in F i g . 1. ' O n l y this D R R S fit had a L o r e n t z - L o r e n z and P-wave absorpt ion fit dif ferent f r o m 0. A l l others had best Bis with these p a r a m e t e r s equal to z e r o . 846 VOLUME 43, NUMB EM 12 PHYSICAL REVIEW LETTERS 17 SEPTEMBER 1979 of the model independence. The absence of the Lorentz-Lorenz effect in all but one of the DRRS calculations is particularly significant because Gibbs, Gibson, and Stephenson14 have shown that this correction changes the effective nuclear size 1 A particularly interesting feature of the analy-sis with both models is that even if the proton ra-dii are allowed to vary far outside the region con-sistent with electron-scattering experiments for "C and leO, the neutron radius which gives the best fit to the ratio data does not change appreci-ably. We are consequently making direct meas-urements of the neutron radius relative to "C and 180, respectively, and not the difference between a neutron and a proton radius. The statistical un-certainty in the determination of a neutron radius for each set of parameters is very small. A more meaningful estimate of both the statistical errors and the parameter dependence is given by the standard deviation of these radii from the mean. This is quoted as our final error. There are no known experimental determina-tions cf the neutron rms radius for 1SC. For leO there have been several recent determinations. A total-cross-section measurement reported a difference of rms neutron and proton radii for laO of 0.19± 0.02 fm.1! Note that they assumed r,(180) =r,(leO) =r„(180). This corresponds to (rn»)'« = 2.86±0.03 fm when the laO <r,»)lrt of 2.67 fm is used, and is in agreement with our re-sults. The elastic-scattering results cf Iversen et al." at 164 and 230 MeV indicate that the neu-tron-proton radius difference for leO is * 0.03 ± 6.04 fm. The results cf Jansen et al." at 160 MeV are not inconsistent18 with ours. It should be noted that the various experiments were done at energies varying from 30 to 230 MeV and con-sequently sampled different regions cf the nucle-ar density.1"'20 Our method for determining neutron radii from the ratio of IT" elastic-scattering differential cross sections is relatively insensitive to the pro-ton radius of the isotope and to the model used for the calculation. For the cases studied, this experiment gives the best measurement of neu-tron radii of neighboring isotopes known to date. This research was supported in part by the Na-tural Sciences and Engineering Research Council of Canada. "TRITJMF visitor. 'R. C . Barrett and D. F . Jackson, Nuclear Sizes and Structure. (Clarendon, Oxford, 1977); M . M . Sternhelm and Kwang-Bock Too, Phys. Rev. Lett. 41, 1784 (1978). 'S. A . Dytman, J . F . Amann, P. O. Barnes, J . N. Craig, K. G. R. Doss, R. A. Elsenstein, J . D. Sherman, W. R. Wharton, G. R. Burleson, S. L. Verbeck, R. J . Peterson, and H. A . Thlessen, Phys. Rev. C 18, 2316 (1978). S K. Strieker, H. McManus, and J . Carr, Phys. Rev. C 1£, 929 (1979). *RT H. Landau and A. W. Thomas, Nucl. Phys. A302. 461 (1978). S N . J . DIGlacomo, A. S. Rosenthal, E . Rost, and D. A. Sparrow, Phys. Lett. 66B. 421 (1977). •M. Thies, Phys. Lett. 63B, 43 (1976). ' B . Preedom, in Proceedings of the Seventh Inter-national Conference on High Energy Physics and Nu -clear Structure, edited by M . T . Locker (Birkhauser, Basel, 1977), p. 119. " M . Ericson and T . E . O. Ericson, Ann. Phys. (N.Y.) 36, 323 (1966). ~ ~ * M . Krell and T . E . O. Ericson, Nucl. Phys. B l l , 521 (1969). 1 0 A . S. Rosenthal, Ph.D. dissertation, University of ' Colorado, 1978 (unpublished). " C . W. De Jager et al.. At. Data Nucl. Data Tables 14, 479 (1974). 1 2 R. R. Johnson, B. Bassalleck, K. Erdman, B. Gyles, T . Marks, T . Master son, D. R. Gil l , and C. Sabev, Phys. Lett. 78B. 560 (1978). 1 3 R. R. Johnson, T . Marks, T . G. Masterson, B. Bas-salleck, K. L . Erdman, W. Gyles, D. Gil l , and C. Sabev, Can. J . Phys. 56, 6, 775 (1978). " W . R. Gibbs, B. F. Gibson, and G. J . Stephenson, Jr . , Phys. Rev. Lett. 39, 1316 (1977). 1 S M . D. Cooper, ln Meson -Nuclear Physics—1976, edited by P. D. Barnes, R. A. Elsenstein, and L. S. Kisslinger, AIP Conference Proceedings No. 33 (American Institute of Physics, New York, 1976), p. 237. " S . Iversen, A . Obst, H. Nana, K. K. Seth, C. L. Mor-r is , N. Tanaka, R. A. Thlessen, K. Boyer, W. Cottin-game, E. Moore, R. Boudrle, and D. Dehnhard, Phys. Lett. 62B. 51 (1979). " J . Jansen, J . Zichy, J . P. Albanese, J . Arvieux, J . Bolger, E . Boschitz, C. H. Q. Ingram, and L. Pflug, Phys. Lett. 77B. 359 (1978). " J . P. Maillet, J . P. Dedonder, and C. Schmit, Instltut de Physique Nucleaire, Orsay, Report No. IPNO/TH 79.8, 1979 (unpublished). " M . Johnson and R. Bethe, Comm. Nucl. Part. Phys. 8, 75 (1978). I 0We have repeated this analysis using a Fermi dis-tribution for " C . Though unrealistic for a light nucleus, this calculation offers a severe test of what neutron-distribution parameter we are measuring. The neutron rms radius deduced with this form agreed with those in Table I to within 0.01 fm. 847 Appendix 6 Target Ladder The targets were mounted on a target ladder, allowing the targets to be changed quickly from the experiment control room. The t o t a l distance of t r a v e l was 16 cm, which allowed up to f i v e targets, each 4 cm i n height, to be used. The target ladder was driven by a chain powered by a synchronous motor with a clutch and brake. The target p o s i t i o n was given by the voltage from a potentiometer which was turned by the drive chain. This voltage was compared, with the voltage comparator c i r c u i t , to one of a set of reference voltages provided by h e l i c a l potentiometers. The voltage comparator output was fed to two t r i g g e r s , one of which switched on the motor, depending on whether the comparator output was above 0.5 V or less than -0.5 V. The sign of the voltage comparator output therefore determined the d i r e c t i o n of motion of the ladder. The target moved i n the righ t d i r e c t i o n u n t i l the comparator output f e l l between -0.5 V and 0.5 V, at which point the power to the motor and the brake was switched o f f . The brake was applied when no current was supplied to the brake solenoid. The IV 'window' i n which current to the motor was switched off was needed to prevent the ladder from o s c i l l a t i n g about the required p o s i t i o n due to time delays i n stopping. The ladder p o s i t i o n was reproducible to about 0.2 mm independent, of the d i r e c t i o n of t r a v e l . The supply voltage to a l l of the potentiometers was the +12V from a NIM bin. Since the same supply voltage was used for both potentiometers i n the comparison, the target p o s i t i o n was i n s e n s i t i v e to gross changes i n t h i s voltage. Each target p o s i t i o n was adjusted by means of the relevant reference potentiometer. Remote switching of the reference voltage, through an analogue switch, i n i t i a t e d a target change which took about 2 seconds for 4 cm t r a v e l . An LED was l i t when current was applied to the motor so that a s t i c k i n g mechanism would be signaled by the l i g h t remaining on. Since power i s applied to the motor for short periods only, i t i s possible to run the ladder i n a vacuum without overheating problems. The whole ladder assembly was rotated by turning a s t a i n l e s s s t e e l supporting rod. The angle with respect to the spectrometer was read from a pointer fixed to the rod and a c a l i b r a t e d scale fixed to the top of the scattering chamber. The angle could be read to better than 1° accuracy. PUBLICATIONS WILLIAM GYLES R.R. Johnson, B. Bassaleck, K. Erdman, W. Gyles, T. Marks, T. G. Masterson, D.R. G i l l and C. Sabev, Elastic Sattering of it" from 1 2 C at 29 MeV, Phys. Lett. 75B, 560 (1978). R.R. Johnson, B. Bassaleck, K. Erdman, W. Gyles, T. Marks, T.G. Masterson, D.R. G i l l and C. Sabev', %~ Elastic Scattering on 2 u 8Pb at 29 MeV, Can. J. Phys. 56, 775 (1979). R.R. Johnson, T.G. Masterson, B. Bassaleck, W. Gyles, T. Marks, K.L. Erdmari, A.W. Thomas, D.R. G i l l , E. Rost, J.J. Kraushaar, J. Alster, C. Sabev, J. Arvieux and M. Krell, Neutron Radii Determinations from the Ratio of it" Scattering from 1 2> 1 3C and 1 6> 1 80, Phys. Rev. Lett. 43, 844 (1979). B.M. Barnett, W. Gyles, R.R. Johnson, K.L. Erdman, J.J. Kraushaar, S. Lepp, T.G. Masterson, E. Rost, D.R. G i l l , J. Alster, I. Navon, R.H. Landau and A.W. Thomas, Proton Radii Determinations from the Ratio of T& Elastic Scattering from A iB and 1 2B, Phys. Lett. 97B, 45 (1980). D. R. G i l l , K.L. Erdman, E.W. Blackmore, W. Gyles, B.M. Barnett, C. Oram, R.R. Johnson, T.G. Masterson and N. Grion, Elastic Scattering of 13.9 Mev Positive Pions from i 2C, Phys. Rev. C26, 1306 (1982). I. Navon, D. Ashery, J. Alster, G. Azuelos, B.M. Barnett, W. Gyles, R.R. Johnson, D.R. G i l l , T.G. Masterson, True Absorption and Scattering of 50 Mev Pions, Phys. Rev. C28 2548 (1983). 

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