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UBC Theses and Dissertations

29 MeV elastic scattering differential cross section ratio of 12C/13C 1979

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2 9 M e V TT 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 1 2 C / 1 3 C . by WILLIAM GYLES B.Sc. U n i v e r s i t y of Manchester, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 1 9 7 9 @ W i l l i a m Gyles, 1979 I n 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 t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e 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 , I a g r e e t h a t t h e 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 t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f fHYS ICS The 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 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 i i ABSTRACT The large fluxes of positive and negative pions available in meson f a c t o r i e s over the l a s t few years have lead to an accumulation of precise n-nuclear data over wide ranges in energy and atomic number. L i t t l e nuclear structure information has been extracted from the data since a microscopic model f o r the pion i n t e r a c t i o n in the nucleus i s not f u l l y developed. However the e l a s t i c scattering d i f f e r e n t i a l cross sections for. low energy pions are predicted well over a wide range of atomic mass using a potential i n which some parameters are empirically derived., since the. potential i s semi-empirical some nuclear structure information can not r e l i a b l y be derived d i r e c t l y from the data. Measurement of differences i n the nuclear structure between neighbouring nuclides, however, should be r e l i a b l e i f the potential produces the correct variation of d i f f e r e n t i a l cross section in t h i s mass region. In the experiment reported here the d i f f e r e n t i a l cross section r a t i o for e l a s t i c scattering of 29 MeV tr - on 1 3 C / l 2 C i s measured using s c i n t i l l a t o r range telescopes. Solid carbon targets of pressed powder were used.. since only a r e l a t i v e measurement i s made the errors in the r a t i o are s t a t i s t i c a l only. A large peak i n the d i s t r i b u t i o n of the cross section i i i r a t i o i s produced by changes i n the s-p i n t e r f e r e n c e minimum between 1 2 C and » 3 C . The c r o s s s e c t i o n r a t i o i s s e n s i t i v e to the neutron d i s t r i b u t i o n of 1 3 C because of the l a r g e s-wave i n t e r a c t i o n of the pion with neutrons i n the n u c l e u s . A measurement of the neutron rms. r a d i u s of 1 3 C and some t e s t s of dependence on the shape of the neutron d i s t r i b u t i o n and o p t i c a l p o t e n t i a l parameters are made. The neutron d i s t r i b u t i o n rms. r a d i u s of 1 3 C i s found t o be 2.365±.025 fm. iv Table of Contents Chapter I. Introduction 1 Chapter I I . Apparatus and Experimental Technique 8 1. a) Apparatus 8 b) Targets 20 2. Experimental Procedure - 21 Chapter I I I . Analysis and Results 24 1. Analysis 24 2. Results • 32 Chapter IV. Theoretical Interpretation 38 1. Point Nucleus Impulse Approximation 38 2. Optical Potential Calculations 42 a) Core + Valence Neutron Model 47 b) Neutron Distribution rms. Radius 51 Chapter V. Summary and Discussion 59 Bibliography 61 V L i s t of Tables Table I. Eesults for Ar=r n-rp of * 8Ca 3 ( Taken from Varma and Zamick8>. ) Table I I . Dimensions of s c i n t i l l a t o r s . 16 Table I I I . I n e l a s t i c scattering contribution to R. ...... 33 Table IV. Results. - 34 Eesults ( continued ) 35 Table V. E l a s t i c scattering cross section r a t i o s 36 of » 3 C / i 2 c Table VI. Optical potential parameters 44 Table VII. Eesults of c a l c u l a t i o n s using d i f f e r e n t ...... 53 neutron d i s t r i b u t i o n s with equal rms. radius. v i L i s t of Figures Figure 1. E l a s t i c scattering d i f f e r e n t i a l 6 cross section r a t i o s . Figure 2. M8 beamline 9 Figure 3. Time of f l i g h t spectrum of incident beam .... 10 Figure 4. Eeam p r o f i l e s 12 Figure 5. Experimental layout 12 Figure 6. Range telescopes 14 Figure 7. Range d i s t r i b u t i o n s of pions i n telescopes .. 17 Figure 8. Experimental and calculated ., 18 stopping patterns Figure 9. Experimental and calculated 20 stopping patterns at 0°. Figure 10. Electronic logic 22 Figure 11. Time of f l i g h t spectrum from arm 1 25 Figure 12. E vs. dE/dX scatterplot showing cuts made ... 26 Figure 13. E vs. dE/dX scatterplot at 30° 28 Figure 14. E l a s t i c scattering cross section 37 r a t i o s of i 3 C / i 2 C Figure 15. Comparison of * 3C and 1 2 C d i f f e r e n t i a l 39 cross sections. Figure 16. Impulse approximation calculations of fi ..... 41 Figure 17. Calculations of 1 2 C cross sections .......... 45 with parameter sets A, B and C. Figure 18. Optical potential calculations of R. ........ 49 Valence neutron + core model. Figure 19. X z curves for valence neutron + core model .. 50 v i i F i g u r e 20. 1 3 C neutron d i s t r i b u t i o n s with equal 52 rms. r a d i u s Figure 21. tz contour p l o t from o p t i c a l p o t e n t i a l 54 c a l c u l a t i o n s using parameter set ft. F i g u r e 22. R e s u l t s c f o p t i c a l p o t e n t i a l ................ 56 c a l c u l a t i o n s with l 3 r p =2.306 fm. F i g u r e 23. B e s u l t s cf o p t i c a l p o t e n t i a l 57 c a l c u l a t i o n s with * 3r p=2.240 fm. F i g u r e 24. X 2 curves of o p t i c a l p o t e n t i a l .............. 58 c a l u l a t i c n s with parameter s e t s A, B and C. v i i i ACKNOWLEDGEMENTS I am pleased to have t h i s opportunity of thanking my Research Supervisor, Dr. Richard R. Johnson, f o r his guidance and encouragement during the course of t h i s work. I would also l i k e to express my thanks to the other members of the PISCAT group, for t h e i r valuable assistance , and to the Batho Memorial Biomedical F a c i l i t y , f o r the use of the i r pion channel. 1 CHAPTER I Introduction Since the advent of the meson fa c t o r i e s ( LAMPF, TRIUMF, SIN) a great deal of data on low energy pion-nucleus scattering has accumulated . Pion-nucleus o p t i c a l potentials have been developed to the extent that they can reasonably predict the cross sections f o r e l a s t i c scattering over a wide energy range and over a large spread i n atomic weight . Part of the impetus f o r t h i s e f f o r t has been the p o s s i b i l i t y of studying nuclear structure with pions, but so far t h i s goal has hardly been achieved. The pion-nucleus interaction has proved to be a complex mechanism requiring many factors to be taken into account to produce f i t s to the data. This, coupled with the lack of structure i n the cross sections, makes i t d i f f i c u l t to separate nuclear structure e f f e c t s from the uncertainties in the i n t e r a c t i o n . In p a r t i c u l a r , i f o p t i c a l potential parameters are f i t t e d to experimental pion-nucleus data, the nuclear structure information that can then be extracted from the f i t i s l i m i t e d . Ideally, a potential would be derived s t r i c t l y from consideration of the multiple scattering of a pion i n the nucleus and a l l parameters f o r such a potential derived from the pion-nucleon scattering amplitudes . I t i s necessary, however , to include such nuclear effects as absorption on two nucleons 1) and short range correlations between nucleonsi»2> ( the Ericson-Ericson/Lorentz-Lorenz e f f e c t ) . This introduces parameters that are usually determined by pion-nucleus 2 experiments 3**' 5' 6' 7>. Since the c r o s s s e c t i o n s g e n e r a l l y show l i t t l e s t r u c t u r e , v a r i a t i o n s i n the parameters may mimic n u c l e a r s t r u c t u r e d e t a i l s such as d e n s i t y d i s t r i b u t i o n s , so these d e t a i l s can not r e l i a b l y be d e r i v e d d i r e c t l y from the experimental c r o s s s e c t i o n . The experiment reported here measures 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 of i r - on 1 2 C and 1 3 C at 29 MeV. By making a r e l a t i v e measurement i t i s suggested t h a t n u c l e a r s t r u c t u r e e f f e c t s can be i s o l a t e d . I n a c c u r a c i e s i n the p o t e n t i a l used should c a n c e l to a l a r g e extent. Any s y s t e m a t i c measurement e r r o r s w i l l a l s o c a n c e l each other out, thus removing n o r m a l i s a t i o n e r r o r s and v i r t u a l l y e l i m i n a t i n g the problem of long term i n s t a b i l i t i e s i n the apparatus. Systematic measurement e r r o r s i n the r a t i o are thus n e g l i g i b l e compared to s t a t i s t i c a l e r r o r s . In Chapter IV ,a comparison of the 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 of * 2C and 1 3 C shows the neutron d i s t r i b u t i o n of 1 3 C . D i s c r e p a n c i e s e x i s t at present between neutron rms. r a d i i measured using d i f f e r e n t techniques. For example, i n Table I i s a summary of r e c e n t experimental values of A r = r n - r p i n * 8 C a ( t a b l e taken from r e f e r e n c e 8 ). H a r t r e e - f o c h c a l c u l a t i o n s 1 6 > p r e d i c t A r ~ 0.2 fm. , which agrees o n l y with the high energy proton and <* p a r t i c l e r e s u l t s . Accurate and r e l i a b l e values of r^ would be a u s e f u l t e s t f o r t h e o r e t i c a l approximations used i n d e n s i t y dependent Hartree Foch c a l c u l a t i o n s . The pion might seem t o be an obvious choice f o r s t u d y i n g neutron d i s t r i b u t i o n s because o f i t s d i f f e r e n t i n t e r a c t i o n s with neutrons and protons. Indeed s e v e r a l attempts have been Table Eesults for A r = r n - r ^ of * 8Ca. Taken from Varma and Zamick. 8* Method | A r (fm) | Bef. | j -+ 1 T — p scattering <! 1.05 GeV | 1.0 GeV | 10.8 to | 16.3 MeV | .19±.05 .21±.05 .39±.10 I 9 I I 1 0 1 1 1 1 1 oi e l a s t i c < scattering 79 MeV | 166 MeV | 1.37 GeV | .03±.08 .38±.12 .20±.06 j 12 | 1 1 3 1 1 1 4 1 TI t o t a l cross sections 90 to | 240 MeV | L .08±.05 1 1 5 1 i . i 4 made to e x p l o i t t h e i s o v e c t o r n a t u r e of 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 n t h i s way. , B a t i o s of c+ and fl- t o t a l c r o s s s e c t i o n s were m e a s u r e d 1 7 * 1 8 * a t e n e r g i e s near 1 GeV ..At t h i s energy, t h e c-p ( or ti+n) i n t e r a c t i o n i s about 3 t i m e s s t r o n g e r t h a n the n-n ( o r fl+p ) i n t e r a c t i o n . The r a t i o 6*/6~ # however, i s i n s e n s i t i v e t o r e a s o n a b l e v a r i a t o n s i n the neut r o n d e n s i t y d i s t r i b u t i o n s . Because o f t h i s i n s e n s i t i v i t y and because o f t h e dependence on c a l c u l a t i o n s of Coulomb e f f e c t s , t h e d e n s i t y d i s t r i b u t i o n s e x t r a c t e d from t h e s e measurements are not r e l i a b l e . When measuring the d i f f e r e n c e s i n r a d i i between n e i g h b o u r i n g i s o t o p e s many u n c e r t a i n t i e s i n t h e exp e r i m e n t and t h e o r y a r e c a n c e l l e d . T h i s method was used by C o o p e r l 9 > t o e x t r a c t n e u t r o n rms r a d i i f o r l 8 0 and 4 8 C a by comparing t h e i r p i o n t o t a l c r o s s s e c t i o n s i n the 100—200 MeV range w i t h t h o s e of 1 6 0 and *°Ca r e s p e c t i v e l y . At t h e s e e n e r g i e s , the n u c l e u s i s e s s e n t i a l l y a b l a c k d i s c because o f the s t r o n g 3,3 r e s o n a n c e , t h u s the r e s u l t s were c o n s i d e r e d i n s e n s i t i v e t o t h e i n t e r a c t i o n model used. Jansen e t a l 2 0 > have compared T T * and T f - 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 s from 1 8 0 and * 8 C a a t t h e 3,3 resonance and e x t r a c t e d n e u t r o n r a d i i . As t h e i n t e r a c t i o n i s so s t r o n g the p i o n samples o n l y the s u r f a c e of the n u c l e u s , so t h e c r o s s s e c t i o n s a r e not s e n s i t i v e t o low moments o f the d i s t r i b u t i o n such as the rms. r a d i u s . A m a t t e r d i s t r i b u t i o n model must be used t o r e l a t e t h e e f f e c t i v e r a d i u s t o t h e rms. v a l u e . As p o i n t e d out by Ster n h e i m and Y o o 2 l > , however, t h e n e u t r o n rms. r a d i i e x t r a c t e d depend upon the p a r t i c u l a r s e t of o p t i c a l parameters used. T h i s i s caused by t h e s i m p l e 5 d i f f r a c t i v e nature of the s c a t t e r i n g , which i s determined l a r g e l y by the product of k, the wavenumber i n the nucleus, and r , the e f f e c t i v e r a d i u s at which the a b s o r p t i o n o c c u r s . Changes i n the a b s o r p t i v i t y cause changes i n the e f f e c t i v e r a d i u s of the n u c l e u s , thus l e a d i n g t o f i t s f o r d i f f e r e n t neutron rms. r a d i i . Changes i n the r e a l p a r t of the p o t e n t i a l a l s o a f f e c t the value of k, again g i v i n g f i t s f o r d i f f e r e n t r a d i i . A necessary requirement i s that the p o t e n t i a l should a l s o p r e d i c t the measured t o t a l c r o s s s e c t i o n i f i t i s to independantly f i x the p o t e n t i a l parameters. One can a l l e v i a t e t h i s problem by using pipns of lower energy away from the 3,3 resonance. Here the nucleus i s not so a b s o r p t i v e , and the c r o s s s e c t i o n data are not g e n e r a l l y d i f f r a c t i v e . Dytman et a l . 2 2 > measured the 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 cf 1 2 C and * 3C with 50 MeV n+. F i g . 1 shows t h a t the d i s t r i b u t i o n i s q u i t e f l a t with only a s m a l l peak at low angles caused by a s h i f t i n the s,p i n t e r f e r e n c e minimum. I t may be a n t i c i p a t e d , by c o n s i d e r i n g the f r e e n-nucleon s c a t t e r i n g amplitude, t h a t a l a r g e r e f f e c t would be produced with n~. The s c a t t e r i n g amplitude may be parametrised f(6) = b 0 +bj X.x* (c 0 +c x t v t ) k.k« n e g l e c t i n g a s m a l l s p i n dependent term, and where the c o e f f i c i e n t s s-wave i s o s c a l e r s c a t t e r i n g length .005 fm s-wave i s o v e c t o r s c a t t e r i n g l e n g t h . 13 fm p-wave i s o s c a l e r s c a t t e r i n g volume .64 fm 3 p-wave i s o v e c t o r s c a t t e r i n g volume .43 fm 3 are determined from the tr-N s c a t t e r i n g l e n g t h s 2 3 * . F i g u r e 1. E l a s t i c S c a t t e r i n g D i f f e r e n t i a l Cross S e c t i o n E a t i o s of 50MeV n + on » 3 C / » 2 C . Data from D y tman. 2 2 ) 7 Tc and T c 1 are the wavevectors of the incoming and outgoing pion. "t and X are the isospin vectors of the nucleon and pion. t.T =+1 for n +p or n~n and -1 for n-p or -n+n. For the i s o s c a l e r nucleus, 1 2 C , only the i s o s c a l e r terms contribute . For H+n, the is o s c a l e r and isovector amplitudes cancel to a large extent, and the extra neutron of * 3C w i l l not have a large effect on the T I + cross section. With n~, however, the i s o s c a l e r and isovector amplitudes add. The r e l a t i v e proportions of s and p wave are altered from the completely i s o s c a l e r case because of the large s-wave isovector amplitude, therefore a large change i n the s-p interference minimum may be expected. Since the isovector amplitudes are e f f e c t i v e only where there i s a neutron or proton excess , the if should be sensit i v e to the difference between proton and neutron r a d i i . 8 CHAPTEB I I Apparatus and Experimental Technique. 1. a) Apparatus The experiment was performed on the TBIUMF b i o m e d i c a l pion beamline , M8. A sketch of the beamline i s shown i n F i g . 2. The pion k i n e t i c energy of the beam was 30 MeV with Ap/p=4.2%. P a r t i c l e f r a c t i o n s were pions=13%, muons=3% and electrons=84%. The proton c u r r e n t d u r i n g the week's run was t y p i c a l l y 7uA, g i v i n g a pion f l u x of about 1.4x10 s s — * . The beam f l u x was monitored p r i m a r i l y with a gas i o n i z a t i o n chamber, open t o the atmosphere, p l a c e d d i r e c t l y a f t e r the l a s t quadrupole. Although the i o n i z a t i o n chamber was c a l i b r a t e d to 1% a c c u r a c y 2 * * , the c a l i b r a t i o n was not needed f o r the r a t i o measurement, the only requirement being t h a t i t should be s t a b l e over the time r e q u i r e d t o perform the measurements on the three t a r g e t s ( about 3 t o 5 hours ) . The beam was a l s o monitored a f t e r the t a r g e t by a p a i r of l a r g e p l a s t i c s c i n t i l l a t o r s , M1 and M2, used i n c o i n c i d e n c e to e l i m i n a t e protons, a r i s i n g from the t a r g e t , which stop i n the f i r s t c o u n t e r . The s c i n t i l l a t o r s a l s o determined the muon and e l e c t r o n contaminations by measuring the f l i g h t time of the p a r t i c l e s from a c a p a c i t i v e pickup placed i n f r o n t of the pion p r o d u c t i o n t a r g e t , T2, i n the primary proton beam. A t i m e - o f - f l i g h t spectrum taken t h i s way shows n-'s, u's and e's c l e a r l y separated i n time ( F i g . 3 ). A p a i r of wire chambers g i v i n g X and Y beam p r o f i l e s were mounted behind the s c i n t i l l a t o r s M1 and M2. They we.re t r i g g e r e d 9 Figure 2 . M8 Beamline. C h a n n e l l e n g t h Momentum r a n g e Momentum a c c e p t a n c e Momentum r e s o l u t i o n P o l a r i t y TT f l u x a t 180 M e V / c , f u l l momentum a c c e p t a n c e 10 cm Be t a r g e t C o n t a m i n a t i o n a t 180 M e V / c i r " 10 cm c o o l e d B e t a r g e t D o s e r a t e t o a 5 x 5 x 5 c c f i e l d 8 m t a k e - o f f a n g l e 30" 0 - 2 2 0 M e V / c ±6.7% A P / P FWHM 1 . 5 * A P / P FWHM i r + o r IT" J 8 * 1 0 s i r + / s e c / y A p r o t o n ! 1 . 3 x 1 0 6 i r " / s e c / y A p r o t o n ~2k% e l e c t r o n s ~ 1 1 * muons ~ 0 . 2 r a d / m i n / u A p r o t o n F i g u r e 3. Time of F l i g h t Spectrum of I n c i d e n t Beam. 11 by the coincidence M1»M2 . The wire chambers and s c i n t i l l a t o r s , mounted on r a i l s , could be s l i d along the beam d i r e c t i o n . Beam p r o f i l e s were taken, with the corresponding beam tune, f o r both target positions. The beam p r o f i l e s at the target positions are shown i n F i g . 4 along with a contour plot generated by an on-line program. P r o f i l e s were also taken at i n t e r v a l s along the beam to estimate the.beam divergence , which was found to be approximately ±1.4°. F i g . 5 i s a diagram of the experimental layout. The detector arms, mounted on a table, pivot to rotate around the target centre though angles up to 90° for Arm 1 and 70° f o r Arm 2. Because of space r e s t r i c t i o n s in the experimental area, a table which would allow rotation of the arms to angles greater than 90° could not be used. To make measurements at these angles the table was turned around and the beam refocussed on the new target position. Negative pions cannot be recognised s o l e l y by dE/dX and t o t a l energy measurement as nuclear disintegration ( star formation ), cn stopping, destroys the energy signature of the pion . Tie pion range, however, i s a function of energy, and the e l a s t i c a l l y scattered pions may be separated by th e i r range i n a s c i n t i l l a t o r range telescope. The two range telescopes used ( see F i g . 6 ) differed only in the elements that stopped the e l a s t i c a l l y scattered pions. The f i r s t two counters i n each arm define the s o l i d angle and select only p a r t i c l e s coming from the di r e c t i o n of the target. The next thick counter almost stops the pions and defines the energy A E / A X . In Arm 1, the pions stop i n the next counter, whose thickness i s designed to F i g u r e H. Beam P r o f i l e s . Contour plot of the beam at downstream target po sition 13 F i g u r e 6. Range Telescopes. v; s;;;;x C1 V //// / A C 1 Arm1 Arm 2 r W / / / / / l C 2 C 3 I /// //A C 2 C 3 15 stop 28.4±3 MeV pions ( 30 MeV i n c i d e n t pion beam and 1.6MeV energy l o s s i n the t a r g e t ). In Arm 2 t h i s counter i s r e p l a c e d by f i v e t h i n counters , the t h i c k n e s s of each being e q u i v a l e n t to approximately 1 . 9 MeV d i f f e r e n c e i n pion energy. The f i n a l counter i n each arm serves as a veto f o r p a r t i c l e s t h a t pass through the stack. The dimensions of the s c i n t i l l a t o r s are given i n Table I I . The e f f e c t i v e s o l i d a ngles of the two t e l e s c o p e s were approximately 6 mb/sr (Arm 1) and 8 mb/sr (Arm 2) . The energy r e s o l u t i o n i s l i m i t e d by the t h i c k n e s s o f the stopping c o u n t e r s , range s t r a g g l i n g , r e s o l u t i o n of the i n c i d e n t beam and the energy spread caused by s c a t t e r i n g from d i f f e r e n t depths i n the t a r g e t f o r some geometries. The c o n t r i b u t i o n s to the range d i s t r i b u t i o n from these f a c t o r s superimposed on the s t o p p i n g counters , are shown i n F i g . 7. The r e s u l t i n g s topping p a t t e r n , along with the experimental s t o p p i n g p a t t e r n i s shown i n F i g . 8 . The c o n v o l u t i o n to combine the e f f e c t s on the range d i s t r i b u t i o n was done with a Monte-Carlo program 2 5> . A Gaussian shape was assumed f o r the range s t r a g g l i n g with A r = . 042 r 0 , where r„ i s the mean range. The r a t i o A r / r 0 has been t a b u l a t e d f o r protons over a wide energy r a n g e 2 6 * . The corresponding values f o r pions may be e v a l u a t e d from those of protons with the same range by accounting f o r the d i f f e r e n c e i n masses of pions and protons A r / r 0 f o r 30 MeV pions i s found to be .032. Using A E = A r d E / d r an i n t r i n s i c r e s o l u t i o n of 1.2 MeV FWHM i s obtained f o r the See, f o r example, E. D. Evans, The Atomic Nucleus, pg. 664. Table I I . Dimensions of S c i n t i l l a t o r s in Telescopes. .. ARM 1 i ! ARM 2 (Counter| i _ _ _ _ _ _ j _ _ Diameter (cm) |Thickness I (cm) I|Counter I + + + - | | C1 | Diameter (cm) |Thickness| I (cm) | •\ -i | 0.33 | 1 1 — | C1 | 3.0 l_ | 0.33 3.0 I C2 | 3.0 t 0.33 | | C2 | 3.0 | 0.33 | I C3 | 5.0 | 1.97 i i C3 I 5.0 | 1.27 | i s | 5.0 I 1-27 || S1-S5 | 5.0X5.0 | 0.425 | | VETO | 10.0X7.0 | 0,-640 || VETO | 10.0X7.0 | 0.640 | L —1 . j J _ _ x_ .j. J 17 F i g u r e 7. Range D i s t r i b u t i o n of Pions i n Telescope, Range d i s t r i b u t i o n s including effects of fa Range straggling B + 4.2% Ap/p C + target thickness Arm 1 stopping counter Arm 2 stopping counters F i g u r e 8 . Experimental and C a l c u l a t e d Stopping P a t t e r n s . Arm 2 at 60° -Target at 45° Experimental Calculated 51 S2 S3 S5 19 t e l e s c o p e s . F i t t i n g to the st o p p i n g p a t t e r n s of Arm 2 r e s u l t s i n the value of A r / r 0 =.042 used, corresponding t o an i n t r i n s i c r e s o l u t i o n o f 1.6 MeV FWHM. F i g u r e 9 compares the measured and c a l c u l a t e d stopping p a t t e r n s f o r 30MeV pions with A p / p=4.2% with the t a r g e t i n f r o n t of the t e l e s c o p e . The measured d i s t r i b u t i o n has a long range t a i l caused by charged n u c l e a r fragments from s t a r formation t r a v e l l i n g beyond the pion range, f u r t h e r l i m i t i n g the r e s o l u t i o n , b) Targets The 1 3 C and * 2C t a r g e t s were made by compressing carbon powders i n t o t h i n mylar frames. An empty frame was a l s o made, the t h r e e frames being as s i m i l a r as p o s s i b l e . The 5.17 cm. diameter c i r c u l a r rim of each t a r g e t , o f 0.019 cm. t h i c k mylar, was 0.85 cm. deep and the two windows were of 0.00025 cm. t h i c k mylar. The t h i c k n e s s of the windows represented l e s s than 0.2% of the mass/cm? of the t a r g e t s . Over 99% of the t a r g e t frame mass was at the cir c u m f e r e n c e , where the pion f l u x d e n s i t y was l e s s than 5% of the peak. The t a r g e t d e n s i t y was l i m i t e d by the requirement t h a t the u n c e r t a i n t y i n the pion energy on s c a t t e r i n g be l e s s than ±2 MeV. The area was then r e s t r i c t e d by the amount ( 7gm. ) of * 3C a v a i l a b l e . The t a r g e t d e n s i t i e s were .330 gm./cm.2 ( 1 3C) and .327 gm./cm.2 ( 1 2C) . The * 3C t a r g e t was of 99.7% i s o t o p i c p u r i t y while the 1 2 C t a r g e t was of n a t u r a l i s o t o p i c composition. Thin p l a s t i c h o l d e r s were glued t o the base of each t a r g e t to allow p r e c i s e r e l o c a t i o n on the s c a t t e r i n g t a b l e . 20 ( F i g u r e 9. Experimental and C a l c u l a t e d Stopping P a t t e r n s at 0°. r • I I l Experimental V Calculated a I 1 I I I S1 S 2 S 3 S 4 S 5 V E T O 21 2. Experimental Procedure Laser reference beams, which had previously been aligned by the biomedical group, were used to align our apparatus. V e r t i c a l and horizontal planar beams were aligned with t h e i r i n t e r s e c t i o n along the beam di r e c t i o n and close to the beam centre. A 3" thick lead block with a 3/8" hole d r i l l e d through was placed at the target position with the hole along the laser beam int e r s e c t i o n . A wire chamber p r o f i l e of the beam just behind the lead was taken, c l e a r l y showing the position of the hole. The lead was removed and another p r o f i l e taken so now the beam centre could be related to the laser i n t e r s e c t i o n . The table and target were aligned using the l a s e r system and aluminum blocks with relocating s l o t s were fix e d to the f l o o r to provide automatic alignment i f the table was moved.. An add i t i o n a l check could be made on the table alignment with the beam dire c t i o n using the approximate 1/sin*(9/2) dependence of the d i f f e r e n t i a l cross section at small angles. This i s discussed in Chapter I I I . A diagram of the elec t r o n i c l o g i c i s given i n F i g . 10. Many of the electrons were rejected by the discriminators, which were adjusted to accept a l l pions. In each arm an event was defined by the coincidence C1*C2«C3. The C2 counters were connected to constant f r a c t i o n discriminators and the timing of the event was defined by the output pulse of t h i s discriminator. The coincidence C1«C2«C3«BF, timed by the EF ( the output pulse of the capacitive pickup ), was used to st a r t the TDC's, which were stopped by the event pulse C1«C2«C3, and also to interrupt the computer. Time-of-flight / CN CN U •H m o U •rH c o 1-1 +J o 0) r H W o 01 M 3 cn • H ARM 1 Cl C 2 C 3 V E T O ARM. 2 S S1 S2 S3 S5 - C D - V E T O •CD- -czt 23 and analogue signals were recorded for each event i n CAMAC modules.,Bits were set i n a 24 b i t C212 pattern unit for each element of the two arms which triggered i n the event..The data were transmitted via a 2MHz s e r i a l CAMAC transmission l i n k to a NOV A 1200 computer where they were buffered and then recorded on a magnetic tape. A real-time FOETBAN program interrogated the buffer and on command displayed t i m e — o f - f l i g h t , pulse height or stopping pattern information while the experiment was i n progress. Each angle involved runs with each of the two carbon targets and with the empty frame. Since the width of the stopping d i s t r i b u t i o n depended on the target angle and the angle of scatter, the target angle was adjusted to the best compromise f o r the two arms. Each i n d i v i d u a l run was li m i t e d to less than 4 hours ( mostly l e s s than 2 hours ) so that any long term d r i f t s i n detector e f f i c i e n c i e s , ion chamber e f f i c i e n c y or pion f r a c t i o n would not be appreciable i n the: time taken for the two carbon targets. Near the minimum of the cross section (60°—70°) the targets were interchanged u n t i l enough events were c o l l e c t e d . 24 CHAPTER I I I A n a l y s i s and R e s u l t s 1. A n a l y s i s A modified KIOWA histogramming r o u t i n e on t h e OBC IBM 370 computer was used i n the a n a l y s i s of the da t a . T e s t s were made on each event and parameters of the event d i s p l a y e d on a histogram or s c a t t e r p l o t i f the t e s t s were s u c c e s s f u l . In t h i s way cuts could be made on t i m e - o f - f l i g h t , energy deposited i n C3 ( A E / A x ) , energy d e p o s i t e d i n the stop p i n g counters ( E ), or the absence of a veto s i g n a l . A t i m e - o f — f l i g h t spectrum from Arm 1 ( F i g . 11 ) shows the pions c l e a r l y separated from muons and e l e c t r o n s produced i n the pion production t a r g e t . The pion peak however c o n t a i n s protons from the carbon t a r g e t and muons from pion decays between the t a r g e t and d e t e c t o r s . Also a t low angles (<40°) many muons from pion decays i n the beam are d e t e c t e d . These p a r t i c l e s may be separated by c u t s on the E and A E / A X s i g n a l s . The stopping counter i s composed of 5 elements i n arm 2 and the s i g n a l s from these were added, weighted a c c o r d i n g t o the r e l a t i v e g a i n s . The r e l a t i v e g a i n s were found from the s i g n a l s t r e n g t h s of the 96.3 MeV/c e l e c t r o n s from the channel. Since these have a range of 39 cm. i n the s c i n t i l l a t o r they were assumed to d e p o s i t the same energy i n each of the sto p p i n g elements. S c a t t e r p l o t s of E vs. A E / A X of p a r t i c l e s i n the pion t i m e - o f - f l i g h t peak were produced, with the c o i n c i d e n c e requirement C1»C2»C3«C4»VETO. Muons and protons were separated by cuts on A E / A X ( F i g . 12) . Note t h a t the pions have a well 2 3 7 9 . 2 3 1 3 . 2 1 9 6 . 21 ii. 2 C 1<*. 2 0 1 3 . 1 9 5 2 . 1 0 4 1 . 1 1 . 3 0 . 1 7 t ; . 1 7 D P . 1 6 4 7 . 15 3 6 . 1 5 2 5 . 1 4 6 4 . 1 4 0 3 . 1 3 4 2 . 12 81 . 1 2 2 0 . 1 1 5 9 . 1 0 9 b . 1 0 3 7 . 9 7 6 . 9 1 5 . R54 . 7 e 3 . 732 . 67L . 5 4 9 . 4 3 d . * 2 7 . 3fc6. 3 0 5 . 2 4 4 . 1 fc 3 . 12 2 . 61 , 30l e l e c t r o n s i l l 11 3 P X X + X X X X X + 1 • X X X X + 2 2 2 4 C O X X X X X X X + P 4 unions • • X X X X X X X X X X X X xx X X » x x X X X A X X * x x x x A X X X A X X X x x x x A X X X A X X X A X X X A X X X A X X X G A X X X F x x x x x x X A X X X X X A X X X X X A X X X X X A X X X X X A X X X X X A X X X X X A X X X X • X A X X X X X X A X X X X * X X A X X X X X X X A X X X X X pions ••>. X X A X X X X X X X X A X X X X X X X X A X X X X X + 11 1 1 1 3 5 3 L X X X A X X X X X X X X X X A X X X X X X 4 1 5 Z 1 2 9 D A 5 4 2 5 7 5 5 & J P » X X X X A X X X X X X X r 6 2 2 3 L O l r . i F . B! N FOGF i i i m i i i i i i i u m m n m i m i n m i l i u i i i i m i | n : 5 4 4 4 ^ 5 S < ^ 5 6 6 f c 6 6 7 7 7 7 7 6 3 3 3 S 9 < : 9 9 9 0 0 0 C 01 1 1 1 1 2 2 2 2 2 5 3 3 3 3 4 4 4 4 4 5 5 5 5 1 6 6 6 66 7 7 7 7 7 3 3 8 e S ? 9 ? ^ 5 7 9 1 3 ^ 7 " 1 3 5 7 9 1 3 < 7 9 1 3 5 7 0 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 * 3 5 7 9 1 3 5 7 9 1 3 5 * Yc ̂ oooconorooooooooccoocoooorooccccccoocoooooooocooooooooojoooooojoocoocooooooootl c H fD H i H) M H- O B 5 (D I 3> O n H I B I N - » H • H> i Q r+ W T3 rt) O r+ H C B t o Ul 26 F i g u r e 12- E vs. A E / A X S c a t t e r p l o t Showing Cuts Made. J -« rg tr» I M rJ • O. f- f— «M . . <_ — e» *oW MA ^if\J - INJ x L U ! < ! (j. (T w CT- CO O ^ r- L, j> >o o ;y tn o o cr m c CT> tvJ o o> — o • o- o o CC ? O IC 13 C X! r»- O I tr >o O a •« o >̂ «J" o t) f* o CC — C • cc i" o f* (?• o i r— o I N h O I f- -G 0 r- m c r~ o f— fi o p» !*J o f- — o . o o 43 0s o n o T — ~> k i o o Lo m o _ O I ,o o in _ O r— O i in <o o tn x» ~• ir\ o tn •*» o in rsi o • in O O J c o s f x> o - o» r~ O «4- in O 4- — CD • 4- o n Im J u m a o o •0 o im ir. o . -4- C \ O m rj o i m —* o • r*i c- O M U D <M CD O CM f- C r\j C rsi m f~ (si -r T , :M f\* O i fV -* c > r v v O •i l_T O H N C -« «o o • in o i O i i*1. O I-* (Nt O . o c cr o CO c r- o -o o m c o m c ( M C J L U ir. ^ c ^ 3* CT- C  — u"\ c if. C> x 1!>n c m i. • *nio C if' c • j i.i rsj c_- r- tr. f N J ' f — J~. l_' f~ "-^ ^ C ^ ^ x to s r— r- r-jr— <c -o vO o ini*.* m ir 4 *r i -, * O o O ir. c IA c i^io in ^ if\ c ir> o • • • , t~ ir. c- o f-''-*\ rv n r- if. m o «n c> fnrnf^r-ir\Iirsjr>JfNJ—«—««j--if-inf\) • 27 27 d e f i n e d A E / A X but a l a r g e spread i n E, because of energy from n u c l e a r d i s i n t e g r a t i o n on s t o p p i n g . At angles l e s s than 45° mucns from pion decay i n f l i g h t are r e j e c t e d by making a cut on E to i n c l u d e only picns producing n u c l e a r d i s i n t e g r a t i o n s i n the s t o p p i n g counter ( F i g . 13). The p o s i t i o n of the c u t to be made i s found by examining the A E / A X vs. E s c a t t e r p l o t of the empty t a r g e t frame run which c o n t a i n s mostly muons at these an g l e s . In a l l cases the same c u t s were made f o r 1 2 C , * 3C and empty t a r g e t frame runs. The e f f i c i e n c i e s of the d e t e c t o r s f o r d e t e c t i n g e l a s t i c a l l y s c a t t e r e d pions was the same f o r the three t a r g e t s , assuming p h o t o m u l t i p l i e r gains d i d not change a p p r e c i a b l y d u r i n g the time f o r the three runs. 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 may be c a l c u l a t e d from the data using doydft = (HF /IC F -NB /IC 8 ) cos VS) SJ (0) / (Fft ej f DE^T ) where NF and Ng are the number of pions detected during t a r g e t i n and t a r g e t out runs, IC F and ICj are the corresponding i o n chamber readings, F i s a c o n v e r s i o n f a c t o r from the i o n chamber reading to the number o f pions i n c i d e n t on the t a r g e t , % i s the angle between the t a r g e t normal and the beam, Slejtis the e f f e c t i v e s o l i d angle, D i s the f r a c t i o n of pions r e a c h i n g the d e t e c t o r from the t a r g e t without decaying, En i s the e f f i c i e n c y of the d e t e c t o r arm f o r p i o n s , S i s the f r a c t i o n of the range d i s t r i b u t i o n of the e l a s t i c a l l y P I O N S V E T O , H M S C A f. I N 1111 i 11 11 12<!ii22?.'Z2i33}3333 iih','-^'. <,<,5555555 5 5 5 6 6 6 6 6 66 j 6 6 77 7 7 777 7 7 7 8 C o ? 3 OS 8 6 9 9 9 9 9 9 9 * 3 9 1 2 3 i « 5 6 7 H ' : " ~ 1 2 3 ' » 5 6 7 B < : f : i 2 3<f5t 7 8 ^ 0 1 2 3 4 5 6 78 901 2 3 4 5 6 7H901 2 345671) 701 2 34 56 / 8 9 0 1 2 3*56 7 8 9 0 1 , : 3 4 5 6 7 3 5 0 1 2 3 ^ 6 7 8 9 O O 3 C 0 0 C W O O J O C O O J O C J C O C )OOOCOOOOOCCCOOCOOOOOtOOO3OO3OO0OO 3 0 C 0 0 0 0 0 0 J O O O O O S C P O O r O O O w C O O O O O O O O O O W n O 0 0 0 0 3 0 C C O 0 00 0 300 C O O C O 0 0 0 0 O O O C O O C C C C O O J O 0 0 0 0 0 1CO0O0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O C C O 0 0 0 0 O J O O O O O O O O O O C ' 0 0 0 0 0 co 29 scattered pions covered by the stopping element J (0) i s the Jacobian dJl^j,/d5lt 0 m T i s the number of nuclei/cm 2 in the target. In c a l c u l a t i n g the d i f f e r e n t i a l cross section r a t i o however most factors cancel , leaving i 3 ( J / i 2 < j = B = [ i 3 N ( : / i s ^ - t y / i c , 3 / [ i 2 N F / i 2 I C F - % / I C j , ] * 2 T / i 3 T Superscripts 1 2 i 1 3 s i g n i f y quantities attributable to the run with 1 2 C , 1 3 C target respectively. J(8) i s the same for 1 2 C and * 3C to better than 0.02% at a l l angles . Uncertainties i n factors such as F »Jleff A N < * ETT therefore do not a f f e c t the r e s u l t s . S t a t i s t i c a l errors were calculated from &E/E= { [ 8 1 3N F M i n E ) / 6 1 3 N F J 2 + [ S L 2 N F M i n E)/d* 2N F J 2 +1 8N3 i(lnH)/dN 8 323 1 ^ 1 3 N F 12N F * 3N F - i 2 N f i 3 I C , 1 3N F - NB 1 3 i c F ICj 1 2 I C F 1 2 NF - N8 1 Z I C F I C 3 L 1 3IC f * 2 I C F Y 1 3N, \13X F _ - N6 V* 2N F ~ CF I C B ] i i 2 I C p where ox s i g n i f i e s the s t a t i s t i c a l error i n x. Where more than one set of runs had been taken at one angle the s t a t i s t i c a l l y weighted mean of the r a t i o s was calculated. In a l l such cases the d i s t r i b u t i o n of values for the r a t i o was consistant with s t a t i s t i c a l errors. The angle of the beam r e l a t i v e to the 0° l i n e of the 30 s c a t t e r i n g t a b l e was e v a l u a t e d f r o m t h r e e p a i r s o f r u n s w i t h t h e same arm a n d t a r g e t a t 30° l e f t a n d 30° r i g h t . A s s u m i n g t h e c r o s s s e c t i o n a t 30° v a r i e s as 1 / s i n * ( 0 / 2 ) f r o m E u t h e r f o r d s c a t t e r i n g , t h e n t h e r a t i o s o f l e f t a n d r i g h t c r o s s s e c t i o n s may be w r i t t e n (5 (9-C+X) /o*(9-C-X) = s i n * ( (9-C-X) /2) / s i n * ( (0-C+X) /2) C= t h e c o r r e c t i o n t o t h e mean s c a t t e r i n g a n g l e f o r f i n i t e s o l i d a n g l e ( s e e b e l o w ) . X= t h e a n g l e b e t w e e n t h e beam d i r e c t i o n a n d 0° on t h e s c a t t e r i n g t a b l e . W r i t i n g 9-C a s 9 C and l e f t , r i g h t 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 a s <5L i 6% r e s p e c t i v e l y one g e t s lOL /6* 31/4= s i n [ (0 C - X ) / 2 ] / s i n [ (0 C +X)/2} =[ s i n (9 C /2) - c o s (0 C /2) X/2 J/[ s i n (0 C /2) +cos (0 C /2) X/2 ] X=[ 2 s i n (9 C /2) / c o s (9 C /2) J£ 1- (6 L /6% ) 1 / 4 M 1 + (6 L / d * ) *'4 J 6L/6K i s e v a l u a t e d i n t h e same way a s E. T h r e e v a l u e s o f X (0. 4 3 , 0 . 61,0. 37) a r e o b t a i n e d f r o m t h r e e 30° l e f t , r i g h t r u n s , g i v i n g X= 0.5°±.1° where t h e e r r o r i s s i m p l y t h e s t a n d a r d d e v i a t i o n o f t h e t h r e e v a l u e s . The a n g l e X i s a d d e d t o a l l t h e a n g l e s f o r arms on t h e l e f t a n d s u b t r a c t e d f r o m t h o s e on t h e r i g h t . An a d d i t i o n a l p r o b l e m r e m a i n s i n d e f i n i n g t h e mean s c a t t e r i n g a n g l e o v e r t h e d e t e c t o r a t s m a l l a n g l e s , s i n c 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 v a r i e s r a p i d l y a n d n o n - l i n e a r l y h e r e . The mean s c a t t e r i n g a n g l e s h o u l d be e v a l u a t e d by 31 averaging the d i s t r i b u t i o n of s c a t t e r i n g angles produced by the f i n i t e beam spot s i z e and s o l i d angle subtended by the d e t e c t o r s , weighting each angle with the c r o s s s e c t i o n at t h a t angle, or to a good approximation by 1 / s i n * ( 9 / 2 ) . S i n c e , however, the r a t i o of c r o s s s e c t i o n s does not vary r a p i d l y i n t h i s r e g i o n , the c o r r e c t i o n was only e v a l u a t e d approximately by c o n s i d e r i n g a p o i n t beam spot. The i n t e g r a t i o n was then performed ever the second d e t e c t o r which d e f i n e s the s o l i d angle f o r a p o i n t source. The r e s u l t s f o r 30° and 40° are 29.6° and 39.8. Because of the broadening of the range d i s t r i b u t i o n and the i n h e r e n t width due to range s t r a g g l i n g i t i s p o s s i b l e f o r i n e l a s t i c a l l y s c a t t e r e d pions to be detected i n the s t o p p i n g c o u n t e r s . The e f f i c i e n c i e s f o r d e t e c t i n g the pions from the 4.4 MeV s t a t e of * 2C and the 3.68 MeV s t a t e of * 3C were e v a l u a t e d using the Monte C a r l o program mentioned p r e v i o u s l y to s i m u l a t e the two arms . The c r o s s s e c t i o n s f o r the e l a s t i c s c a t t e r i n g and to these e x c i t e d s t a t e s were c a l c u l a t e d by E. R o s t 2 7 > . The small e f f i c i e n c i e s combined with a s m a l l r a t i o of i n e l a s i c to e l a s t i c c r o s s s e c t i o n and a tendency f o r the i n e l a s t i c c o n t r i b u t i o n s to c a n c e l i n the r a t i o a l l c o n t r i v e to g i v e a very s m a l l e f f e c t , kn estimate of the s i z e of the i n e l a s t i c c o n t r i b u t i o n to R was made f o r l a r g e angles where the r a t i o 0 ( i n e l a s t i c ) /<$ ( e l a s t i c ) i s g r e a t e s t . The r e s u l t s are shown i n Table I I I . 32 2. R e s u l t s The 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 r e s u l t s f o r each s e t of runs and the weighted means f o r each a n g l e are g i v e n i n Ta b l e i v . In cases where two a n g l e s were v e r y c l o s e ( < 1° ) t h e r a t i o s were combined by e v a l u a t i n g the weighted mean and w e i g h t i n g the a n g l e s s i m i l a r l y t o f i n d t h e mean a n g l e ( T a b l e V and F i g u r e 14 ) . The r e s u l t s i n Table V were used i n t h e a n a l y s i s d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s . 33 Table I I I . I n e l a s t i c S c a t t e r i n g C o n t r i b u t i o n t o B . I 1 I I |Angle | I I I I I 12c Cross S e c t i o n s (mb/sr) |Elastic|4.4MeV r - 130 | I 8.4 ~[o.49 | 140 | 9.1 |0.62 I I | 150 | 9.6 |0.73 I I i 3 c Cross | S e c t i o n s I (mb/sr) I E l a s t i c ! ! . 6 8 M e v J I I E*2 | I I E* 3 I B / I f | I I I 11.6 | 0.40 | .61 | I I I 34|.998| I I 12.5 | 0.49 | .59 | .34|1.00| I I 13.1 | 0.57 | .62 | I I I I 36 | 1.00 | ( I i E 1 2 and E 1 3 are the r a t i o s of the e f f i c i e n c i e s f o r e l a s t i c / i n e l a s t i c s c a t t e r i n g of * 2 c and 1 3 C . E' = the measured r a t i o with no c o r r e c t i o n f o r i n e l a s t i c s c a t t e r i n g . Table IV. Results. Angle| c o . m | r h (B) + 29. 5 r +- 130. 5 0. 953±. 0.971±, 093 065 1.018±.10 0.965±.054 + 1 139-5 1.052±. 0.888±. 065 041 +— Weighted Mean —I 0.935±.035 161.2 166- 2 1.64 ±. 1.81 ±. 1.34 ±. 1.53 ±. 22 35 16 13 170- 3 1.59 ±. 1.59 ±. 32 22 1. 51U.089 + — 1.59 ±.18 |71. 3 +- 180. 3 1.63 ±. 12 1.83 ±. 1.70 ±. 1.79 ±. 14 13 14 + — -Hi 1.77±. 11 r 85.3 [90.4 I T H [91.4 1.46 ±. 10 1.47 ±. 1.427±. 1.46 ±. 1.37 ±. 11 084 11 13 1.442±.066 1.421±.084 35 Table IV. (continued) [ T T | Angle | l * G / 1 2 6 I Weighted jc.o.ml (R) j Mean 100.5| 1.58 ±.13 | | 1.56 ±.13 | 1.575±.094 111.31 1.217±.073 | 115.2I 1.222±.0eV~[ 131. i t 1.2811.069 1 141. 11 1. 098±.064 | 149.91 1.134±.090 J j 150-91 1.26 ±.13 | | | 1.153±.064 | 1.175±.057 I I I 36 Table V. E l a s t i c S c a t t e r i n g Cross S e c t i o n R a t i o s of 1 3 C / 1 2 C . Angle Cm O. m. 29.8 39.8 45.0 50. 1 60.6 66.2 71.0 80.3 90.7 101.3 110.3 115.2 130.5 140.5 150.6 i 136/lz6 (K) 4 0.976±.047 0.9351.035 0.93U.041 0.985±.066 1.2521.075 1.5111.089 1.61 ±.10 1.77 ±.079 1.434±.052 1.575±.094 1.217±.073 1.222±.061 1.28H.069 1.098±.064 1.1631.048 F i g u r e 1 4 . E l a s t i c S c a t t e r i n g Cross Section E a t i o s of 1 3 C / 1 2 C . 1 - 8 " 1 - 7 - 1 - 6 - V 5 - 1 - 4 • 1 - 3 - 1 - 2 - 1-1 • 1 0 - 0 - 9 - 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 ©cm. 38 CHAPTER IV T h e o r e t i c a l I n t e r p r e t a t i o n I t can be seen immediately from F i g . 14 t h a t the n~ i s more s e n s i t i v e t o the e f f e c t s of the valence neutron i n 1 3 C than i s v+ a t 50 MeV. ( F i g . 1 ) . The l a r g e peak c e n t r e d at 70°-80° i s due to a change i n the s t r u c t u r e i n the s-p i n t e r f e r e n c e minimum between l 2 C and * 3C. The 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 2 8 * of 29 MeV „- on l 2 C i s shown i n F i g . 15 together with the r e s u l t s of an 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 ( see s e c t i o n 2 ) . M u l t i p l y i n g the 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 by the values of R i n Table V gi v e s the l i n e i n d i c a t e d . I t can be seen t h a t the minimum of the 1 3 C c r o s s s e c t i o n has been s h i f t e d s l i g h t l y and r a i s e d above t h a t of » 2C. 1« Point Nucleus Impulse Approximation Model A simple c a l c u l a t i o n was made to see how f a r t h i s s t r u c t u r e i s p r e d i c t e d by the pion-nucleon amplitudes. The r a t i o o f c r o s s s e c t i o n s was e v a l u a t e d by adding together the s c a t t e r i n g amplitudes of the twelve nucleons f o r 1 2 C and adding the e x t r a n~n amplitude f o r 1 3 C . T h i s g i v e s f o r the r a t i o R=[ 13b0 +b, +(13c 0 + C i ) k 2 c o s (9) 3 2 / t 12b0 +12c0 k 2 c o s (9) 3 2 where k=.48 f m - 1 f o r 30 MeV pions O p t i c a l p o t e n t i a l parameters b 0 ,ba , c 0 and C j were used r a t h e r 39 F i g u r e 15. Comparison of 1 3 C and * 2C D i f f e r e n t i a l Cross S e c t i o n s . i i i 1 1 i 1 1 1 i 1 i • i 1 1 1 1 1 1 — 2 0 4 0 6 0 8 0 100 120 140 ®c.m. 40 than f r e e nN amplitudes to give a b e t t e r approximation to s c a t t e r i n g i n the nucleus. The main d i f f e r e n c e i s i n an i n c r e a s e i n b 0 . The parameters were taken from S t r i e k e r , McManus and C a r r 5 > , (SMC), Set 1 (Table VI, Set B). Imaginary terms were added to b 0 and c 0 - The imaginary terms were Im bo= .020 fm.; Im c 0 = .047 fm? The s i z e s of the imaginary terms were chosen to g i v e approximately the c o r r e c t value f o r B at 80°, but the r e l a t i v e s i z e s of Im b 0 and Im c 0 were taken from the r a t i o of the imaginary p a r t s to the s and p wave a n n i h i l a t i o n s t r e n g t h s of SMC Set 1 . The r e s u l t i s shown as the dashed l i n e i n F i g . 1 6 . The maximum i s p r e d i c t e d at the c o r r e c t p o s i t i o n and the trend at l a r g e a n g l e s i s roughly c o r r e c t . At low a n g l e s , however, the agreement i s not good, although the d i p below B = 1 i s p r e d i c t e d . The same c a l c u l a t i o n with imaginary terms set to zero i s a l s o shown ( dotted l i n e ) . By i n c r e a s i n g the a b s o r p t i o n the d i p i s g r e a t l y reduced s i n c e now the c r o s s s e c t i o n minimum of 1 3 C i s not o n l y s h i f t e d i n angle but i s a l s o r a i s e d by the imaginary terms. The imaginary terms are such t h a t the s-p i n t e r f e r e n c e minimum becomes l e s s deep as i t i s moved to a s m a l l e r angle. Since t h i s model e s s e n t i a l l y uses a p o i n t n u c l e u s , one would expect the r e l a t i v e amounts of s and p waves to d i f f e r from those of a f i n i t e s i z e nucleus. A l s o m u l t i p l e s c a t t e r i n g i n the nucleus may have the e f f e c t o f reducing the p-wave s c a t t e r i n g , by the Lorenz-Lorentz e f f e c t . Reducing the p-wave amplitude by p r o p o r t i o n a l y the same amount i n both n u c e i , by simply changing k to .32 fm-i , produces the s o l i d curve; T h i s curve r e p r e s e n t s 41 F i g u r e 16- Impulse Approximation C a l c u l a t i o n s of R. 42 the best f i t t o the data when onl y k i s v a r i e d . A l l the e s s e n t i a l f e a t u r e s of the d i s t r i b u t i o n of E can be seen i n t h i s curve although q u a n t i t a t i v e l y the agreement i s not good. The same c a l c u l a t i o n f o r 50 MeV it* with k reduced by the same f a c t o r i s shown i n F i g . 1. Again q u a l i t a t i v e l y t h e agreement i s q u i t e good and i t i s seen t h a t the s m a l l e r s i z e of the r a t i o near the c r o s s s e c t i o n minimum i s due t o the minimum being s h i f t e d by a s m a l l e r amount and i n the opposite d i r e c t i o n than i n the 30 MeV fl- case* The imaginary terms then produce a decrease i n the c r o s s s e c t i o n which i s c a n c e l l e d p a r t l y by the i n c r e a s e i n number of nucleons. Such reasonable q u a l i t a t i v e agreement using so simple a model suggests t h a t c a l c u l a t i o n s using an o p t i c a l p o t e n t i a l i n c o r p o r a t i n g the c o r r e c t k i nematics and f i n i t e s i z e e f f e c t s should be r e l i a b l e ! 2« 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 The c p t i c a l p o t e n t i a l used was t h a t of S t r i e k e r , McManus and C a r r 5 > and was of the form 2w001,t=-4n{b(r)+pj, B0f* (r)+ <c0 ( P j - 1 ) / 2 P l V 2 f ( r ) ) + ( C 0 (P £-1)/2p e § 7 2j)2 (r)} +4n [V.L(r)c(r )V + C 0 / p , V - f 2 ( r ) \ 7 3 +2wVc (r) where b ( r ) = P l (b 0 ;f(r)-£ T Tb 18/>(r) ) c ( r ) = 1 / P l ( c 0 f ( r ) - t ^ C l i f ( r ) ) ty(r)=ft<r)-fl,(r) ETT=- 1 f o r rr- and +1 f o r -a* L (r) = (1+4nA(A-1) c (r)/3A) — 1 i s the E r i c s o n - E r i c s o n f a c t o r 43 Vc = Coulomb p o t e n t i a l w=w/(1+E/A) i s the pion reduced energy, where w i s the pion t o t a l energy i n the pion-nucleus c e n t r e of mass frame, E=w/M and M=nuclear mass/A. The f a c t o r s P l = (1 + E)/(1+E/A) and p,, =(1+E/2)/(1 + E/A) are reduced mass f a c t o r s . The terms, i n v o l v i n g V zp and V 2 ^ 2 are t o account f o r tr a n s f o r m a t i o n from the nH c e n t r e of mass system to the rtk s y s t e m 2 9 * . T h i s p o t e n t i a l i s d e s c r i b e d f u l l y i n SMC. The wave eguation used t o c a l c u l a t e the 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 was of the form ( V 2 +k 2+2wU o pt )J0'(r) =0 where k i s the wave vector of the pion-nucleus c e n t r e of mass system. A program developed by A. Thomas and M. K r e l l was used to e v a l u a t e the p a r t i a l wave phase s h i f t s and hence the 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 from the wave e q u a t i o n . The s e t of o p t i c a l p o t e n t i a l parameters used (set A, Table VI) was taken from SMC s e t 1, with the a b s o r p t i o n parameters, B 0 and C 0 , adjus t e d t o f i t the 1 2 C 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 i g . 17 ). The d a t a 2 8 > i n F i g . 17 are the combined r e s u l t s of s e v e r a l runs. The r e a l and imaginary p a r t s of B 0 and C 0 were kept e q u a l i n accordance with p i o n i c atom data**. A m o d i f i e d Gaussian form, j>(*)=J>0l 1 t " ( r / a ) 2 3 expl - (r/a) 2 3 where f0 =nucleon d e n s i t y a t the c e n t r e , was used f o r the proton Table VI. O p t i c a l P o t e n t a l Parameters. l ~ 1 | SET A | SET B | SET C I U n i t s | ho | -.040+i.002 j | -.040+i.002 | -.040+.002 1 f I bi | -. 11 -i.001 | -.11 -i.001 | -.11 -i.001 1 f m, I Bo I --17 +i.17 | -.13 +i.13 | -.17 +i.17 1 fm.* 1 Co I • 75 +i.007 | .75 +i.007 | .75 +i.007 |fm.3 | C 1 i .62 +i.004 | .62 +i.004 | .62 +i.004 |fm.3 | Co | -.79 + i . ? 9 | -.75 +i.75 | -.79 +i.79 1 f m.6 1 A | 1 .0 1 L O | 0.6 F i g u r e 17. C a l c u l a t i o n s of 1 2 C Cross S e c t i o n with Parameter Sets A, B and C. 46 and n e u t r o n d i s t r i b u t i o n s of 1 2 C . The parameter , <x , was s e t t o the s h e l l model v a l u e , oc = (Z,N-2)/3 f o r n e u t r o n , p r o t o n d i s t r i b u t i o n s . The rms. r a d i u s i s th e n g i v e n by <r f 2 n>i/2 = I 2.5-2/Z,N i]»/2 a The p r o t o n rms. r a d i i , r p , were o b t a i n e d by s u b t r a c t i n g o u t t h e p r o t o n s i z e from the charge r a d i i , <r,2>V2 g i v e n by e l e c t r o n s c a t t e r i n g , u s i n g <r 2> = <rf>- . 82 The n e u t r o n and pr o t o n d i s t r i b u t i o n s f o r l2C were assumed e q u a l a s i s r e a s o n a b l e from t h e i r c l o s e d s h e l l n a t u r e and from H a r t r e e - F o c h c a l c u l a t i o n s 1 6 * . The charge r a d i u s 3 0 * o f l2C, 1 2r c=2.46±.025 fm., g i v e s p r o t o n and n e u t r o n r a d i i o f i 2 r.=2.326±.026 fm. There i s some d i s c r e p e n c y i n the l i t e r a t u r e on t h e rms. charge r a d i u s of 1 3 C . The r a t i o of 1 3 C / 1 2 c rms. charge r a d i i was measured t o be .96±.01 by C r a n n e l l 3 * * e t a l . a t S t a n f o r d . The momentum t r a n s f e r range covered i n t h e expe r i m e n t was .7 t o 1.7 f m - 1 . ¥ang et a l . a t S a s k a t o o n 3 2 * l a t e r measured the r a t i o t c be .975± .02, c o v e r i n g t h e momentum t r a n s f e r range of .2 to 1.1 fm-*. They combined t h e two r e s u l t s t o g i v e a r a t i o o f .968±.015. I n a h i g h e r energy e l e c t r o n s c a t t e r i n g experiment H e i s e n b e r g e t a l . 3 3 * found a r a t i o of .9907±.0004 They a t t r i b u t e t h e d i s c r e p e n c y w i t h t h e r e s u l t s of C r a n e l l as b e i n g due t o the d i f f i c u l t y i n d e t e r m i n i n g the t a r g e t t h i c k n e s s w i t h t h e . l o w p u r i t y l i q u i f i e d methane t a r g e t used by C r a n e l l . T h i s i s suggested by t h e d i f f e r e n c e i n c r o s s s e c t i o n r a t i o s of 47 13C/12C measured by the two experiments at the same momentum t r a n s f e r . Yang used a gas methane t a r g e t , e n r i c h e d to 84%. The high momentum t r a n s f e r r e s u l t , 1 3 r c = 2 . 4 4 fm., has been used i n t h i s a n a l y s i s , but checks are made of the dependence on the proton r a d i u s . Two d i f f e r e n t approaches were used t o d e s c r i b e the 1 3 C neutron d i s t r i b u t i o n . F i r s t l y the d e n s i t y d i s t r i b u t i o n of a neutron with a p-wave harmonic o s c i l l a t o r wavefunction was added to the Gaussian d i s t r i b u t i o n of a 6 neutron core. S e n s i t i v i t y t o the s i z e of the neutron core was i n v e s t i g a t e d . In the second case a modified Gaussian d i s t r i b u t i o n was used f o r the 7 neutrons and a determination of the neutron rms. r a d i u s of 1 3 C was made. Since the p o i n t at 100° must be i n disagreement with any reasonable f i t , t h i s p o i n t was l e f t out of the X 2 e v a l u a t i o n s so t h a t i t would not e f f e c t the p o s i t i o n s of X 2 minima. The d i f f e r e n t c a l c u l a t i o n s , however, tended t o converge a t t h i s p o i n t so the i n f l u e n c e on the X 2 minima would not be g r e a t , a) Core + Valence Neutron Model The wave f u n c t i o n of the p-wave neutron was averaged over a l l a n g l e s . The harmonic o s c i l l a t o r w e l l s t r e n g t h , V , was l e f t as a parameter t o be v a r i e d . The wavefunction of a p a r t i c l e i n a s p h e r i c a l harmonic w e l l i s given by jarnI|n(r)=Hn[ (r) Yf (6) where n i s the number of i n t e r n a l r a d i a l nodes and 1 i s the angular momentum guantum number. 48 B-n i ( r ) = / 3/22 1-n+2 (2n+21+1)! ( v r 2)l / 2 L 1 + » 2 ( V r z ) e - v r V 2 -Jm n! ((21+1) !!) 2 and n L i , Ll+i/2(/3)=XI(-1) 2 R(n\ (21 + 1)11 >3ft k=0 \kj (2l+2k+1)!» •v=mW/-n , where m= the reduced mass and U) = the o s c i l l a t o r frequency. For the p-wave neutron o f 1 3 C , n=0, 1=1. The rms. r a d i u s i s given by <nl|r2|nl>=(2n+l+3/2)V~ l= 5/2"V For the 6 neutrons of 1 2 C the rms. r a d i u s i s then [ (2x3/2 +4x5/2)/6VJ V i assuming a harmonic w e l l . For an rms. r a d i u s of 2.326 fm. the w e l l depth, v» f o r 1 2 c i s .40 fm~ 2. i t might be expected t h a t the e f f e c t i v e w e l l depth f o r the e x t r a neutron of 1 3 C w i l l be s m a l l e r than t h i s s i n c e the neutron i s unpaired. Using e q u a l neutron and proton d i s t r i b u t i o n s of rms. radius= 2.306 fm. f o r the core of 1 3 C (reduced core) c a l c u l a t i o n s of R were made f o r v a r y i n g w e l l s t r e n g t h , V . The c a l c u l a t i o n s were repeated f o r 1 3 r =2.306 fm and neutron core r a d i u s of 2.326 fm. (equal to 1 2 c r a d i u s ) . The best f i t s i n the two cases were e s s e n t i a l l y i d e n t i c a l , ( F i g . 18) and the agreement with the data i s good, except perhaps f o r s m a l l an g l e s . The ~XZ f o r the best f i t s are the same , although the minimum i s sharper f o r the reduced core case ( F i g . 19). Also shown i n F i g . 19 i s the X 2 curve f o r a neutron c o r e r a d i u s of 2.360 fm. I t can be seen t h a t as the core r a d i u s i n c r e a s e s the best f i t w e l l s t r e n g t h i n c r e a s e s . The TC2 curves are g e n e r a l l y not p a r a b o l i c . The p o s i t i o n s o f the minima are t h e r e f o r e F i g u r e 18. 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 of E. Valence Neutron + Core Model. F i g u r e 19. Xz Curves f o r Valence Neutron + Core Model. 51 d e f i n e d as midway between the two p i o n t s where X 2 i s 1 g r e a t e r than the lowest value. E v a l u a t i n g the rms. r a d i i , f o r the t o t a l neutron d i s t r i b u t i o n s gives * 3r R=2.354±.014, 2.365±.019 and 2.370±.020 fm. fox r (core)=2.306, 2.326 and 2.360 fm. R e s p e c t i v e l y . The best f i t rms. r a d i u s remains almost c o n s t a n t . T h i s suggests that the s e n s i t i v i t y i s to the rms. r a d i u s of the neutron d i s t r i b u t i o n and not t o the d e t a i l e d s t r u c t u r e . C a l c u l a t i o n s of R were t h e r e f o r e made with d i f f e r e n t neutron d i s t r i b u t i o n s of the same t o t a l rms. radius=2.36 fm. The d i s t r i b u t i o n s were formed by combining p-wave neutron wavefunctions of d i f f e r e n t w e l l s t r e n g t h with modified Gaussian d i s t r i b u t i o n s of d i f f e r i n g rms. r a d i i . The proton d i s t r i b u t i o n of * 3C measured by Heisenberg i s represented as a m o d i f i e d Gaussian d i s t r i b u t i o n , but oc d i f f e r s by about 6% from the s h e l l model va l u e . The v a r i a t i o n s i n the d i s t r i b u t i o n f o r 1 3 r ^ ( F i g . 20) used here are approximately e q u i v a l e n t t o v a r y i n g c< by ±10%. The r e s u l t i n g d i s t r i b u t i o n s f o r R are very s i m i l a r and the t o t a l X 2 v a r i e s by l e s s than 1.5 ( Table VII ) . T h i s suggests t h a t i t i s p o s s i b l e t o o b t a i n a measurement of l 3 r n which i s not s i g n i f i c a n t l y dependent on u n c e r t a i n t i e s i n the form of the neutron d i s t r i b u t i o n . Combination with i n f o r m a t i o n from higher energy pion s c a t t e r i n g and with p i o n i c atom data may permit measurement of more d e t a i l e d s t r u c t u r e i n the d i s t r i b u t i o n . b) Neutron D i s t r i b u t i o n rms. Radius C a l c u l a t i o n s of R were made, using a modified Gaussian d i s t r i b u t i o n f o r both the neutrons and protons, varying both 1 3 r p and l 3 r n . A X2 contour p l o t ( F i g 21) was produced from the 52 F i g u r e 20. * 3C Neutron D i s t r i b u t i o n s with Equal rms. Badius <N O CO CD < t ^ *~ ~" o o o o ( £ _ w j s u o j j n e u ) ( / Table VII. Results of Calculations Osing Different Neutron Distributions with Equal rms. Radius. |Line |Fig 20 B I r n (core) | (fm.) j (fm7 2) 2.280 2.306 2.340 .32 .353| .4 1 Gaussian d i s t r i b u t i o n I 2.380 | .50 I 20.77 19.81 19.40 19.30 19.93 F i g u r e 21. "X2 Contour Plot from 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 using Parameter Set A. 55 r e s u l t s (Figs 22 and 23). The value of * 3 r n at the X 2 minimum changes from 2.370±.022 fm. at * 3rp = 2.306 fm. to 2.337±.025 fm. at * 3 r p = 2.240 fm. The guantity i 3 r n - i 3 r p correspondingly changes from .064 fm. to .103 fm. The measurement of 1 3 r n i s therefore s l i g h t l y dependent On the value of 1 3 r ^ assumed. The HLZ contour shows, however, that the XT- i s more sensitive to the neutron d i s t r i b u t i o n than the proton d i s t r i b u t i o n . Some tests were made of the dependence of the measurement of 1 3 r n on the o p t i c a l potential parameters. The proton radius was fixed at 2.306 fm'. Two a d d i t i o n a l parameter sets were used ( B and C, Table VI ). Set B i s SMC Set 1 (no variation of B0 and C 0 ) . Set C i s SMC Set 1 with the Ericson-Ericson parameter, A , changed to 0.6. The f i t s to the 1 2 C cross section are shown in F i g . 17. The resulting T C 2 (Fig. 24) give 1 3 r n =2.374±. 015 fm. and 2.387±.016 fm. f o r sets B and C respectively. The dependence on the value of 1 2 r c used was investigated by performing the calculations with l 2 r t l = 1 2rp=2.300 fm. Parameter set A was used wiith *3rj,=2. 280. fm. The r e s u l t i n g X 2 minimum (Fig. 24, curve D) i s at l 3 r n = 2.344 fm--. The two cases, 1 2 r p =2.326 fm. and i 2rp=2.305 fm,. therefore both give values of i 3 r n - i 2 r p = .44 fm. . This demonstrates the measurement of 1 3 r - n i s r e l a t i v e to the size 1 2 C assumed. F i g u r e 22. R e s u l t s of 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 with *3r„=2.306 fm. 57 F i g u r e 23. E e s u l t s of 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 with 1 3 r D =2.240 fm. 58 F i g u r e 24. "X2 curves of 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 with Parameter Sets k, B and C. 59 CHAPTER V Summary and Discussion The e l a s t i c scattering d i f f e r e n t i a l cross section r a t i o s of 29 MeV n~ on " c / i a c were measured at angles from 30° to 150°. Since the measurement was a r e l a t i v e one the errors on the r a t i o are purely s t a t i s t i c a l . A large peak i n the r a t i o at about 80° shows that the n- i s more sensitive.to the e f f e c t s of the valence neutron than tr + at 50 MeV, where the d i s t r i b u t i o n i s quite f l a t . The peak in the d i s t r i b u t i o n i s due to a change i n the structure of the s-p interference minimum between * 2C and 1 3 C , which i s due mainly to the large s-wave n~n in t e r a c t i o n . , In Chapter IV, section 1, the n-nucleus in t e r a c t i o n was represented by a sum of AN scattering amplitudes. I t was shown that the data are f i t quite well with t h i s simple model i f one parameter , the r e l a t i v e amount of s and p wave scattering, i s allowed to vary. This suggests that in using a detailed o p t i c a l potential c a l c u l a t i o n any >errors or uncertainties i n the potent i a l w i l l cancel to a large extent since they w i l l produce the same e f f e c t i n both nu c l e i . In t h i s way, nuclear structure information such as density d i s t r i b u t i o n s may be i s o l a t e d i n the interpretation i f the c a l c u l a t i o n produces the correct behaviour for variations i n the nuclear structure.. A 6 neutron + harmonic o s c i l l a t o r valence neutron model was used for the neutron d i s t r i b u t i o n of 1 3C. It was found that the o p t i c a l potential calculations were i n s e n s i t i v e to the size of the core but depended larg e l y on the neutron rms. radius. A 60 measurement of the 1 3 C neutron rms. r a d i u s was made and some t e s t s were made f o r dependence on the o p t i c a l p o t e n t i a l parameters. For the t h r e e s e t s of parameters used, values of i 3 C rms. r a d i u s of 2.370±.022, 2.374±.015 and 2.387±.016 fm. were found. E r r o r s due to the u n c e r t a i n t y i n 1 3 r p - 1 2 r p , as measured by Heisenberg are n e g l i g i b l e . The v a l u e s of 1 3r^ found using d i f f e r e n t o p t i c a l p o t e n t i a l parameters and d i f f e r e n t neutron d i s t r i b u t i o n s span from 2.354 fm. t o 2.387 fm. The value 2.387 has a much l a r g e r % z than the o t h e r s . Leaving out t h i s measurement the neutron rms. r a d i u s i s measured to be 2.365±.025 fm. where the e r r o r s guoted are the range covered by standard d e v i a t i o n s of the remaining measurements i n c l u d i n g those with the neutron + core model. The r e l i a b i l i t y of t h i s method of measuring neutron r a d i i should be t e s t e d . The proton r a d i u s d i f f e r e n c e s between neighbouring i s o t o n e s may be measured where accurate v a l u e s a l r e a d y e x i s t from e l e c t r o n s c a t t e r i n g . In t h i s case u + would be used and the s i t u a t i o n i s then analogous to the present experiment apart from Coulomb e f f e c t s . Secondly a measurement of the neutron r a d i u s should be made on an i s o t o p e such as C a 4 8 where p r e v i o u s measurements and Hartree Foch c a l c u l a t i o n s e x i s t . By combining with i n f o r m a t i o n from higher energy pion s c a t t e r i n g and p i o h i c atom data , where the s u r f a c e d e n s i t y only i s sampled , i t may be p o s s i b l e to o b t a i n more d e t a i l e d i n f o r m a t i o n on the neutron d e n s i t y d i s t r i b u t i o n . 61 BIBLIOGRAPHY M. Ericson and T.E.O. Ericson, Ann. Phys. 36, 323 (1966) J . Eisenberg, J. Hufner and E.J. Moniz, Phys. Lett. .47B, 381 (1973) M. K r e l l and T.E.O. Ericson, Nucl. Phys. JBJL1, 521 (1969) J . Hufner, Physics Reports 2JG, 1 (1975) K. Strieker, H. McManus and J. Carr, i Phys. Rev. £19, 929 (1979), W. Gibbs, B.F. Gibson and G.J. Stephenson, Phys. Rev. Lett. 39, 1317 (1977) N.J. Digiacomo, Phys. Lett. 66B, 421 (1977) G.K. Varma and L. Zamick, Nucl. Phys. A306, 343 (1978) G.K. 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