ABSTRACT. Spin Alignment in 12C+12C Inelastic Scattering. Stephen Joseph Willett. Yale University 1981

Transcription

1 ABSTRACT Spin Alignment in 12C+12C Inelastic Scattering Stephen Joseph Willett Yale University 1981 The excitation function of 12c+l2C scattering has been characterized by numerous resonances (total width r= kev) of unique spin in the energy range E_=5-20 MeV and broader structures (T=2-4 MeV) at CM E>10 MeV. The latter have been proposed to be a continuation of the former: molecular states of narrow width near and above the Coulomb barrier. However, other researchers have proposed alternate, nonresonant mechanisms that might also give rise to the broad maxima. To help establish the physics underlying the gross structures, we have undertaken a measurement of the spin alignment in the inelastic scattering. These measurements were performed over a wide range of particle scattering angles (53 <0CM<1O1 ) and at 21 energies (15<E <33 MeV) chosen to map over both the broad and narrow CM

2 structures observed in the cross section data o(e). Our method employed particle-gamma radiation correlations in the scattering plane, so that both alignment A(E,0) and phase angle 0(E,8) information could be extracted. Separate experiments established an absolute calibration for the alignments. Our results demonstrate that the alignment is only weakly a function of scattering angle. Furthermore, the angle-averaged alignment <A> is enhanced at energies where either broad or narrow maxima occur in o(e) for E<26 MeV, but above that energy A(E) becomes relatively constant with respect to both E and 0. The phase angle 0 is approximately linear in 0, and exhibits a trend of increasing slope of 0(0) with increasing interaction energy. Deviations from this general behavior at certain energies may be interpreted as signs of true quasimolecular resonances. We also compared theoretical predictions to our results. Resonant theories, such as the single particle or the band crossing models, do not reproduce the alignment data. Non-resonant theories, such as the DWBA or the diffraction model, can simultaneously reproduce gross features of both A(E) and o(e), and also satisfactorily predict 0(0). These predictions fail at higher energies, indicating that their simplifying assumptions are no longer valid in that regime. The behavior correlated with the narrow structure still awaits detailed explanation, most probably incorporating a resonance origin.

3 SPIN ALIGNMENT IN l2c+12c INELASTIC SCATTERING A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Stephen Joseph Willett December 1981

4 ii ACKNOWLEDGEMENTS It is a pleasure to acknowledge the contributions of many people to this work and to my graduate education. I am most grateful to my advisor, Karl Erb. He has given generously of his time, patience, and experience to teach me the ways of experimental physics. Karl's initial stimulus and constant encouragement during this project, even after his departure for Oak Ridge National Laboratory, have been vital. I value his guidance, his support, and his friendship over the past five years. I also appreciate D. Allan Bromley's role in my professional training. His enthusiasm for this work, and indeed for all of physics, was contagious. In his dual role as laboratory director and (in the past year) dissertation director, he has made it possible through conferences and workshops elsewhere and with visitors to the laboratory for me to interact with nuclear physicists from around the world. He has also given me invaluable advice on the art of presenting my research, both orally and in writing. I also thank the other members of my thesis committee-- Peter Parker, Steve Sanders, Denise Caldwell, and Tom Appelquist-- for their attentive reading of this manuscript and suggestions for its improvement. It has been said that students learn as much from their colleagues as from the faculty. Among my many teachers I would like to single out Steve Korotky and Munther Hindi. As a member of the heavy ion group, Steve has been generous in his assistance with this research and with other phases of my education; for that and for his friendship I thank him. I am also grateful to two research fellows, Bob Phillips and Yosio Kondo, for the immense help they lent to the experimental and theoretical aspects of this research. In addition to Karl, Steve, and Bob, who shared the bulk of the data acquisition tasks, I acknowledge Ed Schloemer, Ahmed Jabbarian, Hoshe Gai, and Mark Anthony for their assistance in gathering the data. I am also indebted to Jay Petersen, David Radford, and Peter Parker, upon whose expertise I have relied on

5 Ill countless occasions. WNSL has been fortunate to have a small army of support personnel, whose diverse skills are readily available to the research staff. During my stay here, the accelerator was maintained and operated by Kenzo Sato, Dick D'Alexander, Phil Clarkin, Ted Duda, Bob Harrington, Mike Sivertz, and Mike Bekker. The electronics staff included Charlie Gingell, Dick Wagner, Jack Burton, and Tom Barker. Expert help in the machine shop came from Al Jeddry, Joe Cimino, and Ray Comeau. Mary Anne Schulz, Susan Anderson, Rita Bonito, Ann Dallavalle-Schombert, and Fran DeGreiner provided secretarial support. Sandy Sicignano was the technical illustrator, and Steve Pitcher and Neil Bauer have maintained the computer system. Business affairs were handled by Fred Curtis, Marvin Curtis, and Karen DeFelice. I am indebted to each of these people for his/her cheerful assistance. No list of acknowledgements from a Yale physics graduate would be complete without the name of Sara Batter. As the DGS secretary,'sara has taken a personal interest in every entering class, making us feel welcome at Yale from the minute we arrived and taking our side in occasional battles with the bureaucracy. Although at times I felt the contrary to be true, I did not in fact spend all my waking hours in New Haven on physics. On the whole, graduate school has been an enjoyable experience, due chiefly to activities and friendships with the people mentioned above, fellow physics students, and their counterparts in other departments at Yale. Their companionship is deeply treasured. Finally, I wish to express my gratitude to my parents for their love and guidance (and genes), and to my wife and best friend, Diane Pike, for editing this thesis and for her love and good humor that have kept me entertained and motivated during the darkest hours of this work.

6 CONTENTS Abstract... Acknowledgements Figures Tables ii vi viii 1. Introduction P r e f a c e Nuclear molecular phenomena... 2 a) Early evidence and models... 2 b) Later experiments and theories... 3 c) Recent experimental data... 8 d) Searches in other systems Theoretical models a) D o o rways b) Extensions of the Imanishi m o d e l c) Spectrum g e n e r a t i o n...19 d) Microscopic models Inelastic scattering Direct reaction models a) D W B A b) D i f f r a c t i o n c) Barrier t o p d) Wave i n t e r f e r e n c e Alignment R e s u l t s F o r m a l i s m Conservation principles and definition of alignment Particle-gamma correlation function Identical particle symmetry Experimental measurement of the correlation function Absolute normalization Experimental Technique and Data Reduction Beam and t a r g e t s Detection of particles and gamma radiation Electronics Event a n a l y s i s Experimental correlation function Least squares fit to the correlation function Absolute Normalization Inelastic proton scattering on 12C Inelastic alpha particle scattering on 12C Complete i2c+12c* angular correlation Determination of the normalization c o n stant

7 V 5. Results and comparison with other d a t a A lignment a) Relative normalization b) Energy dependence " P olarization" Phase a n g l e Comparison with previous d a t a Theoretical calulations Rotation of the quantization axis and the statistical prediction Resonant m o d e l s a) Single-particle shape resonances b) Band crossing m o d e l Non-resonant models a) Plane w a v e s b) Distorted w a v e s c) Diffraction model d) Barrier top m o d e l Phase a n g l e Conclusion Summary of present w o r k Extensions to other w o r k Appendix A. Tables of D a t a Appendix B. Angular dependence of alignment and phase angle R e f e r e n c e s

8 vi FIGURES Schematic potential for quasimolecular interactions... 5 i 2 c + i 2 c gamm a r a d i a t i o n y i e l d s... 9 R e s o n a n c e e n e r g i e s v s. J ( J + 1 ) 1 6 I n e l a s t i c s c a t t e r i n g e x c i t a t i o n f u n c t i o n s C o m p a r is o n o f r e s o n a n c e m o d e ls w i t h i n e l a s t i c s c a t t e r i n g I n e l a s t i c s c a t t e r i n g p r e d i c t i o n s o f n o n - r e s o n a n t m o d e ls S c h e m a t i c d i a g r a m o f 1 2 C + l 2 C i n e l a s t i c s c a t t e r i n g E 2 r a d i a t i o n p a t t e r n s E f f e c t o f n o n - d i s t i n c t i o n o f f i n a l p r o d u c t s i n e l a s t i c s c a t t e r i n g o f i d e n t i c a l p a r t i c l e s T y p i c a l i n - p l a n e c o r r e l a t i o n f u n c t i o n D e f i n i t i o n o f a n g l e s a n d d e t e c t o r s I n - p l a n e c o r r e l a t i o n e x p e r i m e n t s Tw o b o d y k i n e m a t i c s... > 6 5 E l e c t r o n i c s A n a l y s i s o f p - p e v e n t s T y p i c a l E x v s. 0 t s p e c t r u m T y p i c a l a n g u l a r d i s t r i b u t i o n s p e c t r u m A n a l y s i s o f p - y e v e n t s T y p i c a l T j v s. 0 * s p e c t r u m I n e l a s t i c p r o t o n s c a t t e r i n g n o r m a l i z a t i o n e x p e r i m e n t Gamma r a d i a t i o n s p e c t r u m P r o t o n a n g u l a r d i s t r i b u t i o n Gamma r a d i a t i o n a n g u l a r d i s t r i b u t i o n V a r i a t i o n o f y - y i e l d w i t h d i s t a n c e C o m p a r is o n o f 1 2 C ( o, o ' ) 1 2 C ( 2 + ) a l i g n m e n t m e a s u r e m e n t s

9 Complete angular correlation experiment... A n a l y s i s o f A E - E - y e v e n t s... A b s o l u t e n o r m a l i z a t i o n o f t h e M e V a l i g n m e n t d a t a.... A l i g n m e n t v s. a n g l e f o r e a c h e n e r g y... A l i g n m e n t v s. e n e r g y f o r e a c h a n g l e... <A > a n d o v s. e n e r g y... V a r i a t i o n o f t h e c o r r e l a t i o n f u n c t i o n w i t h e n e r g y a n d a n g l e. P h a s e a n g l e v s. p a r t i c l e a n g l e... C o m p a r is o n w i t h t h e M u n ic h d a t a... D i f f r a c t i o n m o d e l : o p r e d i c t i o n s v s. A... B r e i t - W i g n e r r e s o n a n c e s : A ( E ) a n d o ( E ) B a n d c r o s s i n g m o d e l : A ( E ) a n d o ( E )... B a n d c r o s s i n g m o d e l : P a r t i a l w a v e c o n t r i b u t i o n s... DWBA: A ( E ) a n d o ( E )... A ( E, 6 ) f o r BCM, DWBA, a n d D i f f r a c t i o n... D i f f r a c t i o n m o d e l : A ( E ) a n d o ( E )... S l o p e s o f 6 ( 6 )... P h a s e a n g l e s v s. p a r t i c l e s c a t t e r i n g a n g l e s... A ( E ) : D a t a a n d p r e d i c t i o n s... v i i

10 Vlll TABLES I ) P a r a m e t e r s o f i n - p l a n e r u n s I I ) F r o n t e n d c o n f i g u r a t i o n I I I ) P h a s e a n g l e : E x p e r i m e n t a n d t h e o r y A. l I n - p l a n e a n g u l a r c o r r e l a t i o n s A. 2 A l i g n m e n t s, " p o l a r i z a t i o n s ", a n d p h a s e a n g l e s A. 3 C o m p le t e a n g u l a r c o r r e l a t i o n s

11 1. INTRODUCTION 1. 1 P r e f a c e C h a r a c t e r i s t i c o f m o s t f i e l d s i n p h y s i c a l a n d b i o l o g i c a l s c i e n c e s i s t h e c l o s e i n t e r p l a y b e t w e e n t h e o r y a n d e x p e r i m e n t. T h e i n i t i a l m o t i v a t i o n m a y co m e f r o m e i t h e r s o u r c e, b u t i t s r a p i d a n d c o n t i n u a l d e v e l o p m e n t d e p e n d s u p o n a s y n e r g i s t i c e f f o r t f r o m b o t h a r e a s. O b s t a c l e s a r e i n e v i t a b l e i n e i t h e r t h e o r y o r e x p e r i m e n t - - t o o m a n y o r t o o f e w c h o i c e s, t o o l i t t l e o r t o o m u c h i n f o r m a t i o n b u t i n p u t f r o m o n e a r e a c a n h e l p o v e r c o m e d i f f i c u l t i e s i n t h e o t h e r. A l t e r n a t i v e l y, i n a c t i v i t y i n o n e a r e a m a y l e a d t o d i s i n t e r e s t i n t h e o t h e r, a n d t h e e n t i r e f i e l d w i l l l i e f a l l o w, p e r h a p s f o r y e a r s. B o t h m o d e s o f i n t e r a c t i o n a r e s e e n i n t h e h i s t o r y o f t h e s t u d y o f n u c l e a r m o l e c u l e s, a s c a n b e d i s c e r n e d i n t h e r e v i e w i n t h i s c h a p t e r. 1

12 2 I n f a c t, t h e p r e s e n t s t u d y w a s s t i m u l a t e d b y t h e m u l t i t u d e o f t h e o r i e s p r o p o s e d t o e x p l a i n c a r b o n - c a r b o n i n e l a s t i c s c a t t e r i n g w i t h i n ( o r w i t h o u t ) t h e q u a s i m o l e c u l a r p i c t u r e. We h o p e t h a t t h e p r e s e n t a t i o n o f n e w d a t a i n t h i s f i e l d w i l l s e r v e t o c o n t i n u e t h e i t e r a t i v e p r o c e s s b e t w e e n t h e o r y a n d e x p e r i m e n t, a n d l e a d t o g r e a t e r u n d e r s t a n d i n g o f t h i s s m a l l s e g m e n t o f n a t u r a l p h e n o m e n a N u c l e a r m o l e c u l a r p h e n o m e n a a ) E a r l y e v i d e n c e a n d m o d e ls W i t h t h e a d v e n t o f h i g h r e s o l u t i o n d e t e c t o r s a n d a c c e l e r a t o r s f o r h e a v y i o n s tw o d e c a d e s a g o c a m e t h e f i r s t e v i d e n c e f o r n u c l e a r m o l e c u l a r p h e n o m e n a. B r o m l e y, K u e h n e r, a n d A l m q v i s t o b s e r v e d s t r u c t u r e i n t h e l 2 C C 9 0 e l a s t i c s c a t t e r i n g e x c i t a t i o n f u n c t i o n ( B r 6 0 ) a n d c o r r e l a t e d s t r u c t u r e s n e a r t h e C o u lo m b b a r r i e r M e V ) i n t h e p r o t o n, n e u t r o n, a l p h a, a n d gamma e x i t c h a n n e l s o f t h e sam e r e a c t i o n ( A ). T h i s b e h a v i o r i s c o n t r a s t e d w i t h t h e s t r u c t u r e l e s s r e s u l t s o b t a i n e d f o r i 6 q + i 6 o. T h e c o r r e l a t e d b e h a v i o r m a d e a s t a t i s t i c a l e x p l a n a t i o n ( E r 6 3, M o 6 7 ) u n l i k e l y, a n d s u g g e s t e d t h a t s p e c i a l s t a t e s i n 2 4 Mg n e a r E x = 2 0 M ev w e r e b e i n g p o p u l a t e d. T h e s h a r p n e s s o f t h e s t a t e s i n d i c a t e d t h a t t h e y d i d n o t r e a d i l y m i x w i t h t h e n u m e r o u s ( h u n d r e d s p e r M e V ) n e a r b y c o m p o u n d n u c l e a r l e v e l s, a n d t h e c h a r a c t e r i s t i c w i d t h T < k e V i n d i c a t e d t h e s t a t e s h a d a l i f e t i m e i ~ h / r ~ 1 0 " 2 C s e c. T h i s c o m p a r e s w i t h a t r a n s i t t i m e o f ~ s e c f o r t h e p a s s i n g i o n s. S e m i c l a s s i c a l a r g u m e n t s w o u l d i n d i c a t e a p e r i p h e r a l i n t e r a c t i o n : w h i l e e a c h c a r b o n

13 n u c l e u s h a s a r a d i u s o f a p p r o x i m a t e l y 3 f m, t h e d i s t a n c e o f c l o s e s t a p p r o a c h u n d e r C o u lo m b r e p u l s i o n i s m o r e t h a n 8 fm a t 6 M e V. T h e C h a l k R i v e r g r o u p p r o p o s e d t h a t tw o c a r b o n n u c l e i e n t e r e d a " m o l e c u l a r " s t a t e d i s t i n c t f r o m a f u s e d 2 4 M g, b u t t h e b i n d i n g m e c h a n is m w a s s t i l l t o b e e x p l a i n e d. O n e e a r l y t h e o r y b y D a v i s ( D a 6 0 ) p r o p o s e d t h a t t h e r e s o n a n c e s w e r e r o t a t i o n - v i b r a t i o n s t a t e s i n t h e p o t e n t i a l o f t h e r e l a t i v e m o t i o n o f t h e t w o i o n s : i. e., t h e o p t i c a l p o t e n t i a l. I n a s e n s e t h i s b e g g e d t h e q u e s t i o n : a n o p t i c a l p o t e n t i a l c a n h i d e a l o t o f p h y s i c s. M o r e o v e r, n o p h y s i c a l l y r e a s o n a b l e c h o i c e o f o p t i c a l m o d e l p a r a m e t e r s w a s a b l e t o r e p r o d u c e t h e e x p e r i m e n t a l l y - o b s e r v e d s p e c t r u m. V o g t a n d M c M a n u s ( V o 6 0 ) s a w t h e g r o u n d s t a t e d e f o r m a t i o n a n d e a s y d e f o r m a b i l i t y o f 1 2 C a s e s s e n t i a l i n g r e d i e n t s ; t h e r i g i d c l o s e d - s h e l l n u c l e u s w a s s e e n t o b e i n c a p a b l e o f m o l e c u l a r b i n d i n g. D e c a y t o c o m p o u n d n u c l e a r s t a t e s w a s t h o u g h t t o b e i n h i b i t e d b y t h e c e n t r i f u g a l b a r r i e r f o r h i g h s p i n s t a t e s, a n d b y a h i g h n u c l e o n i c r e a r r a n g e m e n t e n e r g y r e f l e c t i n g t h e P a u l i p r i n c i p l e ( W i 6 1 ). b ) L a t e r e x p e r i m e n t s a n d t h e o r i e s A d d i t i o n a l d a t a u p o n w h i c h t o j u d g e t h e t h e o r i e s w e r e c a l l e d f o r. F o r e a c h r e s o n a n c e o n e c o u l d i n p r i n c i p l e m e a s u r e t h e c r o s s s e c t i o n o, t h e i n t r i n s i c s p i n J, t h e t o t a l w i d t h r, a n d t h e p a r t i a l d e c a y w i d t h s r i t o v a r i o u s e x i t c h a n n e l s i. I n p a r t i c u l a r, l a r g e w i d t h s t o t h e e l a s t i c a n d s p e c i f i c r e a c t i o n c h a n n e l s c o u l d i n d i c a t e t h a t t h e e x c i t a t i o n f u n c t i o n f l u c t u a t i o n s w e r e n o t m e r e l y s t a t i s t i c a l a r t i f a c t s f r o m c o m p o u n d n u c l e a r l e v e l s. F u r t h e r e x p e r i m e n t a l w o r k a t C h a l k R i v e r

14 4 ( B r 6 1, A 1 6 3, K u 6 3 ) e s t i m a t e d t h e c a r b o n p a r t i a l w i d t h a n d e s t a b l i s h e d t h e s p i n s o f t h e r e s o n a n c e s t o b e f a i r l y s m a l l ( 2 +, 4 + ). T h e s e n e w d a t a d i d n o t g i v e a r i n g i n g e n d o r s e m e n t t o a n y e x t a n t t h e o r y. T h e c e n t r i f u g a l b a r r i e r w a s n o t h i g h e n o u g h t o p r e v e n t s p r e a d i n g t o n e a r b y c o m p o u n d s t a t e s, a n d t h e e f f e c t o f t h e e x c l u s i o n p r i n c i p l e w a s n o t e a s i l y t e s t e d. D u r i n g t h i s p e r i o d n e w m o d e ls w e r e n e e d e d t o e x p l a i n t h e d a t a, a n d t h e f i e l d l a y r e l a t i v e l y i n a c t i v e. T h e t o p p a n e l o f f i g i l l u s t r a t e s t h e d i f f i c u l t i e s i n t h e m o d e l s. D e p i c t e d i s t h e r e a l p a r t o f t h e p o t e n t i a l (C o u lo m b p l u s n u c l e a r p l u s c e n t r i f u g a l c o n t r i b u t i o n s ) f o r t h e o n e - b o d y d e s c r i p t i o n o f t h e r e l a t i v e m o t i o n o f t h e tw o c a r b o n n u c l e i. S t r u c t u r e s s e e n i n t h e d a t a w e r e n o t b r o a d e n o u g h ( a f e w M e V ) t o b e v i r t u a l s h a p e r e s o n a n c e s ( E 2 ) a b o v e t h e p o t e n t i a l w e l l. T h e n a r r o w w i d t h ( i. e., l o n g l i f e t i m e ) m i g h t d i c t a t e q u a s i b o u n d s t a t e s w i t h i n t h e w e l l, b u t d i r e c t p o p u l a t i o n a t E x s e e m e d t o r e q u i r e t u n n e l i n g t h r o u g h a s u b s t a n t i a l b a r r i e r, i m p l y i n g a s m a l l c r o s s s e c t i o n. A c t i n g u p o n a s u g g e s t i o n b y N o g a m i, I m a n i s h i p r o p o s e d ( I m 6 8, I m 6 9 ) t h a t t h e u n d e r l y i n g b i n d i n g m e c h a n is m w a s a v i r t u a l e x c i t a t i o n o f o n e o f t h e c a r b o n i o n s t o t h e 2, + s t a t e. T h i s l e v e l a t E = M e V i s k n o w n 1 x t o b e s t r o n g l y c o l l e c t i v e a n d a m e m b e r o f t h e g r o u n d s t a t e b a n d o f q u a d r u p o l e s h a p e d e f o r m a t i o n s. T h e tw o c a r b o n i o n s, a p p r o a c h i n g e a c h o t h e r a t E 3 i n t h e l o w e r p a n e l o f f i g , a r e d r o p p e d b y t h e Q - v a l u e o f t h e v i r t u a l i n e l a s t i c s c a t t e r i n g i n t o q u a s i - b o u n d s t a t e s o f t h e i n t e r a c t i o n p o t e n t i a l. T h i s m e c h a n is m r e p r o d u c e s t h e o b s e r v e d n a r r o w w i d t h s w i t h o u t r e q u i r i n g t u n n e l i n g, a n d w i t h a s u i t a b l e s e t o f p a r a m e t e r s r e p r o d u c e d t h e e n e r g i e s a n d s p i n s o f t h e t h r e e r e s o n a n c e s

15 ) S C H E M A T IC P O T E N T IA L FOR Q U A SIM O LE C U LA R IN T E R A C T IO N S T h e r e a l p a r t V ( r ) o f t h e o p t i c a l p o t e n t i a l i s p l o t t e d v e r s u s t h e r e l a t i v e d i s t a n c e r b e t w e e n t h e tw o n u c l e i. I n t e r a c t i o n e n e r g y E j c o r r e s p o n d s t o a q u a s i b o u n d s t a t e i n t h e p o t e n t i a l w e l l. A v i r t u a l, o r s h a p e, r e s o n a n c e o c c u r s a t E 2. E 3 i s a t a n e n e r g y E r g r e a t e r t h a n a q u a s i b o u n d s t a t e, w h e r e E ^ i s t h e e x c i t a t i o n e n e r g y o f o n e o r b o t h o f t h e i n d i v i d u a l n u c l e i. E 4 c o r r e s p o n d s t o a s h a p e r e s o n a n c e w h i c h i s E x a b o v e a q u a s i b o u n d s t a t e. T h e r e l a t i o n o f t h e s e f o u r c a s e s t o q u a s i m o l e c u l a r m o d e ls i s d i s c u s s e d i n s e c t s b a n d 1. 3 b.

16 QUASIMOLECULAR INTERACTIONS ENERGY ENERGY SEPARATION DISTANCE Fig. 101

17 s i n g l e a n d m u t u a l 2* a n d s i n g l e 3 " ( E x = M e V ) e x c i t a t i o n s d e n o t e d r e s p e c t i v e l y a s ( 0 +, 0 + ), (2*, 0 + ), ( 2 +, 2 + ), a n d ( 3 ', 0 + ). T h i s c a l c u l a t i o n t a k e s t h e I m a n i s h i m o d e l t o i t s l o g i c a l c o n c l u s i o n b y i n c l u d i n g a l l p o s s i b l e e x c i t a t i o n m o d e s o f t h e c a r b o n n u c l e i t h o u g h t t o b e i m p o r t a n t i n t h i s e n e r g y r e g i o n. I n y e t a n o t h e r e x t e n s i o n o f N o g a m i 's o r i g i n a l i d e a, T a n i m u r a ( T a 7 8, T a 8 0 ) h a s c o u p l e d t h o s e sam e f o u r c h a n n e l s a n d c a l c u l a t e d l 2 C + l 2 C s c a t t e r i n g u s i n g a m o d i f i e d a d i a b a t i c p o t e n t i a l ( M A P ), i n c o n t r a s t t o t h e s u d d e n p o t e n t i a l o f t h e d o u b l e - r e s o n a n c e m o d e l. I n s e c t t h e p r e d i c t i o n s o f BCM a n d MAP c a l c u l a t i o n s f o r i n e l a s t i c s c a t t e r i n g a r e c o m p a r e d. c ) S p e c t r u m g e n e r a t i o n O t h e r a p p r o a c h e s h a v e b e e n f o l l o w e d i n g e n e r a t i n g a s p e c t r u m o f r e s o n a n c e s t h a t m i g h t r e p r o d u c e t h e e x p e r i m e n t a l s e q u e n c e. F o r e x a m p l e, C o s m a n, e t a l., ( C o 8 1 ) h a v e r e c e n t l y p r o p o s e d t h a t t h e 1 2 C + l 2 C r e s o n a n c e s b e g r o u p e d i n t o f i v e c o l l e c t i v e b a n d m u l t i p l e t s, a n d s u g g e s t e d t h e i r b a s i s t o b e s h e l l m o d e l s h a p e i s o m e r s o f 2 4 M g. F ro m a d i f f e r e n t t h e o r e t i c a l b a s i s, I a c h e l l o ( I a 8 1 ) h a s a r g u e d t h a t q u a s i m o l e c u l a r r e s o n a n c e s, i n s t e a d o f b e i n g g o v e r n e d b y t h e q u a d r u p o l e d e g r e e s o f f r e e d o m t h a t d o m i n a t e l o w - l y i n g e x c i t e d s t a t e s o f t h e n u c l e i, s h o u l d b e g o v e r n e d b y d i p o l e o p e r a t o r s c h a r a c t e r i s t i c o f d i a t o m i c m o l e c u l e s. A p p r o a c h i n g t h e s u b j e c t i n a n a l g e b r a i c m a n n e r, h e p o i n t s o u t t h a t t h e a p p r o p r i a t e g r o u p s y m m e tr y o f t h e p r o b l e m s h o u l d b e U ( 4 ), a n d i t s d e c o m p o s i t i o n i n t o t h e c h a i n U ( 4 ) 3 0 ( 4 ) 3 0 ( 3 ) 3 0 ( 2 ) w o u l d

18 20 generate a spectrum of resonances with energies E(J,v )= C q + C 1(v+^)+ C2(v + )2 + C 3J(J+1), i n w h i c h v a n d J a r e t h e v i b r a t i o n a l a n d s p i n q u a n tu m n u m b e r s, r e s p e c t i v e l y. E r b a n d B r o m l e y ( E r 8 1 ) h a v e a p p l i e d t h i s f o r m u l a, w h i c h c o r r e s p o n d s t o a p p r o x i m a t e s o l u t i o n s o f t h e M o r s e p o t e n t i a l f o r m o l e c u l e s ( S c 6 8, s e c t. 4 9 ), t o t h e 2 8 c o r r e l a t e d l 2 C + l 2 C r e s o n a n c e s b e l o w 1 3 M ev w h o s e s p i n s a r e e s t a b l i s h e d. T h e y d i s c o v e r e d t h a t t h e e n e r g i e s c a n b e r e p r o d u c e d w i t h f o u r f r e e p a r a m e t e r s CL ( o n e o f w h i c h, C3, i s i n v e r s e l y p r o p o r t i o n a l t o t h e m o m e n t o f i n e r t i a ) a n d t w o q u a n tu m n u m b e r s, J a n d v. E r b a n d B r o m le y a l s o s u g g e s t e d t h a t t h e s t a t e a t M ev m a y p r o v e t o b e m o r e i m p o r t a n t i n t h e m o l e c u l a r b i n d i n g t h a n p r e v i o u s l y s u s p e c t e d. P r e v i o u s l y, T o h s a k i - S u z u k i ( T o 7 8 ) h a d p r o p o s e d t h a t c o u p l i n g t o t h i s " 3 a " s t a t e w o u l d e x p l a i n w h y t h e 8 B e + l 6 0 e x i t c h a n n e l i s s o p r o m i n e n t i n t h e 1 2 C C i n t e r a c t i o n. d ) M i c r o s c o p i c m o d e ls I n c o n t r a s t t o t h e p r e c e d i n g m a c r o s c o p i c a p p r o a c h e s, o t h e r r e s e a r c h e r s h a v e t r i e d t o u n d e r s t a n d t h e e x p e r i m e n t a l d a t a o n m i c r o s c o p i c b a s e s. T w o b a s i c a p p r o a c h e s h a v e b e e n a t t e m p t e d : s h e l l m o d e l a n d s e l f - c o n s i s t e n t c a l c u l a t i o n s. W i t h i n t h e f o r m e r, L e a n d e r a n d L a r s s o n ( L e 7 5 ) h a v e c a l c u l a t e d p o t e n t i a l e n e r g y s u r f a c e s u s i n g a m o d i f i e d N i l s s o n o n e - c e n t e r s h e l l m o d e l. F o r 2 *M g t h e y o b t a i n s e c o n d a r y m in im a a t l a r g e d e f o r m a t i o n s, a n d s u g g e s t t h a t m o l e c u l a r s t a t e s m a y b e a s s o c i a t e d w i t h t h e s e, i n a n a l o g y

19 21 with fission isomers. M o r e r e c e n t l y, C h a n d r a a n d M o s e l ( C h 7 8 ) h a v e o b t a i n e d, f r o m t w o - c e n t e r s h e l l m o d e l c a l c u l a t i o n s, a n i n t e r a c t i o n p o t e n t i a l w i t h a s h o u l d e r n e a r 4. 5 f m. T h e y a r g u e t h a t t h i s i s a s s o c i a t e d w i t h a n a v o i d e d l e v e l c r o s s i n g b e t w e e n t h e f _. _ a n d p l e v e l s i n t h e //fc X /c c o m p o u n d n u c l e u s, w h i c h r e a r r a n g e s t h e n u c l e o n i c c o n f i g u r a t i o n i n 2 4 M g t o m i n i m i z e e n e r g y i n t h e s e p a r a t e d s y s t e m. T h e a u t h o r s d r a w p a r a l l e l s t o t h e p r e c e d i n g w o r k, b u t a r g u e t h a t a f u l l t w o - c e n t e r c a l c u l a t i o n i s n e e d e d t o i d e n t i f y w h i c h p o t e n t i a l m in im u m o f L e a n d e r a n d L a r s s o n c o r r e s p o n d s t o t h e m o l e c u l a r s t a t e s. M i c r o s c o p i c c a l c u l a t i o n s s t a r t i n g f r o m t h e n u c l e o n - n u c l e o n i n t e r a c t i o n h a v e u s e d s e l f - c o n s i s t e n t m e t h o d s s u c h a s t i m e - d e p e n d e n t H a r t r e e - F o c k ( T D H F ). E a r l y s t u d i e s b y M a r u h n a n d C u s s o n ( M a 7 6 ) c o u l d c o n s i d e r o n l y h e a d - o n c o l l i s i o n s ( L = 0 ), b u t d e n s i t y p r o f i l e s i n d i c a t e d t h a t a f t e r t h e c o l l i s i o n t h e s y s t e m s p e n t a s i g n i f i c a n t t i m e i n a m o l e c u l e - l i k e c o n f i g u r a t i o n. L a t e r t h r e e - d i m e n s i o n a l c o n s t r a i n e d T D H F c a l c u l a t i o n s b y t h e O r s a y - L i v e r m o r e g r o u p ( C u 7 9, F ) d e m o n s t r a t e d t h a t t h e o r i e n t a t i o n o f t h e o b l a t e c a r b o n g r o u n d s t a t e s h a s a l a r g e e f f e c t o n t h e s c a t t e r i n g. I n p a r t i c u l a r, t h e a x i a l c o n f i g u r a t i o n ( f a c e - o n - f a c e s c a t t e r i n g ) g i v e s r e s u l t s s i m i l a r t o t h e u s u a l o p t i c a l p o t e n t i a l i n t h i s m a s s r e g i o n. I n c o n t r a s t, t r i a x i a l s c a t t e r i n g ( e d g e - o n - e d g e ) i s s t r i k i n g l y d i f f e r e n t, g i v i n g a s h a l l o w p o t e n t i a l s h a p e w i t h a m in im u m a t a l a r g e r r a d i u s t h a t i s n e e d e d t o s u p p o r t t h e c o l l e c t i v e m o l e c u l a r s t a t e s. 2 D e n s i t y c o n t o u r s i n t h i s w o r k s u g g e s t e d a l o n g - l i v e d m o l e c u l a r 2 C h a n d r a a n d M o s e l h a d r e a c h e d s i m i l a r c o n c l u s i o n s.

20 22 c o n f i g u r a t i o n. T h e s e a u t h o r s a l s o d e m o n s t r a t e d, b y c a l c u l a t i n g a c o l l e c t i v e m a s s p a r a m e t e r, t h a t t h e n u c l e a r s t r u c t u r e o f t h e i n d i v i d u a l c a r b o n i o n s c h a n g e s r a p i d l y i n t h e c o l l i s i o n. T h i s i m p l i e s t h a t m o d e ls b a s e d o n s t a t i c p r o p e r t i e s o f t h e 1 2 C n u c l e u s c a n b e o n l y a p p r o x i m a t e a t b e s t, a n d a f u l l d y n a m i c a l u n d e r s t a n d i n g o f t h e r e a c t i o n r e l i e s o n f u t u r e m i c r o s c o p i c c a l c u l a t i o n s o f t h i s k i n d I n e l a s t i c s c a t t e r i n g A s t h e I m a n i s h i m o d e l a n d i t s e x t e n s i o n s r e c o g n i z e d, t h e w a v e f u n c t i o n f o r t h e 2 + s t a t e h a s a l a r g e o v e r l a p w i t h t h e g r o u n d s t a t e o f 1 2 C. I f i t i s e n e r g e t i c a l l y p o s s i b l e, w e w o u l d e x p e c t a l a r g e c r o s s s 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 t o t h i s s t a t e. M o r e o v e r, b e c a u s e o f i t s p r e s u m e d r o l e i n t h e b i n d i n g m e c h a n is m o f t h e q u a s i m o l e c u l e s, i t m a y b e e x p e c t e d t h a t t h i s e x i t c h a n n e l w i l l d e m o n s t r a t e r e s o n a n t p h e n o m e n a. B u t a t h i g h e r e n e r g i e s a n d s p i n s i t b e c o m e s i n c r e a s i n g l y d i f f i c u l t t o e s t a b l i s h a n o n - s t a t i s t i c a l o r i g i n t o s t r u c t u r e s i n a n e x c i t a t i o n f u n c t i o n. I n p a r t i c u l a r, n o t a l l e x i t c h a n n e l s m a y h a v e a n a p p r e c i a b l e b r a n c h i n g f r o m e a c h r e s o n a n c e, a n d i n t e r f e r e n c e f r o m a r e s o n a n c e b a c k g r o u n d a m p l i t u d e m a y s h i f t a p e a k som e f r a c t i o n o f i t s w i d t h. I n p a r t i c u l a r, S h a p i r a, e t a l., ( S h 7 4 ) d e m o n s t r a t e d t h a t d a t a k n o w n a t t h a t t i m e ( ) i n e l a s t i c, i n e l a s t i c, a n d a l p h a p a r t i c l e c h a n n e l s a b o v e M ev w e r e c o n s i s t e n t w i t h - - a l t h o u g h t h e y d i d n o t r e q u i r e a H a u s e r - F e s h b a c h s t a t i s t i c a l e x p l a n a t i o n. W i t h t h i s m o t i v a t i o n, C o r m i e r, e t a l., ( C o 7 7, C o 7 8 ) m e a s u r e d t h e t o t a l c r o s s s e c t i o n f o r s i n g l e a n d m u t u a l i n e l a s t i c s c a t t e r i n g t o t h e 2 + s t a t e

21 23 as a function of beam energy; their results are shown in fig The p e r s i s t e n c e o f t h e n a r r o w (r~ a f e w h u n d r e d k e V ) s t r u c t u r e s u p t o a b o u t 2 0 M ev b e a r s n o t i c e. I n a d d i t i o n, s t r i k i n g a n d r e g u l a r l y - s p a c e d b r o a d s t r u c t u r e s (r~ a f e w M e V ), t h a t c o n t i n u e t o t h e h i g h e s t e n e r g i e s m e a s u r e d, a l s o a p p e a r. T h e e n e r g i e s o f t h e c e n t r o i d s o f t h e s e s t r u c t u r e s a r e r o u g h l y c o r r e l a t e d a m o n g t h e t w o i n e l a s t i c c h a n n e l s a n d t h e f u s i o n d a t a o f S p e r r, e t a l., ( S p 7 6 ). B a s e d o n e n e r g y s y s t e m a t i c s o f t h e b r o a d s t r u c t u r e s, C o r m i e r i n t e r p r e t e d th e m a s a c o n t i n u a t i o n o f t h e d i n u c l e a r r e s o n a n c e s t a t e s s e e n a t t h e l o w e r e n e r g i e s. I f o n e w e r e t o p l o t t h e c e n t r o i d s o f t h e s e s t r u c t u r e s v e r s u s t h e p o s t u l a t e d s p i n s i n d i c a t e d i n f i g , t h e p o i n t s w o u l d f a l l o n t h e s am e l o c u s i n f i g t h a t e n c o m p a s s e d t h e l o w e r e n e r g y r e s o n a n c e s. A p a r t i a l w i d t h d e c o m p o s i t i o n ( C o 7 8 ) o f t h e g r o s s f e a t u r e s i n d i c a t e d r ^ / r fc o t = f o r t h e p r e s u m e d r e s o n a n c e s, w h i l e r e d u c e d w i d t h s f o r t h e ( 2 +, 0 + ) a n d ( 2 +, 2 + ) c h a n n e l s w e r e f o u n d t o b e c o m p a r a b l e 3 t o t h o s e o f t h e ( 0 +, 0 + ) c h a n n e l. T h e s e f a c t s a r e e v i d e n c e o f t h e m o l e c u l a r n a t u r e o f t h e s e s t r u c t u r e s. C o r m i e r i n t e r p r e t e d t h e b r o a d s t r u c t u r e s a s d o o r w a y s t a t e s, s l i g h t l y m o re c o m p l i c a t e d t h a n s i m p l e s h a p e r e s o n a n c e s, w i t h c o u p l i n g l e a d i n g t o m o r e c o m p le x c o n f i g u r a t i o n s a n d f r a g m e n t a t i o n b e l o w 2 0 M e V. A f u r t h e r d i s c u s s i o n o f t h i s i n t e r p r e t a t i o n o f t h e d a t a c a n b e f o u n d i n r e f. C o 8 0. K o n d o, e t a l., ( K o 7 9 a ) a n d T a n i m u r a ( T a 8 0 ) h a v e r e p r o d u c e d t h e e n e r g i e s o f t h e b r o a d s t r u c t u r e s v i a c o u p l e d - c h a n n e l t r e a t m e n t o f t h e 3 I n f a c t, t h e i n e l a s t i c c r o s s s e c t i o n s a r e h u g e ; t h e sum o f t h e s i n g l e a n d m u t u a l 2 + e x c i t a t i o n a r e % o f t h e t o t a l r e a c t i o n c r o s s s e c t i o n i n t h i s e n e r g y r e g i o n, s o a n y s u c c e s s f u l t h e o r y m u s t a c c o u n t f o r t h e k e y r o l e t h e i n e l a s t i c c h a n n e l s p l a y i n t h e s c a t t e r i n g.

22 104) INELASTIC SCATTERING EXCITATION FUNCTIONS T h e u p p e r p a n e l i n d i c a t e s t h e i n t e g r a t e d c r o s s s e c t i o n f o r l 2 C C s c a t t e r i n g w h i c h l e a v e s o n e o f t h e n u c l e i i n t h e 2 + s t a t e ; t h e l o w e r p a n e l s h o w s o f o r t h e c a s e w h e n b o t h e x i t i n g i o n s a r e i n t h a t s t a t e. T h e d a t a f o r E > 1 5 M ev a r e f r o m C o 7 7 a n d C o 7 8. T h e d a t a f o r E < 1 5 M ev a r e f r o m E m 7 3, w i t h t h e r e l a t i v e n o r m a l i z a t i o n a s n o t e d i n F u 8 0. A n e a r l i e r v e r s i o n o f t h i s f i g u r e ( C o 7 8 ) h a d t h e E m 73 c r o s s s e c t i o n s t o o l a r g e b y n e a r l y a f a c t o r o f ) C O M PARISO N O F RESONANCE M ODELS W IT H IN E L A S T IC S C A T T E R IN G T h e s o l i d l i n e r e p r e s e n t s t h e c r o s s s e c t i o n f o r t h e 1 2 C ( 1 2 C, 1 2 C ) l 2 C ( 2 + ) r e a c t i o n, a s p r e s e n t e d i n f i g T h e o t h e r l i n e s a r e c o u p l e d - c h a n n e l p r e d i c t i o n s f o r t h i s e x c i t a t i o n f u n c t i o n : t h e d o t t e d l i n e u s i n g t h e b a n d c r o s s i n g m o d e l ( K o 7 9 a ) a n d t h e d a s h e d l i n e u s i n g t h e m o d i f i e d a d i a b a t i c p o t e n t i a l ( T a 8 0 ).

23 or(mb) E-cm (MeV) Fig. 104

24 <x ( m b ) E C M ( M e V ) Fig. 105

25 21 with fission isomers. M o r e r e c e n t l y, C h a n d r a a n d M o s e l ( C h 7 8 ) h a v e o b t a i n e d, f r o m t w o - c e n t e r s h e l l m o d e l c a l c u l a t i o n s, a n i n t e r a c t i o n p o t e n t i a l w i t h a s h o u l d e r n e a r 4. 5 f m. T h e y a r g u e t h a t t h i s i s a s s o c i a t e d w i t h a n a v o i d e d l e v e l c r o s s i n g b e t w e e n t h e f? ^ 2 a n d p ^ 2 l e v e l s i n t h e c o m p o u n d n u c l e u s, w h i c h r e a r r a n g e s t h e n u c l e o n i c c o n f i g u r a t i o n i n 2 4 M g t o m i n i m i z e e n e r g y i n t h e s e p a r a t e d s y s t e m. T h e a u t h o r s d r a w p a r a l l e l s t o t h e p r e c e d i n g w o r k, b u t a r g u e t h a t a f u l l t w o - c e n t e r c a l c u l a t i o n i s n e e d e d t o i d e n t i f y w h i c h p o t e n t i a l m in im u m o f L e a n d e r a n d L a r s s o n c o r r e s p o n d s t o t h e m o l e c u l a r s t a t e s. M i c r o s c o p i c c a l c u l a t i o n s s t a r t i n g f r o m t h e n u c l e o n - n u c l e o n i n t e r a c t i o n h a v e u s e d s e l f - c o n s i s t e n t m e th o d s s u c h a s t i m e - d e p e n d e n t H a r t r e e - F o c k ( T D H F ). E a r l y s t u d i e s b y M a r u h n a n d C u s s o n ( M a 7 6 ) c o u l d c o n s i d e r o n l y h e a d - o n c o l l i s i o n s ( L = 0 ), b u t d e n s i t y p r o f i l e s i n d i c a t e d t h a t a f t e r t h e c o l l i s i o n t h e m o l e c u l e - l i k e c o n f i g u r a t i o n. s y s t e m s p e n t a s i g n i f i c a n t t i m e i n a L a t e r t h r e e - d i m e n s i o n a l c o n s t r a i n e d TD H F c a l c u l a t i o n s b y t h e O r s a y - L i v e r m o r e g r o u p ( C u 7 9, F ) d e m o n s t r a t e d t h a t t h e o r i e n t a t i o n o f t h e o b l a t e c a r b o n g r o u n d s t a t e s h a s a l a r g e e f f e c t o n t h e s c a t t e r i n g. I n p a r t i c u l a r, t h e a x i a l c o n f i g u r a t i o n ( f a c e - o n - f a c e s c a t t e r i n g ) g i v e s r e s u l t s s i m i l a r t o t h e u s u a l o p t i c a l p o t e n t i a l i n t h i s m a s s r e g i o n. I n c o n t r a s t, t r i a x i a l s c a t t e r i n g ( e d g e - o n - e d g e ) i s s t r i k i n g l y d i f f e r e n t, g i v i n g a s h a l l o w p o t e n t i a l s h a p e w i t h a m in im u m a t a l a r g e r r a d i u s t h a t i s n e e d e d t o s u p p o r t t h e c o l l e c t i v e m o l e c u l a r s t a t e s. 2 D e n s i t y c o n t o u r s i n t h i s w o r k s u g g e s t e d a l o n g - l i v e d m o l e c u l a r 2 C h a n d r a a n d M o s e l h a d r e a c h e d s i m i l a r c o n c l u s i o n s.

26 22 c o n f i g u r a t i o n. T h e s e a u t h o r s a l s o d e m o n s t r a t e d, b y c a l c u l a t i n g a c o l l e c t i v e m a s s p a r a m e t e r, t h a t t h e n u c l e a r s t r u c t u r e o f t h e i n d i v i d u a l c a r b o n i o n s c h a n g e s r a p i d l y i n t h e c o l l i s i o n. T h i s i m p l i e s t h a t m o d e ls b a s e d o n s t a t i c p r o p e r t i e s o f t h e 1 2 C n u c l e u s c a n b e o n l y a p p r o x i m a t e a t b e s t, a n d a f u l l d y n a m i c a l u n d e r s t a n d i n g o f t h e r e a c t i o n r e l i e s o n f u t u r e m i c r o s c o p i c c a l c u l a t i o n s o f t h i s k i n d I n e l a s t i c s c a t t e r i n g A s t h e I m a n i s h i m o d e l a n d i t s e x t e n s i o n s r e c o g n i z e d, t h e w a v e f u n c t i o n f o r t h e 2* s t a t e h a s a l a r g e o v e r l a p w i t h t h e g r o u n d s t a t e o f 1 2 C. I f i t i s e n e r g e t i c a l l y p o s s i b l e, w e w o u l d e x p e c t a l a r g e c r o s s s 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 t o t h i s s t a t e. M o r e o v e r, b e c a u s e o f i t s p r e s u m e d r o l e i n t h e b i n d i n g m e c h a n is m o f t h e q u a s i m o l e c u l e s, i t m a y b e e x p e c t e d t h a t t h i s e x i t c h a n n e l w i l l d e m o n s t r a t e r e s o n a n t p h e n o m e n a. B u t a t h i g h e r e n e r g i e s a n d s p i n s i t b e c o m e s i n c r e a s i n g l y d i f f i c u l t t o e s t a b l i s h a n o n - s t a t i s t i c a l o r i g i n t o s t r u c t u r e s i n a n e x c i t a t i o n f u n c t i o n. I n p a r t i c u l a r, n o t a l l e x i t c h a n n e l s m a y h a v e a n a p p r e c i a b l e b r a n c h i n g f r o m e a c h r e s o n a n c e, a n d i n t e r f e r e n c e f r o m a r e s o n a n c e b a c k g r o u n d a m p l i t u d e m a y s h i f t a p e a k som e f r a c t i o n o f i t s w i d t h. I n p a r t i c u l a r, S h a p i r a, e t a l., ( S h 7 4 ) d e m o n s t r a t e d t h a t d a t a k n o w n a t t h a t t i m e ( ) i n e l a s t i c, i n e l a s t i c, a n d a l p h a p a r t i c l e c h a n n e l s a b o v e M e V w e r e c o n s i s t e n t w i t h - - a l t h o u g h t h e y d i d n o t r e q u i r e a H a u s e r - F e s h b a c h s t a t i s t i c a l e x p l a n a t i o n. W i t h t h i s m o t i v a t i o n, C o r m i e r, e t a l., ( C o 7 7, C o 7 8 ) m e a s u r e d t h e t o t a l c r o s s s e c t i o n f o r s i n g l e a n d m u t u a l i n e l a s t i c s c a t t e r i n g t o t h e 2 + s t a t e

27 23 as a function of beam energy; their results are shown in fig The p e r s i s t e n c e o f t h e n a r r o w (r~ a f e w h u n d r e d k e V ) s t r u c t u r e s u p t o a b o u t 2 0 M e V b e a r s n o t i c e. I n a d d i t i o n, s t r i k i n g a n d r e g u l a r l y - s p a c e d b r o a d s t r u c t u r e s ( r ~ a f e w M e V ), t h a t c o n t i n u e t o t h e h i g h e s t e n e r g i e s m e a s u r e d, a l s o a p p e a r. T h e e n e r g i e s o f t h e c e n t r o i d s o f t h e s e s t r u c t u r e s a r e r o u g h l y c o r r e l a t e d a m o n g t h e t w o i n e l a s t i c c h a n n e l s a n d t h e f u s i o n d a t a o f S p e r r, e t a l., ( S p 7 6 ). B a s e d o n e n e r g y s y s t e m a t i c s o f t h e b r o a d s t r u c t u r e s, C o r m i e r i n t e r p r e t e d th e m a s a c o n t i n u a t i o n o f t h e d i n u c l e a r r e s o n a n c e s t a t e s s e e n a t t h e l o w e r e n e r g i e s. I f o n e w e r e t o p l o t t h e c e n t r o i d s o f t h e s e s t r u c t u r e s v e r s u s t h e p o s t u l a t e d s p i n s i n d i c a t e d i n f i g , t h e p o i n t s w o u l d f a l l o n t h e sam e l o c u s i n f i g t h a t e n c o m p a s s e d t h e l o w e r e n e r g y r e s o n a n c e s. A p a r t i a l w i d t h d e c o m p o s i t i o n ( C o 7 8 ) o f t h e g r o s s f e a t u r e s i n d i c a t e d p e ^ a s / r t o t = f o r t h e p r e s u m e d r e s o n a n c e s, w h i l e r e d u c e d w i d t h s f o r t h e ( 2 +, 0 + ) a n d ( 2 +, 2 + ) c h a n n e l s w e r e f o u n d t o b e c o m p a r a b l e 3 t o t h o s e o f t h e ( 0 +, 0 + ) c h a n n e l. T h e s e f a c t s a r e e v i d e n c e o f t h e m o l e c u l a r n a t u r e o f t h e s e s t r u c t u r e s. C o r m i e r i n t e r p r e t e d t h e b r o a d s t r u c t u r e s a s d o o r w a y s t a t e s, s l i g h t l y m o re c o m p l i c a t e d t h a n s i m p l e s h a p e r e s o n a n c e s, w i t h c o u p l i n g l e a d i n g t o m o re c o m p le x c o n f i g u r a t i o n s a n d f r a g m e n t a t i o n b e l o w 2 0 M e V. A f u r t h e r d i s c u s s i o n o f t h i s i n t e r p r e t a t i o n o f t h e d a t a c a n b e f o u n d i n r e f. C o 8 0. H o n d o, e t a l., ( K o 7 9 a ) a n d T a n i m u r a ( T a 8 0 ) h a v e r e p r o d u c e d t h e e n e r g i e s o f t h e b r o a d s t r u c t u r e s v i a c o u p l e d - c h a n n e l t r e a t m e n t o f t h e 3 I n f a c t, t h e i n e l a s t i c c r o s s s e c t i o n s a r e h u g e ; t h e sum o f t h e s i n g l e a n d m u t u a l 2 + e x c i t a t i o n a r e % o f t h e t o t a l r e a c t i o n c r o s s s e c t i o n i n t h i s e n e r g y r e g i o n, s o a n y s u c c e s s f u l t h e o r y m u s t a c c o u n t f o r t h e k e y r o l e t h e i n e l a s t i c c h a n n e l s p l a y i n t h e s c a t t e r i n g.

28 ) IN E L A S T IC S C A T T E R IN G E X C IT A T IO N F U N C T IO N S T h e u p p e r p a n e l i n d i c a t e s t h e i n t e g r a t e d c r o s s s e c t i o n f o r 1 2 C C s c a t t e r i n g w h i c h l e a v e s o n e o f t h e n u c l e i i n t h e 2 + s t a t e ; t h e l o w e r p a n e l s h o w s o f o r t h e c a s e w h e n b o t h e x i t i n g i o n s a r e i n t h a t s t a t e. T h e d a t a f o r E > 1 5 M ev a r e f r o m C o 7 7 a n d C o 7 8. T h e d a t a f o r E < 1 5 M ev a r e f r o m E m 7 3, w i t h t h e r e l a t i v e n o r m a l i z a t i o n a s n o t e d i n F u 8 0. A n e a r l i e r v e r s i o n o f t h i s f i g u r e ( C o 7 8 ) h a d t h e Em 73 c r o s s s e c t i o n s t o o l a r g e b y n e a r l y a f a c t o r o f ) C O M PARISO N OF RESONANCE MODELS W IT H IN E L A S T IC S C A T T E R IN G T h e s o l i d l i n e r e p r e s e n t s t h e c r o s s s e c t i o n f o r t h e 1 2 C ( 1 2 C, 1 2 C ) 1 2 C ( 2 + ) r e a c t i o n, a s p r e s e n t e d i n f i g T h e o t h e r l i n e s a r e c o u p l e d - c h a n n e l p r e d i c t i o n s f o r t h i s e x c i t a t i o n f u n c t i o n : t h e d o t t e d l i n e u s i n g t h e b a n d c r o s s i n g m o d e l ( K o 7 9 a ) a n d t h e d a s h e d l i n e u s i n g t h e m o d i f i e d a d i a b a t i c p o t e n t i a l ( T a 8 0 ).

29 cr ( m b ) ECm (MeV) Fig. 104

30 ct ( m b ) E C M ( M e V ) Fig. 105

31 ( 0 +, 0 + ), ( 2 +, 0 + ), ( 2 +, 2 + ), a n d ( 3 ', 0 + ) e x i t c h a n n e l s. T h e i r c a l c u l a t i o n s f o r t h e ( 2 +, 0 + ) f i n a l s t a t e a r e s h o w n i n f i g , t o g e t h e r w i t h t h e d a t a o f C o r m i e r, e t a l. O f c o u r s e t h e s e m o d e ls r e q u i r e t h e r e s o n a n c e s t o b e s t a t e s o f g o o d a n g u l a r m o m e n tu m. I n a n a t t e m p t t o c o n f i r m t h e r e s o n a n c e i n t e r p r e t a t i o n o f t h e s e d a t a, C a n n e l l, e t a l., ( C a 7 9 ) u s e d a p a r t i c l e - g a m m a c o r r e l a t i o n t e c h n i q u e t o m e a s u r e t h e s p i n s o f t h e c o m p o u n d s y s t e m i n t h e i n e l a s t i c c h a n n e l. T h e y d i s c o v e r e d t h a t t h e b r o a d s t r u c t u r e s c o u l d n o t b e c h a r a c t e r i z e d b y a s i n g l e s p i n v a l u e, w h i c h c a l l e d t h e r e s o n a n c e i n t e r p r e t a t i o n i n t o q u e s t i o n. I n f a c t, c o n s i d e r i n g t h a t t h e c h a r a c t e r i s t i c w i d t h T ~ 3 M ev i m p l i e d l i f e t i m e s T ~ 1 0 " 2 2 s e c, r o u g h l y t h e c o l l i s i o n t i m e, t h e p o s s i b i l i t y w a s r a i s e d t h a t t h e b r o a d s t r u c t u r e s w e r e n o t d o o r w a y s t a t e s t o m o l e c u l a r r e s o n a n c e s, b u t s i m p l y k i n e m a t i c a l a r t i f a c t s o f d i r e c t i n e l a s t i c s c a t t e r i n g. I n s u m m a ry, w h i l e t h e m o l e c u l a r c h a r a c t e r o f t h e r e s o n a n c e s i n t h e b a r r i e r r e g i o n i s f i r m l y e s t a b l i s h e d, t h e n a t u r e o f t h o s e a t h i g h e r e n e r g i e s h a s r e m a i n e d a n o p e n q u e s t i o n. I n t h e f o l l o w i n g s e c t i o n w e e x p l o r e t h e v a r i o u s m o d e ls b a s e d o n n o n - r e s o n a n t d i r e c t r e a c t i o n m e c h a n is m s.

32 Direct reaction models a ) DWBA A t p r e s e n t, t h e s t a n d a r d w a y t o t r e a t a n y n u c l e a r r e a c t i o n i s v i a d i s t o r t e d w a v e B o r n a p p r o x i m a t i o n (D W B A ). T h e o p t i c a l p o t e n t i a l o b t a i n e d f r o m a f i t t o e l a s t i c s c a t t e r i n g d a t a i s u s e d t o c a l c u l a t e i n e l a s t i c s c a t t e r i n g. B y t h i s m e th o d C a n n e l l a n d c o w o r k e r s r e p r o d u c e d t h e g r o s s s t r u c t u r e s o b s e r v e d i n t h e i n e l a s t i c e x c i t a t i o n f u n c t i o n. ( M o r e d e t a i l s a r e g i v e n i n c h a p. 6. ) T h e a u t h o r s a l s o f o u n d t h a t a DWBA t r e a t m e n t g a v e a d e q u a t e e x p l a n a t i o n o f t h e p a r t i c l e - g a m m a c o r r e l a t i o n f o r t h e 2* s t a t e m e a s u r e d o u t o f t h e r e a c t i o n p l a n e a t 3 0 M e V, w h i l e t h e c o r r e s p o n d i n g r e s o n a n c e i n t e r p r e t a t i o n p r e d i c t e d a n o u t - o f - p l a n e gamma r a d i a t i o n y i e l d s i x t i m e s l a r g e r t h a n o b s e r v e d. B u t t h e i n - p l a n e / o u t - o f - p l a n e r a t i o f o r t h e o t h e r b r o a d m a x im a a t E = , , a n d M ev s h o w e d t h e a g r e e m e n t b e t w e e n t h e c a l c u l a t i o n s a n d t h e d a t a t o b e w o r s e t h a n t h a t f o r 3 0 M e V. b ) D i f f r a c t i o n A n o t h e r d i r e c t i n t e r a c t i o n a p p r o a c h i s t h a t v i a t h e d i f f r a c t i o n m o d e l o f P h i l l i p s, e t a l ( P h 7 9 ). T h i s a p p r o x i m a t i o n r e p l a c e s t h e DWBA i n e l a s t i c s c a t t e r i n g a m p l i t u d e s b y t h o s e g i v e n b y t h e A u s t e r n - B l a i r r e l a t i o n ( A u 6 5 ). U s i n g a s u i t a b l e p a r a m e t r i z a t i o n t h e s e a u t h o r s w e r e a b l e t o r e p r o d u c e n o t o n l y t h e t o t a l i n e l a s t i c c r o s s s e c t i o n s b u t a l s o t h e i n e l a s t i c a n g u l a r d i s t r i b u t i o n s. W i t h i n t h i s m o d e l, m a x im a i n t h e e x c i t a t i o n f u n c t i o n a r i s e f r o m a n g u l a r m o m e n tu m a n d Q - v a l u e m a t c h i n g,