EVIDENCE FOR INTRINSIC REFLECTION ASYMMETRIC
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- Hugo Clarke
- 5 years ago
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1 A B S T R A C T EVIDENCE FOR INTRINSIC REFLECTION ASYMMETRIC N U C L E A R SHAPES Paul Cottle Yale University 1986 Most of the important advances in the description of collective nuclear phenomena have been based upon the assumption of a reflection symmetry relative to a plane perpendicular to the symmetry axis of the nucleus. It is known, however, that reflection asymmetric shapes, such as alpha clustering configurations and dynamic octupole deformations, can be found in many regions of the periodic table. Isotopes of Ra and Th have long been known to exhibit characteristics suggesting the possibility that static intrinsic reflection asymmetric shapes are present. We examine the nature of reflection asymmetry in Ra and Th by first studying, in detail, three nuclei, 216Rn, 219Ra and 220Ra, produced in the 14C+208Pb fusion reaction using several techniques of gamma ray spectroscopy. Level spectra have been constructed for these nuclei to excitation energies of 2.1 MeV, 4.9 MeV and 3.6 MeV, respectively. Second, we discuss our results within the context of the systematic behavior of other Rn, Ra and Th isotopes and its relationship to predictions of alpha particle cluster, octupole vibration and static octupole deformation models. Third, we find that several observables seen in the Ra-Th-region are qualitatively similar to those found in the
2 Sm-Gd region. Finally, we see that the excitation spectra observed in the mass 217 and 219 Ac and Ra isotopes are described quite well in terms of a weak particle-core coupling model which is generally successful for weakly deformed nuclei. We have found clear evidence for further reflection asymmetric shapes in the light actinide region. We conclude, however, that because of the similarities of the above models we are at present unable to distinguish which model is most appropriate for the description of these nuclei. In addition, we find that the weak coupling model is appropriate for the description of odd-a nuclei in the weakly deformed Ra-Ac region.
3 EVIDENCE FOR INTRINSIC REFLECTION ASYMMETRIC N U C L E A R SHAPES A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Paul Davis Cottle June 1986
4 To Mom, Dad and Tracey
5 ACKNOWLEDGMENTS I have been fortunate enough to have had the support of many outstanding people over the last four years. Unfortunately, I can only mention some of them here. I must first thank Prof. D. Allan Bromley, who contributed in many ways to this work. His endless encouragement, which meant so much coming from a man of his stature, was indispensible in both the research leading to this dissertation and the process of making decisions about my future. Prof. Bromley has my eternal gratitude. Prof. Moshe Gai's wide knowledge of the field of nuclear physics has served as an inspiration to me as well as a valuable resource for my own learning. The contents of this dissertation strongly reflect his "big picture" approach to nuclear physics. I also wish to acknowledge the contributions that Prof. Jolie Ciz- ewski has made to my education and to my self-confidence. I hope that Jolie1s influence on my work, in the future as well as the present, will be obvious to anyone familiar with her accomplishments. I Prof. John Shriner has also played an important role in both my life and my short career. I owe John a great debt for his friendship and help., My fellow graduate students made working in this Laboratory a pleasant and rewarding experience. I have been fortunate to share a hallway with John Ennis, Mario Ruscev, Mark Drigert, Steve Sterbenz, Anna Hayes, Patty Blauner, Laurie Baumel, Kevin Hubbard, Mike Kaurin, John O'Connor, Niki Becker, Bronek Dichter, Steve Rugari, Zhiping Zhao, Paul Magnus,
6 Kooverji Gamadia and (how could I ever forget) Tzu-Fang Wang. Prof. Robert Weller, Dr. Mari Weller, Dr. Nicholas Tsoupas and Prof. Peter Parker have also helped to make this lab a good place to work. I was fortunate to have worked with outstanding collaborators from Brookhaven National Laboratory and SUNY-Stony Brook: Dr. John Olness, Dr. Ernest Warburton, Con Beausang, Dr. Mubina Quader, Dr. Lars Hild- ingsson. Dr. William Piel, Jr., and Prof. David Fossan. I also wish to thank as a group the very helpful people who make up the accelerator operations and support staffs at Brookhaven. Finally, I would like to thank all of. those people who have kept our Laboratory running so smoothly over the past few years, especially Kenzo Sato, Phil Clarkin, Dick D'Alexander, Teddy Duda, Ed Stepensky, Tom Barker, Richard Wagner, Charles Gingell, Al Jeddry, Joe Cimino, Ray Comeau, Nitza Hauser, Rita Bonito, Clare Buckley, Annalee Jacunski, Karen DeFelice, Marvin Curtis, Sandy Sicignano and Mary Anne Schultz. Without these people, the rest of us would have been almost completely lost. /
7 TABLE OF CONTENTS ACKNOWLEDGEMENTS... iii T A B L E O F C O N T E N T S... V T A B L E O F F I G U R E S... v i i i T A B L E O F T A B L E S... x i 1. I N T R O D U C T I O N O V E R V I E W O F T H E O R Y A L P H A P A R T I C L E C L U S T E R I N G I N H E A V Y N U C L E I A N D T H E H Y B R I D M O D E L O C T U P O L E V I B R A T I O N A L B E H A V I O R I N E V E N - E V E N N U C L E I S T A T I C O C T U P O L E D E F O R M A T I O N S I N E V E N - E V E N N U C L E I A F E W C O M M E N T S O N T H E R E L A T I O N S H I P B E T W E E N T H E A L P H A C L U S T E R A N D S T A T I C O C T U P O L E M O D E L S W E A K A N D I N T E R M E D I A T E C O U P L I N G I N T R A N S I T I O N A L O D D - A N U C L E I S T R O N G C O U P L I N G I N O D D - A N U C L E I A N D T H E O C T U P O L E N I L S S O N M O D E L P A R I T Y D O U B L E T S I N R E F L E C T I O N A S Y M M E T R I C O D D - A N U C L E I E X P E R I M E N T A L P R O C E D U R E G E N E R A L F E A T U R E S
8 3. 2 H E A V Y I O N F U S I O N - E V A P O R A T I O N R E A C T I O N S E X P E R I M E N T A L D E T A I L S A N A L Y S I S A N D R E S U L T S P R E V I O U S W O R K O N R n, 2 2 R a A N D R a E X C I T A T I O N F U N C T I O N M E A S U R E M E N T S R a L E V E L S P E C T R U M R n L E V E L S P E C T R U M R a L E V E L S P E C T R U M D I S C U S S I O N S Y S T E M A T I C B E H A V I O R O F E V E N R n, R a a n d T h N U C L E I I N T E R P R E T A T I O N O F R a A N D T h I S O T O P E S I N T E R M S O F A N A L P H A P A R T I C L E C L U S T E R M O D E L S T U D Y O F E V E N R a I S O T O P E S T H R O U G H C A L C U L A T I O N S W I T H T H E U ( 6 ) U ( 4 ) H Y B R I D M O D E L P R E D I C T I O N S F O R T H E O C T U P O L E D E G R E E O F F R E E D O M I N 2 2 R a A C O M P A R I S O N O F T H E Z = I S O T O P E S W I T H T H E Z = O N E S A C O M P A R I S O N O F I - A N D 3 S T A T E S I N T H E R E G I O N S 5 6 < Z < 8 2 A N D Z > S Y S T E M A T I C B E H A V I O R O F O D D - A N U C L E I N E A R A = S U M M A R Y A N D C O N C L U S I O N S
9 B I B L I O G R A P H Y
10 TABLE OF FIGURES Collective Nuclear Motion... P r o p o s e d M o l e c u l a r B a n d i n R e d u c e d A l p h a P a r t i c l e D e c a y W i d t h s A l p h a P a r t i c l e C l u s t e r a n d O c t u p o l e C o n f i g u r a t i o n s P a r t i a l L e v e l S p e c t r u m o f G d U A l p h a P a r t i c l e D e c a y C h a i n P a r t i a l L e v e l S p e c t r u m o f R a P a r t i a l L e v e l S p e c t r u m o f 2 0 N e C l a s s i c a l U ( 6 ) a n d U ( 4 ) R e p r e s e n t a t i o n s. 2 7 S p e c t r o s c o p y o f D y n a m i c a l L i m i t s o f U ( 4 ) 3 1 S c h e m a t i c H y b r i d M o d e l E x a m p l e P a r t i a l L e v e l S p e c t r u m o f E r B ( E 1 ) / B ( E 2 ) v s. J f o r L a n t h a n i d e N u c l e i. 3 8 M o l l e r a n d N i x M a s s C o m p a r i s o n G r o u n d S t a t e P o t e n t i a l E n e r g y S u r f a c e s S p e c t r o s c o p y o f O c t u p o l e C o n f i g u r a t i o n s. 4 4 B a c k b e n d i n g P l o t f o r T h W e a k C o u p l i n g M u l t i p l e t s W o o d s - S a x o n S i n g l e P a r t i c l e L e v e l s L e v e l S p e c t r u m o f A c l 8 i x a ( 1 6 0, x n ) E x c i t a t i o n F u n c t i o n... Compton Suppression Electronics
11 G a m m a - g a m m a T i m i n g E l e c t r o n i c s G a m m a - g a m m a E n e r g y E l e c t r o n i c s i 4 c p b E x c i t a t i o n F u n c t i o n L e v e l S p e c t r u m o f R a R a G a m m a - g a m m a G a t e s R a 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 R n G a m m a - X - r a y G a t e R n G a m m a - g a m m a G a t e L e v e l S p e c t r u m o f R n R a G a m m a - g a m m a G a t e s i g R a 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 L e v e l S p e c t r u m o f R a G r o u n d S t a t e S p i n s o f E v e n - Z, O d d - N Z = N u c l e i S y s t e m a t i c B e h a v i o r o f R a E n e r g y L e v e l s I ( x ) v s. fiw f o r R a, R a S y s t e m a t i c B e h a v i o r o f T h E n e r g y L e v e l s S y s t e m a t i c B e h a v i o r o f E n e r g y L e v e l s i n N = I s o t o n e s a n d R n I s o t o p e s... R a B ( E 1 ) / B ( E 2 ) R a t i o s T h B ( E 1 ) / B ( E 2 ) R a t i o s A l p h a P a r t i c l e H i n d r a n c e F a c t o r s f o r Z = E v i d e n c e f o r A l p h a P a r t i c l e C l u s t e r i n g E p i n H y b r i d M o d e l C a l c u l a t i o n... Hybrid Model Calculation Results
12 P a r t i a l L e v e l S p e c t r u m o f S m P a r t i a l L e v e l S p e c t r u m o f S m P a r t i a l L e v e l S p e c t r u m o f S m P a r t i a l L e v e l S p e c t r u m o f S m P a r t i a l L e v e l S p e c t r u m o f S m P a r t i a l L e v e l S p e c t r u m o f G d P a r t i a l L e v e l S p e c t r u m o f G d P a r t i a l L e v e l S p e c t r u m o f G d P a r t i a l L e v e l S p e c t r u m o f G d S y s t e m a t i c B e h a v i o r o f Sm E n e r g y L e v e l s S y s t e m a t i c B e h a v i o r o f G d E n e r g y L e v e l s S y s t e m a t i c B e h a v i o r o f R a E n e r g y L e v e l s G r o u n d S t a t e A l p h a P a r t i c l e D e c a y W i d t h s f o r L a n t h a n i d e a n d A c t i n i d e N u c l e i B ( E 1 ) / B ( E 2 ) V a l u e s f o r L a n t h a n i d e a n d A c t i n i d e N u c l e i G l o b a l R a n g e o f V a l u e s f o r B ( E 1 ) / B ( E 2 ) E ( 3 " ) v s. N f o r L a n t h a n i d e a n d A c t i n i d e N u c l e i n N i l s s o n D i a g r a m f o r Z = N i l s s o n D i a g r a m f o r N = N i l s s o n D i a g r a m f o r Z > N i l s s o n D i a g r a m f o r N > E ( 3 ) v s. N f o r L a n t h a n i d e a n d A c t i n i d e N u c l e i P E ( 3 _ " ) - E ( l " ) v s. N f o r L a n t h a n i d e a n d A c t i n i d e ' 1 1 n N u c l e i
13 5.33 E(31-)-E(l1") vs. E(41+)/E(21+) for Lanthanide a n d A c t i n i d e N u c l e i P a r t i c l e - C o r e C o u p l i n g f o r A = a n d
14 TABLE OF TABLES 4. 1 C h a r a c t e r i s t i c s o f R a G am m a R a d i a t i o n B ( E 1 ) / B ( E 2 ) V a l u e s f o r R a C h a r a c t e r i s t i c s o f R n G am m a R a d i a t i o n C h a r a c t e r i s t i c s o f R a G am m a R a d i a t i o n B ( E 1 ) / B ( E 2 ) V a l u e s f o r R a P a r a m e t e r s U s e d i n V i b r o n M o d e l F i t s
15 1. INTRODUCTION On a simple level, the low energy spectra of the 2500 experimen t a l l y o b s e r v e d n u c l e i m a y b e c l a s s i f i e d a s b e l o n g i n g t o o n e o f t w o c a t e g o r i e s. M e m b e r s o f t h e f i r s t c a t e g o r y l e n d t h e m s e l v e s t o a s h e l l m o d e l a n a l y s i s r e m i n i s c e n t o f t h a t a p p l i c a b l e t o e l e c t r o n i c b e h a v i o r i n a t o m s. I n c o n t r a s t, e a c h n u c l e a r m a n y - b o d y s y s t e m i n t h e s e c o n d c a t e g o r y s e e m s m o s t n a t u r a l l y t r e a t e d a s a c l a s s i c a l l i q u i d d r o p w h i c h d i s p l a y s v a r i e d v i b r a t i o n a l a n d s t a t i c a l l y d e f o r m e d r o t a t i o n a l m o d e s. C l e a r l y, h o w e v e r, t h i s i s a c a r i c a t u r e ; m o s t n u c l e i d i s p l a y e l e m e n t s o f b o t h t y p e s o f b e h a v i o r. E a r l y s t u d i e s o f t h e n a t u r a l i s o t o p i c a b u n d a n c e s o f h e a v y n u c l e i r e v e a l e d d i s t r i b u t i o n s s t r o n g l y p e a k e d a t n e u t r o n o r p r o t o n n u m b e r s o f 5 0, 8 2 a n d T h e s e " m a g i c " n u m b e r s, a s t h e y w e r e n a m e d, w e r e a l s o h i g h l i g h t e d b y t h e s t r i k i n g b e h a v i o r e x h i b i t e d b y n e u t r o n s e p a r a t i o n e n e r g i e s, n e u t r o n c a p t u r e c r o s s s e c t i o n s a n d e l e c t r i c q u a d r u p o l e m o m e n t s ( M a 4 8 ). M e a s u r e m e n t s o f t h e s e q u a n t i t i e s, a s w e l l a s o t h e r o b s e r v a b l e s, w e r e e x p l a i n e d q u i t e s u c c e s f u l l y i n t e r m s o f a s p h e r i c a l n u c l e a r s h e l l m o d e l a s l o n g a s N a n d Z w e r e w i t h i n a f e w n u c l e o n s o f m a g i c n u m b e r s ( M a 5 5 ). H o w e v e r, n u c l e i i n w h i c h N a n d Z w e r e b o t h q u i t e r e m o v e d f r o m t h e m a g i c n u m b e r s s h o w e d e x c i t a t i o n s p e c t r a w h i c h c o u l d n o t b e r e p r o d u c e d w i t h s i m p l e s h e l l m o d e l s. T h i s w a s f i r s t n o t i c e d i n t h e s t u d y o f n u c l e a r e l e c t r i c q u a d r u p o l e m o m e n t s, w h e r e v a l u e s m u c h l a r g e r t h a n t h o s e p o s s i b l e f o r a s i n g l e v a l e n c e n u c l e o n w e r e o b s e r v e d ( T o 4 9 ). A n u c l e a r c o l l e c t i v e p i c t u r e b a s e d o n t h e c o h e r e n t m o t i o n o f m a n y n u c l e o n s a n d t h e
16 -2- r e s e m b l a n c e o f t h i s s y s t e m t o a c l a s s i c a l l i q u i d d r o p w a s b u i l t u p o n t h i s e x p e r i m e n t a l i n f o r m a t i o n ( R a 5 0, B o 5 2, B o 7 5 ). I n t h e m o s t g e n e r a l s e n s e, t h e s h a p e o f a l i q u i d d r o p c a n b e w r i t t e n i n t e r m s o f s p h e r i c a l h a r m o n i c s a s R = R n { 1 + E a. ( t ) Y (e,</>) } ( 1. 1 ) 0 Jim Urn * w h e r e t h e a. ( t ) a r e i n g e n e r a l t i m e d e p e n d e n t m e a s u r e s o f t h e v a r i o u s Hm m u l t i p o l e d e f o r m a t i o n s o f t h e s h a p e a n d R ^ i s t h e m e a n r a d i u s. B e c a u s e o f t h e r e l a t i v e i n c o m p r e s s i b i l i t y o f n u c l e a r m a t t e r, t h e m o n o p o l e t e r m i s o f l i t t l e i n t e r e s t i n t h e s t u d y o f l o w e n e r g y n u c l e a r s p e c t r a. F u r t h e r, t h e d i p o l e t e r m c o r r e s p o n d s t o a s i m p l e t r a n s l a t i o n o f t h e n u c l e u s w i t h n o c h a n g e i n i n t r i n s i c s h a p e. T h u s, t h e l o w e s t o r d e r c o l l e c t i v e m o d e, a s w e l l a s t h e d o m i n a n t o n e, i s t h e q u a d r u p o l e m o d e. W e w i l l r e s t r i c t o u r d i s c u s s i o n t o a x i a l l y s y m m e t r i c s h a p e s, a n d d e f i n e, i n t h e c o n v e n t i o n a l m a n n e r, a n e w s e t o f s h a p e v a r i a b l e s, 6, b y w R = R q ' { 1 + Z 3 ^ ( 0, * ) J ( 1-2 ) w h e r e R Q i s t h e r a d i u s o f a s p h e r e w i t h t h e s a m e v o l u m e a s t h e n u c l e a r s h a p e. B e c a u s e o f t h e d o m i n a n c e o f t h e q u a d r u p o l e m o d e t h r o u g h o u t t h e p e r i o d i c t a b l e a n d t h e r e l a t i v e w e a k n e s s o f o t h e r m u l t i p o l a r i t i e s, m o s t v e r s i o n s o f t h e c o l l e c t i v e m o d e l d e v e l o p e d i n t h e 3 7 y e a r s s i n c e t h i s
17 -3- p i c t u r e c a m e t o t h e f o r e h a v e d e a l t e x c l u s i v e l y w i t h t h e q u a d r u p o l e d e g r e e o f f r e e d o m. S m a l l c o n t r i b u t i o n s t o t h e s h a p e f r o m h e x a d e c a p o l e t e r m s h a v e b e e n i d e n t i f i e d i n t h e l a s t t w e n t y y e a r s ( A p 7 0, H e 6 8 ). A s a n e x a m p l e o f t h e i r r e l a t i v e i m p o r t a n c e, w e c a n. p o i n t o u t t h a t i n t h e m o s t d e f o r m e d h e a v y n u c l e i a p p r o x i m a t e l y i n t h e g r o u n d s t a t e, w h i l e v a l u e s a s l a r g e a s c a n b e f o u n d. A n i n t r i n s i c s h a p e c o m p o n e n t c a n b e e i t h e r o s c i l l a t o r y ( d y n a m i c a l l y d e f o r m e d, o r v i b r a t i o n a l ) o r n o n - o s c i l l a t o r y ( s t a t i c a l l y d e f o r m e d ). T h e e n e r g y s p e c t r u m c h a r a c t e r i s t i c o f t h e s i m p l e s t c a s e o f v i b r a t i o n a l b e h a v i o r, h a r m o n i c o s c i l l a t i o n, s h o w s a s e r i e s o f e q u a l l y s p a c e d a n d d e g e n e r a t e m u l t i p l e t s. A s t a t i c a l l y d e f o r m e d r o t o r g i v e s r i s e t o a s p e c t r u m o f s t a t e s w h o s e e n e r g y e i g e n v a l u e s o b e y a J ( J + 1 ) r u l e. A n u m b e r o f e x a m p l e s o f c o l l e c t i v e n u c l e a r m o t i o n a r e s k e t c h e d i n f i g u r e B e f o r e , n u c l e a r c o l l e c t i v e m o d e l s w e r e b u i l t u p d i r e c t l y f r o m g e o m e t r i c d e s c r i p t i o n s o f t h e e q u i l i b r i u m n u c l e a r s h a p e s u c h a s t h a t i n e q u a t i o n ( 1. 2 ). I n t h e m i d ' s, a n a l t e r n a t i v e a p p r o a c h t o t h e d e s c r i p t i o n o f n u c l e a r c o l l e c t i v e m o t i o n w a s d e v e l o p e d a r o u n d t h e a l g e b r a i c p r o p e r t i e s o f t h e d y n a m i c s y m m e t r i e s i n h e r e n t i n t h e q u a d r u p o l e m o d e s o f t h e n u c l e u s. T h i s m o d e l, t h e I n t e r a c t i n g B o s o n M o d e l ( I B M ) ( A r 7 5, A r 7 6, A r 7 8, A r 7 9 ), h a s p r o v e n t o b e u s e f u l f o r t h e d e s c r i p t i o n o f a w i d e r a n g e o f n u c l e a r s t r u c t u r e p h e n o m e n a. S i m i l a r a l g e b r a i c m e t h o d s h a v e b e e n s u c c e s s f u l l y a p p l i e d t o m o l e c u l a r b e h a v i o r a t b o t h t h e n u c l e a r a n d a t o m i c l e v e l s ( l a 8 1 a, l a 8 1 b, l a 8 2 a, V a 8 2 ). T h e g e n e r i c n a m e a p p l i e d t o e a c h o f t h e s e m o d e l s i s S p e c t r u m G e n e r a t i n g A l g e b r a ( S G A ). I n t h e c o n t e x t o f t h e s u c c e s s e s o f t h e s h e l l m o d e l f o r s p h e r i c a l n u c l e a r s y s t e m s a n d t h e c o l l e c t i v e p i c t u r e f o r n u c l e i r e m o v e d f r o m m a g i c
18 F i g u r e 1. 1 C l a s s i c a l d e p i c t i o n s o f c o l l e c t i v e n u c l e a r p h e n o m e n a.
19 NUCLEAR C O LLECTIV ITY FOOTBALL DOORKNOBS NEUTRONS PROTONS NEUTRONS PROTONS ROTATIONS TRANSLATIONS TORSIONS DIPOLE QUADRUPOLE OCTUPOLE VIBRATIONS MOLECULAR ị >i
20 n u m b e r s, t h e d e v e l o p m e n t o f a s h e l l m o d e l f o r d e f o r m e d n u c l e i w a s a n a t u r a l o n e. N i l s s o n ' s o r i g i n a l d e f o r m e d s h e l l m o d e l c a l c u l a t i o n s ( N i 5 5 ) m e t w i t h g r e a t s u c c e s s i n p r e d i c t i n g t h e s p i n s o f g r o u n d s t a t e s o f o d d - A n u c l e i a n d c o n t r i b u t e d t o a n u n d e r s t a n d i n g, f r o m a m i c r o s c o p i c p o i n t o f v i e w, o f m a n y o t h e r o b s e r v a b l e s i n d e f o r m e d n u c l e i. O v e r t h e l a s t t w e n t y y e a r s, s i g n i f i c a n t d e v e l o p m e n t s i n h e a v y i o n a c c e l e r a t o r s, h i g h r e s o l u t i o n g a m m a r a y d e t e c t o r s a n d l a r g e s o l i d a n g l e d e t e c t o r a r r a y s f o r t h e s t u d y o f h i g h m u l t i p l i c i t y g a m m a r a y c a s c a d e s h a v e s t i m u l a t e d t h e s t u d y o f t h e n u c l e u s a t h i g h a n g u l a r m o m e n t u m ( d e V o 8 3 ). J u s t a s t h e i n t r o d u c t i o n o f d e f o r m a t i o n i n t h e n u c l e a r s y s t e m c h a n g e s t h e f o r m o f t h e n u c l e a r s h e l l m o d e l, s o a l s o t h e p l a c e m e n t o f t h e n u c l e u s i n a r o t a t i n g f r a m e o f r e f e r e n c e m o d i f i e s t h e r e l a t i v e p o s i t i o n s o f t h e s i n g l e p a r t i c l e o r b i t a l s. T h e m o s t s t r i k i n g e f f e c t o f h i g h a n g u l a r m o m e n t u m o n t h e n u c l e u s i s t h e p r o g r e s s i v e l o s s o f p a i r i n g c o r r e l a t i o n s b e t w e e n o p p o s i t e l y r o t a t i n g n u c l e o n s, r e f l e c t i n g C o r i o l i s e f f e c t s a n d c e n t r i f u g a l l y i n d u c e d d e f o r m a t i o n c h a n g e s. W i t h t h e d e v e l o p m e n t o f t h e C r a n k e d S h e l l M o d e l b y B e n g t s s o n a n d F r a u e n d o r f i n ( B e 7 9 a, B e 7 9 b ), i t b e c a m e p o s s i b l e t o t r a c e t h e e v o l u t i o n o f t h e p a i r i n g f o r c e a n d t h e m o m e n t o f i n e r t i a o f t h e n u c l e u s a t h i g h a n g u l a r m o m e n t a. C o n s e q u e n t l y, a s u b s t a n t i a l k n o w l e d g e o f s i n g l e p a r t i c l e o r b i t a l s a t h i g h a n g u l a r m o m e n t a w a s g a i n e d. A l l o f t h i s w o r k, h o w e v e r, r e l i e d o n t h e p r e s e n c e, i n t h e i n t r i n s i c n u c l e a r s h a p e, o f a s y m m e t r y c o m m o n t o b o t h q u a d r u p o l e a n d h e x a d e c a p o l e m u l t i p o l a r i t i e s a n d i n d e e d i n h e r e n t i n a l l e v e n m u l t i p o l a r i t i e s. I f a p l a n e p e r p e n d i c u l a r t o t h e s y m m e t r y a x i s o f t h e q u a d r u p o l e s h a p e i s d r a w n, i t i s c l e a r t h a t a r e f l e c t i o n t h r o u g h t h i s p l a n e i s a s y m m e t r i c
21 -6- o n e f o r a n y n u c l e u s w h o s e s h a p e c a n b e d e s c r i b e d e n t i r e l y b y e v e n s p h e r i c a l h a r m o n i c s. T h e b r e a k i n g o f s u c h a n i n t r i n s i c r e f l e c t i o n s y m m e t r y i s s i g n a l l e d i n t h e l o w e n e r g y e x c i t a t i o n s p e c t r u m o f a h e a v y n u c l e u s i n t w o w a y s. T h e f i r s t s i g n a t u r e i s t h e a p p e a r a n c e o f n e g a t i v e p a r i t y s t a t e s o f a c o l l e c t i v e o r i g i n ( i n s t e a d o f v i a t h e m e c h a n i s m o f s i n g l e p a r t i c l e p r o m o t i o n w i t h i n t h e s h e l l s t r u c t u r e ). S u c h s t a t e s c a n n o t b e g e n e r a t e d b y s h a p e s c o m p o s e d e x c l u s i v e l y o f e v e n m u l t i p o l a r i t i e s. T h e s e c o n d s i g n a t u r e f o r r e f l e c t i o n a s y m m e t r y i n h e a v y n u c l e i i s t h e p r e s e n c e o f u n u s u a l l y s t r o n g g a m m a r a y t r a n s i t i o n s o f o d d e l e c t r i c m u l t i p o l a r i t i e s, m o s t i m p o r t a n t l y d i p o l e, b e t w e e n q u a n t u m s t a t e s o f t h e n u c l e u s. T h e t w o m e c h a n i s m s f o r p r o d u c i n g r e f l e c t i o n a s y m m e t r i c s h a p e s w h i c h w i l l b e c o n s i d e r e d i n t h i s w o r k l e a d t o t h e s e e n h a n c e d e l e c t r i c d i p o l e ( E l ) t r a n s i t i o n s i n d i f f e r e n t w a y s. S t r o n g E l t r a n s i t i o n s a r e p a r t i c u l a r l y s i g n i f i c a n t g i v e n t h e a b s e n c e o f t h e d i p o l e t e r m i n t h e e x p r e s s i o n f o r t h e n u c l e a r s u r f a c e ( 1. 2 ). B e c a u s e o f t h i s a b s e n c e, t h e e l e c t r i c d i p o l e o p e r a t o r i s d e p e n d e n t i n f i r s t o r d e r u p o n t h e s e p a r a t i o n o f t h e c e n t e r o f n u c l e a r c h a r g e f r o m t h e c e n t e r o f m a s s. S u c h a s e p a r a t i o n o c c u r s i n t h e g i a n t d i p o l e r e s o n a n c e o f t h e n u c l e u s, a p h e n o m e n o n w h i c h i s f o u n d a t e x c i t a t i o n e n e r g i e s o f 1 5 M e V a n d a b o v e ( G o 4 8 ) ; i n t h i s m o d e, t h e p r o t o n a n d n e u t r o n f l u i d s o s c i l l a t e b a c k a n d f o r t h c o h e r e n t l y a n d c o l l e c t i v e l y w i t h r e s p e c t t o o n e a n o t h e r ( s e e f i g u r e 1. 1 ). T h e c e n t e r o f m a s s r e m a i n s s t a t i o n a r y a t t h e g e o m e t r i c c e n t e r o f t h e n u c l e u s, b u t t h e c e n t e r o f c h a r g e i s f o u n d a t t h e c e n t e r o f t h e p r o t o n f l u i d, w h i c h o s c i l l a t e s r e l a t i v e t o t h e c e n t e r o f m a s s. In parallel, the study of molecular resonances in heavy ion inter-
22 a c t i o n s l e d t o t h e d i s c o v e r y t h a t i n t h e c a s e s o f 1 2 C C a n d 1 2 C r e s o n a n c e s a t e n e r g i e s c l o s e t o t h e C o u l o m b b a r r i e r ( t h e e n e r g y r e q u i r e d t o o v e r c o m e t h e e l e c t r o s t a t i c r e p u l s i o n b e t w e e n t w o n u c l e i ), t h e d a t a c o u l d b e c o r r e l a t e d i n t h e f r a m e w o r k o f a n S G A i n w h i c h t h e v e c t o r c o n n e c t i n g t h e c e n t e r s o f t h e t w o i o n s i s t r e a t e d a l g e b r a i c a l l y ( E r 8 1, l a 8 1 a, R u 8 4 ). I n t h i s t r e a t m e n t, c a l l e d t h e V i b r o n m o d e l, t h e s e p a r a t i o n v e c t o r c o n s t i t u t e s a d i p o l e d e g r e e o f f r e e d o m. T h i s m o d e l a l s o m a k e s t h e i n t u i t i v e l y p l a u s i b l e p r e d i c t i o n o f e n h a n c e d E l t r a n s i t i o n s b e t w e e n m o l e c u l a r s t a t e s o f o p p o s i t e p a r i t y i n a n o n - s e l f c o n j u g a t e d i n u c l e a r s y s t e m, a s y s t e m i n w h i c h t h e c h a r g e - t o - m a s s r a t i o s o f t h e t w o c o m p o n e n t s o f t h e m o l e c u l e a r e d i f f e r e n t ( c o n s e q u e n t l y s e p a r a t i n g t h e c e n t e r o f c h a r g e o f t h e s y s t e m f r o m t h e c e n t e r o f m a s s a n d g i v i n g t h e m o l e c u l e a n e l e c t r i c d i p o l e m o m e n t ). H o w e v e r, n e g a t i v e p a r i t y m o l e c u l a r s t a t e s d o n o t e x i s t i n 1 2 C C, a n d t h e s e l f - c o n j u g a t e n a t u r e o f t h e 1 2 C s y s t e m d i s a l l o w s E l t r a n s i t i o n s t o ' f i r s t o r d e r. T h i s l e d G a i a n d c o l l a b o r a t o r s t o e x a m i n e l i g h t n u c l e a r s y s t e m s w i t h n o n - z e r o i s o s p i n i n a s e a r c h f o r t h e p r e d i c t e d e n h a n c e d E l t r a n s i t i o n s. G a i e t a l. o b s e r v e d t h a t E l t r a n s i t i o n s i n a n d 1 0 B e w e r e b y f a r t h e s t r o n g e s t i n e v e n m a s s l i g h t n u c l e i, a n d s u b s e q u e n t l y m a d e a d e t a i l e d s p e c t r o s c o p i c s t u d y o f v i a 1 4 C ( a, 2 T ) a n d 1 4 C ( 7 L i, t j f ) r e a c t i o n s w i t h t h e r e s u l t s s h o w n i n f i g u r e 1. 2 ( G a 8 3 a, R u 8 4 ). T h e i m p o r t a n c e o f p r e - f o r m e d a l p h a p a r t i c l e s i n t h e c o l l e c t i v e b e h a v i o r o f l i g h t n u c l e i ( A < 3 0 ) h a s b e e n k n o w n s i n c e t h e ' s ; t h i s c l u s t e r i n g b e h a v i o r c a n l e a d t o b o u n d n o n - s e l f c o n j u g a t e d i n u c l e a r m o l e c u l a r s y s t e m s. A s a n e x a m p l e w e c a n c o n s i d e r t h e n u c l e u s 1 8 0, i n w h i c h t h e d i n u c l e a r a l p h a p a r t i c l e c l u s t e r i n g c o n f i g u r a t i o n w o u l d b e
23 F i g u r e 1. 2 A p r o p o s e d a l p h a p a r t i c l e c l u s t e r m o l e c u l a r b a n d i n ( G a 8 3 a ).
24 -8-0 cm =l25 (x 2 ) r f ~ I 5 % 4 (T~80 KeV) NONRESONANT T kevx E a (MeV) 4 c - 3 T=8 kev 2+ r= I kev T <2.5keV n + l70 I. 0 2 = 2O% 0 2=O.2% x10 W.u. 15 W.u. t i i i J I L 0.5 do* /d 7i LA STIC^7^^ ^ a +I4C 5.26 r- 0.9x 10"2 W.u W.u x10 2 W.u o$ 4 P - 2h a +,4C / J (J + l)
25 4H e + 14C. F i g u r e 1. 2 sh o w s a p r o p o s e d b a n d o f s t a t e s i n 180 i n w h ic h t h i s d i n u c l e a r a lp h a p a r t i c l e c l u s t e r c o n f i g u r a t i o n may p l a y a m a j o r r o l e (G a 8 3 a ). T h i s b a n d d e m o n s t r a t e s th e tw o s i g n a t u r e s o f r e f l e c t i o n a s y m m e try t h a t we h a v e d i s c u s s e d, lo w l y i n g n e g a t i v e p a r i t y s t a t e s a n d e n h a n c e d i n t r a b a n d E l t r a n s i t i o n s. A t h i r d s i g n a t u r e t h a t f o l l o w s d i r e c t l y fr o m th e a l p h a p a r t i c l e c l u s t e r i n g c o n c e p t i s th e p r e s e n c e o f l a r g e a lp h a p a r t i c l e d e c a y w i d t h s. T h i s s i g n a t u r e i s q u i t e s t r i k i n g l y d e m o n s t r a t e d i n 180, w h e re t h e s e w id t h s h a v e b e e n m e a su re d v i a r e s o n a n t a lp h a p a r t i c l e s c a t t e r i n g (A j 8 3 ). I t b e a r s e m p h a s is, h o w e v e r, t h a t i n n u c l e i s u c h a s 180, t h i s c l u s t e r s t r u c t u r e c o e x i s t s w i t h b o t h s h e l l a n d m ore common r e f l e c t i o n symmet r i c c o l l e c t i v e s t r u c t u r e a n d t h a t r e s i d u a l i n t e r a c t i o n s r e s u l t i n a d m ix t u r e s o f a l l t h r e e i n d i f f e r e n t q uan tu m s t a t e s. T h i s h a s th e c o n s e q u e n c e t h a t a l t h o u g h th e s i g n a t u r e s c h a r a c t e r i s t i c o f e a c h o f th e a b o v e m e n t io n e d s t r u c t u r e s c a n be i d e n t i f i e d, t h e y w o u ld n o t be e x p e c t e d t o a p p e a r i n th e c l a s s i c a l fo rm t h a t w o u ld be p r e s e n t w ere o n l y a s i n g l e ty p e o f s t r u c t u r e i n v o l v e d. P r o m p te d b y th e o b s e r v a t i o n o f n u c l e a r d e c a y i n r a r e e a r t h a n d a c t i n i d e n u c l e i t h r o u g h th e e m i s s i o n o f a n a lp h a p a r t i c l e, a d e b a t e h a s c o n t in u e d t o t h i s d a y a b o u t w h e th e r a l a r g e p r o b a b i l i t y f o r a lp h a p a r t i c l e d e c a y n e c e s s a r i l y i m p l i e s th e e x i s t e n c e o f p r e - f o r m e d a lp h a p a r t i c l e s a t th e n u c l e a r s u r f a c e. F i g u r e 1.3 d i s p l a y s r e d u c e d g r o u n d s t a t e a l p h a p a r t i c l e d e c a y w i d t h s (d e d u c e d fr o m m e a s u r e d l i f e t i m e s ) f o r h e a v y n u c l e i a c r o s s th e p e r i o d i c t a b l e. The l a r g e s t a lp h a p a r t i c l e d e c a y w i d t h s a r e fo u n d j u s t a b o v e th e n e u t r o n m a g ic n um be rs 82 a n d T h e r e f o r e, i t w o u ld be i n t u i t i v e l y p r o b a b le t o f i n d a lp h a p a r t i c l e c l u s -
26 F i g u r e 1.3 C o m p i l a t i o n o f r e d u c e d a lp h a p a r t i c l e d e c a y w id t h s fr o m (R o 8 3 ). The 218Ra w id t h h a s b e e n a d j u s t e d t o c o n fo r m t o th e r e c e n t r e s u l t o f (R a 8 6 ).
27 (A )// (212 p 0 ) CV1CJ N E U T R O N N U M B E R O F A L P H A E M IT T E R DATA C O M P IL E D B Y R O E C K L et o L. G S I 0 1
28 -11- tering configurations in these regions. I t i s i m m e d ia t e ly a p p a r e n t t h a t s u c h a c o n f i g u r a t i o n n e c e s s a r i l y b r e a k s r e f l e c t i o n sy m m e try ( s e e f i g u r e 1. 4 ). F u r t h e r, t h i s c o n f i g u r a t i o n, i n a h e a v y n u c le u s, w o u ld g i v e an i n t r i n s i c e l e c t r i c d i p o l e moment i n a p a r t i c u l a r l y s im p le w ay. The a lp h a p a r t i c l e h a s tw o p r o t o n s a n d tw o n e u t r o n s, g i v i n g a c h a r g e t o m a ss r a t i o o f 1 / 2. The h e a v y c o r e n u c l e u s, h o w e v e r, a lw a y s h a s m ore n e u t r o n s t h a n p r o t o n s a n d a c h a r g e t o m a ss r a t i o o f l e s s t h a n 1 / 2. T h e r e f o r e, th e c e n t e r o f c h a r g e o f th e n u c le u s w i l l be c l o s e r t o th e a lp h a p a r t i c l e c l u s t e r th a n th e c e n t e r o f m a s s. The h i g h e r e l e c t r i c m u l t i p o l a r i t i e s o f s u c h a c o n f i g u r a t i o n c a n be s i m p l y show n t o be n o n - z e r o b y a c l a s s i c a l c a l c u l a t i o n ( A l 8 2 ). The a d d i t i o n o f a n o c t u p o le co m p o n e n t t o th e n u c l e a r s h a p e a l s o r e s u l t s i n th e b r e a k i n g o f r e f l e c t i o n sy m m e try ( s e e f i g u r e 1. 4 ) a n d g i v e s r i s e t o s t r o n g e l e c t r i c o c t u p o le ( E 3 ) t r a n s i t i o n s i n a s t r a i g h t f o r w a r d m a n n e r. B e c a u s e th e f i r s t o r d e r e l e c t r o m a g n e t i c t r a n s i t i o n o p e r a t o r s i m p l y c o n s i s t s o f th e c l a s s i c a l e x p r e s s i o n f o r th e e l e c t r o m a g n e t i c m om ent, an e l e c t r i c o c t u p o le moment i s r e f l e c t e d i n th e o b s e r v a t i o n o f s t r o n g e l e c t r i c o c t u p o le t r a n s i t i o n s b e tw e e n q u a n tu m s t a t e s. I n th e l a t e 's, tw o s e p a r a t e t r e a t m e n t s o f th e o c t u p o l e s h a p e, one b y B o h r a n d M o t t e l s o n (B o 57, Bo 5 8 ), th e o t h e r b y S t r u t i n s k y ( S t 5 7 ), sh ow ed t h a t t h i s s h a p e co m p o n e n t w o u ld l e a d t o a l a r g e r d e n s i t y o f p r o t o n s r e l a t i v e t o n e u t r o n s i n one e n d o f th e n u c le u s t h a n i n th e o t h e r. C o n s e q u e n t ly, th e c e n t e r o f c h a r g e w o u ld be d i s p l a c e d fr o m th e c e n t e r o f m a ss a n d a n e l e c t r i c d i p o l e m om ent, a s w e l l a s s t r o n g E l t r a n s i t i o n s, w o u ld r e s u l t. T he c a l c u l a t i o n s o f r e f e r e n c e s (B o 5 7, Bo 5 8 ) a n d ( S t 5 7 ) a r e b a s e d, h o w e v e r, on d i f f e r e n t e l e c t r o s t a t i c e f f e c t s. I n th e
29 F i g u r e 1. 4 A lp h a c l u s t e r ( a ) a n d o c t u p o le d e fo r m e d ( b ) c o n f i g u r a t i o n s f o r 22 0 R a, d ra w n t o s c a l e. The o c t u p o l e d e fo r m e d sh a p e h a s th e d e f o r m a t i o n p a r a m e t e r s p2= a n d 33= p r e d i c t e d i n (N a 8 4 b ) f o r 222R a.
30
31 -13- B o h r a n d M o t t e l s o n t r e a t m e n t, i t i s a s su m e d t h a t a u n if o r m p r o t o n - n e u t r o n d i s t r i b u t i o n w o u ld e x i s t i n a s p h e r i c a l n u c l e u s, a n d th e p o l a r i z a t i o n a r i s e s e x c l u s i v e l y fro m th e r e f l e c t i o n a s y m m e t r ic d e f o r m a t i o n. I n c o n t r a s t, S t r u t i n s k y t a k e s i n t o a c c o u n t th e m o n o p o le e l e c t r o s t a t i c f i e l d, w h ic h w o u ld f o r c e th e p r o t o n s t o w a r d th e s u r f a c e i n a s p h e r i c a l n u c l e u s. C o n s e q u e n t ly, th e two c a l c u l a t i o n s r e s u l t i n p r e d i c t i o n s o f th e e l e c t r i c d i p o l e moment w h ic h h a v e o p p o s i t e s ig n s, - th e c u r r e n t w e i g h t o f t h e o r e t i c a l e v id e n c e s u g g e s t s t h a t th e S t r u t i n s k y r e s u l t i s th e c o r r e c t o n e. A v e r y r e c e n t c a l c u l a t i o n (D o 8 6 ) a d d s th e e f f e c t s o f a p o s t u l a t e d " n e u t r o n s k i n " on th e n u c l e u s. The r e l a t i v e im p o r t a n c e o f th e s e th r e e te rm s h a s n o t b e e n d e t e r m in e d, a n d p r e s e n t s a f o r m i d a b l e t h e o r e t i c a l o b s t a c l e. J u s t a s f o r sh a p e c o m p o n e n ts o f o t h e r m u l t i p o l a r i t i e s, th e o c t u p o le moment c a n be m a n i f e s t e d i n b o t h v i b r a t i o n a l a n d s t a t i c f o r m s. O c t u p o le v i b r a t i o n a l b e h a v i o r h a s b e e n o b s e r v e d t h r o u g h o u t th e p e r i o d i c t a b l e, a n d i s a s s o c i a t e d w it h e n h a n c e d E l t r a n s i t i o n s i n th e r e g i o n n e a r Z = 6 4. T h e r e i s s t r o n g e v id e n c e ( s e e s e c t i o n 2. 2 ) t h a t th e a l t e r n a t i n g p a r i t y s t r u c t u r e i n th e n u c le u s 150G d ( f i g u r e 1. 5 ) r e s u l t s fr o m th e c o u p l i n g o f a n o c t u p o le p h o n o n c a r r y i n g s p i n - p a r i t y o f 3 t o th e p o s i t i v e p a r i t y s t a t e s o f th e g r o u n d s t a t e b a n d o f t h i s q u a d r u p o le v i b r a t i o n a l n u c le u s (H a 7 7 ). I f a n o c t u p o le p h o n o n i s c o u p le d t o a q u a d r u p o le d e fo rm e d n u c l e u s, a n a d d i t i o n a l 1 s t a t e c o u l d be l o c a t e d b e lo w th e 3 s t a t e. G a m m a -ray b r a n c h i n g r a t i o s s u g g e s t t h a t E l t r a n s i t i o n s fr o m th e r e s u l t i n g n e g a t i v e p a r i t y s t a t e s a r e s i g n i f i c a n t l y s t r o n g e r t h a n t h o s e a r i s i n g fr o m th e t r a n s i t i o n o f a s i n g l e p a r t i c l e fr o m one s h e l l m o d e l o r b i t a l t o a n o t h e r.
32 F i g u r e 1. 5 A p a r t i a l l e v e l s p e c tru m o f 15DG d (H a 7 7 ). The n e g a t i v e p a r i t y s t a t e s show n h a v e b e e n i n t e r p r e t e d i n (H a 7 7 ) t o be th e r e s u l t o f th e c o u p l i n g o f an o c t u p o le p h o n o n h a v i n g J 1^ " t o th e q u a d r u p o le v i b r a t i o n a l c o r e.
33 -14-
34 -15- We w o u ld a l s o e x p e c t t o o b s e r v e l o w - l y i n g n e g a t i v e p a r i t y s t a t e s a n d e n h a n c e d E l t r a n s i t i o n s i n a s t a t i c a l l y o c t u p o le d e fo r m e d n u c le u s. The n e g a t i v e p a r i t y b a n d w o u ld h a v e a 1 " s t a t e a s th e b a n d h e a d, a n d w o u ld fo rm an a l t e r n a t i n g p a r i t y s e q u e n c e a t h i g h e r s p i n s. No o b s e r v a t i o n s o f s t a t i c o c u t p o le d e f o r m a t i o n i n th e g r o u n d s t a t e o f a n u c le u s h a v e y e t b e e n c o n f ir m e d. T h e s t u d y o f r e f l e c t i o n a s y m m e t r ic s h a p e s a l s o g i v e s u s th e o p p o r t u n i t y t o e x a m in e o u r i d e a s c o n c e r n i n g s i n g l e p a r t i c l e m o t io n a n d c o l l e c t i v e b e h a v i o r i n s y s t e m s w i t h d i f f e r e n t s y m m e t r ie s fr o m t h o s e w it h w h ic h m o d e ls s u c h a s th e B o h r - M o t t e l s o n c o l l e c t i v e m o d e l, th e I n t e r a c t i n g B o s o n M o d e l, th e N i l s s o n d e fo r m e d s h e l l m o d e l a n d th e C r a n k e d S h e l l M o d e l w e re o r i g i n a l l y d e v e lo p e d. I n th e p a s t s e v e r a l y e a r s a c o n s i d e r a b l e t h e o r e t i c a l e f f o r t h a s b e e n m o u n te d i n o r d e r t o g e n e r a l i z e th e s e c o n c e p t s to th e c a s e i n w h ic h r e f l e c t i o n sy m m e try i s b r o k e n (N a 8 5, Ro 78, Ro 8 2 a, En 8 5, Da 8 3, l a 8 2 b ). T h e se e f f o r t s h a v e b e e n s y s t e m a t i c a l l y h a n d ic a p p e d b y th e l a c k o f th e l a r g e b o d y o f s y s t e m a t i c d a t a on r e f l e c t i o n a s y m m e tr ic s y s t e m s n e e d e d t o r e f i n e t h e s e i d e a s i n t o r e l i a b l e m o d e l s. F o r o v e r t h i r t y y e a r s i t h a s b e e n known t h a t r e f l e c t i o n a s y m m e tr ic s h a p e s m u st p l a y an im p o r t a n t r o l e a t v e r y lo w e x c i t a t i o n e n e r g i e s i n a num ber o f i s o t o p e s o f R a a n d T h. The f i r s t e v id e n c e t h a t t h i s m ig h t be th e c a s e w as fo u n d b y S t e p h e n s a n d c o l l a b o r a t o r s a t B e r k e l e y i n th e m i d 's ( S t 54, S t 55, S t 5 7 ). I n s p e c t r o s c o p i c s t u d i e s o f th e r a d i o a c t i v e d e c a y c h a i n s o f s e v e r a l U i s o t o p e s, i t w as d i s c o v e r e d t h a t th e a l p h a p a r t i c l e d e c a y s o f e v e n -m a s s U a n d Th i s o t o p e s p o p u l a t e d n e g a t i v e p a r i t y s t a t e s o f v e r y lo w e n e r g i e s i n th e d a u g h t e r n u c l e i. T h e s e d e c a y
35 -16- s t u d i e s w ere e x p a n d e d a t B e r k e le y u n t i l th e e a r l y 's (R u 6 1 ), a n d f u r t h e r r e f i n e d i n th e s t u d i e s o f K u r c e w ic z (K u 7 6, Ku 77, Ku 7 8 ). F i g u r e 1.6 d i s p l a y s r e s u l t s p r e s e n t e d b y K u r c e w ic z on th e d e c a y c h a i n o f U (K u 7 6 ). The l o w - l y i n g n e g a t i v e p a r i t y s t a t e s a r e fo u n d i n b o t h 226Th a n d 22 2 R a a t r o u g h l y e q u a l e n e r g i e s. T e n t a t i v e a s s i g n m e n ts h a v e b e e n g i v e n t o t h r e e s t a t e s s t a t e s i n 21 8 Rn a t 653 k e V, 797 kev a n d 840 k e V ; t h e s e r e q u i r e some e x p l a n a t i o n. I n th e o r i g i n a l w o rk (K u 7 6 ), no s u c h a s s i g n m e n t s w ere r e p o r t e d. I t w as o b s e r v e d i n (P e 8 1 ), h o w e v e r, t h a t th e gamma r a y d e e x c i t a t i o n s o f t h e s e s t a t e s s t r o n g l y s u g g e s t e d th e a s s i g n m e n t s show n i n f i g u r e F u r t h e r, e l e c t r o n c o n v e r s i o n d a t a (L e 6 3 ) c o n f ir m s th e p a r i t y a s s i g n m e n t made f o r th e 797 kev l e v e l. I f t h e s e a s s i g n m e n t s a r e c o r r e c t, th e n e g a t i v e p a r i t y s t a t e s i n 21 8 Rn a r e s i g n i f i c a n t l y h i g h e r i n e n e r g y th a n t h o s e i n 222Ra a n d 226Th. A n o t h e r c h a r a c t e r i s t i c o f t h e s e Rn s t a t e s t h a t d i s t i n g u i s h e s them fro m c o r r e s p o n d i n g s t a t e s i n th e p a r e n t a n d g r a n d p a r e n t n u c l e i i s t h e i r o r d e r i n g. I n b o t h Ra a n d 226T h th e 1 " s t a t e i s l o c a t e d a b o u t 80 kev b e lo w th e 3 " s t a t e. H o w e v e r, th e 3 ' s t a t e f a l l s 43 kev b e lo w th e 1" s t a t e i n 218Rn. S u c h i n f o r m a t i o n may be h e l p f u l i n i n v e s t i g a t i o n s o f d e t a i l s o f th e n u c l e a r s h a p e. R e c e n t l y, s e v e r a l i n v e s t i g a t o r s h a v e s t u d i e d th e h i g h s p i n s t r u c t u r e a s s o c i a t e d w i t h t h e s e l o w - l y i n g n e g a t i v e p a r i t y s t a t e s. An e x a m p le o f t h i s h i g h s p i n s t r u c t u r e i s t h a t o f 218Ra show n i n f i g u r e Two f e a t u r e s s t a n d o u t i n t h i s n u c le u s. The f i r s t i s th e a l t e r n a t i n g p a r i t y b a n d s t r u c t u r e a t s p i n s o f 4 R a n d a b o v e. The s e c o n d f e a t u r e, n o t show n i n th e f i g u r e, i s th e v e r y s t r o n g e n h a n c e m e n t o f th e E l t r a n s i t i o n s l i n k i n g th e o p p o s i t e p a r i t y m em bers o f th e g r o u n d s t a t e b a n d. I t h a s
36 F i g u r e 1.6 A lp h a p a r t i c l e d e c a y c h a i n o f 230U fr o m (K u 7 6 ). T e n t a t i v e s p i n a s s i g n m e n t s i n Rn a r e fr o m (P e 8 1 ).
37 230U D ECAY C H A IN o d 226Th *\ 30.9 m d a 38s 218 Dn a,036s 2l4Po /xs (2+)' (2+ )- (0+ ) -847 '805 (0+) 914 (n- (3 -) (4+) < 609 (5,6+) r Th " Ill o Ra ' Po Rn Pb 82* 128 I
38 F i g u r e 1.7 P a r t i a l l e v e l s p e c tr u m o f Ra c o m p i le d fr o m (G a 8 3 b, Go 8 5 ).
39 (21") (15) ^ T, JT i ». 15 i 2969 (14)] " 'it... 1 (13^) j (I") R a
40 -19- b e e n d e t e r m in e d t h a t th e r e d u c e d m a t r i x e le m e n t s o f t h e s e t r a n s i t i o n s a r e th e l a r g e s t ( a f t e r m a ss d e p e n d e n c e h a s b e e n re m o v e d ) fo u n d i n h e a v y e v e n -m a s s n u c l e i (G a 8 3 b ). The f i r s t o b j e c t i v e o f th e p r e s e n t w o rk i s a n e x a m in a t io n o f th e q u e s t i o n o f r e f l e c t i o n a s y m m e try i n e v e n R n, Ra a n d Th n u c l e i t h r o u g h th e s t u d y o f th e s y s t e m a t i c b e h a v i o r o f t h e i r E l m a t r i x e le m e n t s a s w e l l a s t h e i r e x c i t a t i o n s p e c t r a a t b o t h lo w a n d h i g h s p i n s. The f i r s t w ay i n w h ic h we w i l l do t h i s i s v i a an e x p e r im e n t a l s t u d y o f th e gamma r a y s p e c t r a o f Rn a n d Ra p r o d u c e d i n th e 14C Pb f u s i o n - e v a p o r a t i o n r e a c t i o n a t beam e n e r g i e s o f MeV. E a c h o f t h e s e n u c l e i i s tw o n u c le o n s aw ay fro m 218R a, a n d i n f o r m a t i o n g a i n e d fr o m them w i l l c l a r i f y th e s y s t e m a t i c b e h a v i o r o f o b s e r v a b l e s i n th e m a ss r e g i o n v e r y c l o s e t o 21 8 R a. The s e c o n d s t u d y w i l l i n v o l v e th e t e s t i n g o f n u c l e a r c o l l e c t i v e m o d e ls c o n s t r u c t e d a r o u n d th e s h a p e s d e s c r i b e d a b o v e fro m d a t a a v a i l a b l e o n Ra a n d Th i s o t o p e s. T h i s w i l l i n c lu d e a d i s c u s s i o n o f p r e d i c t i o n s o f a N i l s s o n M o d e l a n d a C r a n k e d S h e l l M o d e l (N a 8 5 ), e a c h o f w h ic h i n c l u d e s an o c t u p o le d e g r e e o f fre e d o m (N a 8 5 ), a s w e l l a s e x a m in a t io n o f c a l c u l a t i o n s o f th e p o t e n t i a l e n e r g y o f th e n u c l e a r g r o u n d s t a t e f o r a r a n g e o f s h a p e s (L e 8 2 a, N a 8 4 b ). We w i l l a l s o p r e s e n t th e r e s u l t s o f a c a l c u l a t i o n o f e n e r g y l e v e l s p e c t r a f o r f o u r Ra i s o t o p e s i n w h ic h th e V i b r o n f o r m a li s m (D a 8 3 ) i s a p p l i e d t o th e a lp h a p a r t i c l e c l u s t e r i n g p i c t u r e. I n a b r o a d e r w ay, we w i l l a l s o e x a m in e th e r e l a t i o n s h i p o f r e f l e c t i o n a s y m m e t r ic b e h a v i o r i n R n, Ra a n d Th n u c l e i t o th e b e t t e r u n d e r s t o o d o c t u p o le v i b r a t i o n a l b e h a v i o r i n P b, P o, U a n d P u i s o t o p e s, a s
41 -20- w e l l a s i n a r a n g e o f e le m e n t s i n th e 5 6 < Z < 8 2, 82<N <126 r e g i o n. By c o m p a r in g th e s y s t e m a t i c b e h a v i o r o f e n e r g y e x c i t a t i o n s p e c t r a i n t h e s e r e g i o n s, we h o p e t o e v o lv e new i n s i g h t s i n t o th e n a t u r e o f th e r e f l e c t i o n a sy m m e try i n R n, Ra a n d Th. The s e c o n d o b j e c t i v e h e re i s t o s t u d y th e s t r e n g t h o f th e c o u p l i n g b e tw e e n an u n p a i r e d v a le n c e n u c le o n a n d th e e v e n -e v e n c o r e i n o d d - A n u c l e i o f t h i s r e g i o n. The p a r t i c l e - c o r e c o u p l i n g i s q u i t e d e p e n d e n t on th e d e g r e e o f d e f o r m a t i o n i n th e c o r e. I n th e c a s e o f an i d e a l s p h e r i c a l n u c l e u s, th e u n p a i r e d n u c le o n a n d th e c o r e i n t e r a c t i n s u c h a w ay t h a t th e s p e c t r u m o f th e o d d -A n u c le u s c o n s i s t s s i m p l y o f a s e q u e n c e o f d e g e n e r a t e m u l t i p l e t s, one m u l t i p l e t f o u n d a t th e e n e r g y o f e a c h s t a t e o f th e e v e n c o r e n u c l e u s. The s t a t e s o f e a c h m u l t i p l e t a r e g e n e r a t e d t h r o u g h th e a d d i t i o n o f th e a n g u l a r momentum o f th e u n p a i r e d n u c le o n, com m on ly d e n o t e d b y j, t o t h a t o f th e c o r r e s p o n d i n g c o r e s t a t e : ii>(j, j, J ) J > x j > rv c o r e J c o r e J f o r J - j < J < J + j. c o r e c o r e T h i s s i t u a t i o n i s known a s th e w eak c o u p l i n g l i m i t. I f th e n u c l e u s h a s a s m a l l d e f o r m a t i o n, th e d e g e n e r a c y i s b r o k e n a n d th e s t a t e s a r e s p r e a d o v e r a s m a l l e n e r g y r a n g e, g e n e r a l l y a s m a l l f r a c t i o n o f th e e n e r g y o f th e c o r e s t a t e. A s th e d e f o r m a t i o n b e c o m e s p r o g r e s s i v e l y l a r g e r, p r o fo u n d c h a n g e s o c c u r. S t a t e s fr o m d i f f e r e n t m u l t i p l e t s a r e m ix e d t o g e t h e r a n d th e w eak c o u p l i n g f o r m a l i s m b e co m e s i n a p p r o p r i a t e. F o r l a r g e d e f o r m a t i o n s, th e s t r o n g c o u p l i n g a p p r o a c h u s e d i n th e N i l s s o n d e fo r m e d s h e l l m o d e l i s c o r r e c t. I n t h i s a p p r o a c h, d e fo rm e d c o r e s t a t e s a r e c o u p le d t o s i n g l e p a r t i c l e e i g e n s t a t e s o f a d e fo rm e d p o t e n t i a l a n d
42 -21- th e s i n g l e p a r t i c l e e n e r g y e i g e n v a l u e s d e p e n d d i r e c t l y on th e c o r e d e f o r m a t i o n. F u r t h e r, th e p a r t i c l e - c o r e i n t e r a c t i o n h a m i l t o n i a n h a s a l a r g e m a g n it u d e. I n th e p a r t i c l e - c o r e s e n s e, 21 9 Ra c a n be r e g a r d e d a s an u n p a i r e d n e u t r o n c o u p le d t o a 21 8 Ra c o r e. B e c a u s e 218Ra b e h a v e s a s a n e a r l y s p h e r i c a l n u c l e u s w i t h v i b r a t i o n a l e x c i t a t i o n m od es, we w o u ld e x p e c t fro m o u r k n o w le d g e o f v i b r a t i o n a l n u c l e i i n o t h e r r e g i o n s o f th e p e r i o d i c t a b l e t h a t Ra w o u ld d i s p l a y p r o p e r t i e s r a t h e r c l o s e t o t h o s e s e e n i n th e w eak c o u p l i n g l i m i t. We t e s t t h a t e x p e c t a t i o n b y e x a m in in g th e e n e r g y s p e c t r u m o f 21 8 Ra b y gamma r a y s p e c t r o s c o p y o f s t a t e s p o p u l a t e d i n f u s i o n - e v a p o r a t i o n r e a c t i o n s o f th e 14C Pb s y s t e m. We a l s o r e v ie w th e s y s t e m a t i c b e h a v i o r o f o d d -A n u c l e i i n th e im m e d ia te v i c i n i t y o f 219Ra i n o r d e r t o e x a m in e th e p a r t i c l e - c o r e c o u p l i n g s t r e n g t h i n n u c l e i o f t h i s r e g i o n. C h a p t e r tw o o f t h i s t h e s i s g i v e s a s c h e m a t ic o v e r v ie w o f t h e o r e t i c a l a p p r o a c h e s t a k e n i n th e t r e a t m e n t o f n u c l e i o f th e R n - R a - T h r e g i o n. T e c h n iq u e s u s e d i n th e e x p e r im e n t a l s t u d i e s p r e s e n t e d i n t h i s w o r k a r e d e s c r i b e d i n c h a p t e r t h r e e, w h ic h i s f o l l o w e d i n c h a p t e r f o u r b y a p r e s e n t a t i o n o f th e e x p e r im e n t a l r e s u l t s. S t u d i e s o f s y s t e m a t i c b e h a v i o r a n d d i s c u s s i o n s o f o u r e x p e r im e n t a l r e s u l t s w i t h i n th e fr a m e w o r k s o f s e v e r a l t h e o r e t i c a l p i c t u r e s a r e t o be fo u n d i n c h a p t e r f i v e. O u r e x p e r im e n t a l f i n d i n g s a n d c o n c l u s i o n s a r e su m m a riz e d i n c h a p t e r s i x. B r i e f l y, t h r o u g h o u r s t u d i e s o f Rn a n d R a we h a v e f o u n d t h a t t r e n d s i n E l m a t r i x e le m e n t s a n d a lp h a p a r t i c l e d e c a y w id t h s a r e c o r r e l a t e d, s u g g e s t i n g t h a t th e e n h a n c e d E l t r a n s i t i o n s a n d l a r g e a l p h a p a r t i c l e d e c a y w id t h s o b s e r v e d i n t h i s r e g i o n r e s u l t fr o m a s i n g l e n u c l e a r
43 -22- ph e no m en on. F u r t h e r, 218R a i s th e l i g h t e s t member o f b o t h i s o t o p i c a n d i s o t o n i c c h a i n s w h ic h d i s p l a y s c o l l e c t i v e n e g a t i v e p a r i t y s t a t e s a l o n g th e y r a s t l i n e. F i n a l l y, 219Ra a n d l i g h t e r o d d - A n e i g h b o r s c a n b e w e l l d e s c r i b e d w i t h i n a n a p p r o a c h b a s e d on w eak c o u p l i n g ; h o w e v e r, i t i s l i k e l y t h a t t h i s i s n o t th e c a s e f o r o d d - A n u c l e i o f m a ss g r e a t e r t h a n 219. We h a v e f o u n d c l e a r e v id e n c e f o r new r e f l e c t i o n a s y m m e t r ic s h a p e s i n th e l i g h t a c t i n i d e s. The p r e s e n t b o d y o f s y s t e m a t i c d a t a o n t h i s m a ss r e g i o n i s u n a b l e, h o w e v e r, t o p e r m it d i s t i n g u i s h i n g am ong th e a lp h a p a r t i c l e c l u s t e r, o c t u p o le v i b r a t i o n a n d s t a t i c o c t u p o le d e f o r m a t i o n m o d e ls b e c a u s e o f t h e i r s i m i l a r i t i e s. H o w e v e r, e n o u g h o f t h i s i n f o r m a t i o n i s now a v a i l a b l e t o s u g g e s t p o s s i b l e e x p e r im e n t a l a p p r o a c h e s t o th e s o l u t i o n o f t h i s p r o b le m. I n a d d i t i o n, o d d -A n u c l e i i n t h i s m a ss r e g i o n a p p e a r t o b e h a v e v e r y m uch l i k e n u c l e i o f s i m i l a r q u a d r u p o le d e f o r m a t i o n i n o t h e r r e g i o n s o f th e p e r i o d i c t a b l e.
44 2. OVERVIEW OF THEORY I n s e c t i o n s we d i s c u s s th e m o t i v a t i o n l e a d i n g t o a t t e m p t s t o u n d e r s t a n d th e o b s e r v a t i o n s on th e R a a n d T h i s o t o p e s w i t h i n th e fr a m e w o r k o f a l p h a p a r t i c l e c l u s t e r i n g, o c t u p o le v i b r a t i o n a n d s t a t i c o c t u p o l e d e f o r m a t i o n m o d e ls. H e re we p r e s e n t a few o f t h e i r m ore im p o r t a n t i m p l i c a t i o n s f o r e v e n - e v e n n u c l e i. E a c h o f t h e s e m o d e ls a r i s e s i n a n a t u r a l w ay fr o m b e h a v i o r o b s e r v e d n o t o n l y i n th e a c t i n i d e r e g i o n b u t a l s o t h r o u g h o u t th e p e r i o d i c t a b l e. The e f f e c t s o f r e f l e c t i o n a sy m m e try i n o d d - A n u c l e i a r e d i s c u s s e d i n s e c t i o n s A L P H A P A R T I C L E C L U S T E R I N G I N H E A V Y N U C L E I A N D T H E H Y B R I D M O D E L The c o n c e p t o f a lp h a p a r t i c l e c l u s t e r i n g i n th e l i g h t a c t i n i d e s i s s u g g e s t e d b y tw o g e n e r a l o b s e r v a t i o n s. F i r s t, r e d u c e d w id t h s f o r th e a l p h a p a r t i c l e d e c a y o f g r o u n d s t a t e s o f Ra a n d Th i n th e 218<A <230 i n t e r v a l a r e l a r g e. I n 218R a, th e g r o u n d s t a t e r e d u c e d a lp h a p a r t i c l e w i d t h i s 20% o f th e W ig n e r L i m i t (Te 5 2 ), a c o n v e n ie n t m e a su re o f m o le c u l a r c h a r a c t e r m o s t o f t e n u s e d i n c o n n e c t i o n w it h h e a v y i o n s c a t t e r i n g. I t i s g e n e r a l l y u n d e r s t o o d t h a t a s t a t e h a v i n g a r e d u c e d w id t h f o r a p a r t i c u l a r n u c l e a r f r a g m e n t o f 100% o f th e W ig n e r L i m i t i s a p u r e c o n f i g u r a t i o n i n w h ic h th e s a i d fr a g m e n t a n d th e r e m a in d e r o f th e n u c le u s a r e th e c o n s t i t u e n t s o f a p u r e m o le c u la r c o n f i g u r a t i o n i n w h ic h th e c o n s t i t u e n t s h a v e t a n g e n t s u r f a c e s. M o r e o v e r, a r e d u c e d w id t h o f e v e n 2%
45 -24- o f th e W ig n e r L i m i t i s s t i l l c o n s i d e r e d t o be " m o l e c u l a r " i n c o l l i s i o n s o f h e a v y i o n s (E b 8 1 ). The s e c o n d m o t i v a t i o n f o r t h i s m o d e l i s th e w e l l e s t a b l i s h e d im p o r t a n c e o f a lp h a p a r t i c l e c l u s t e r i n g f o r l i g h t n u c l e i. We h a v e a l r e a d y i n t r o d u c e d 180 a s one e x a m p le o f a n u c le u s i n w h ic h a lp h a p a r t i c l e c l u s t e r i n g a p p e a r s t o p l a y a r o l e. I n f i g u r e 2. 1, we d i s p l a y a m uch s t u d i e d e x a m p le, 20N e. R e d u c e d a lp h a p a r t i c l e w id t h s o f 20% a n d 50% o f th e W ig n e r L i m i t f l a g t h o s e s t a t e s i n w h ic h th e e f f e c t o f a lp h a p a r t i c l e c l u s t e r i n g i s m o st p r o n o u n c e d. A lt h o u g h th e a r r a n g e m e n t o f s t a t e s i s n o t a s s im p le a s t h a t o f th e p r o p o s e d a lp h a p a r t i c l e c l u s t e r b a n d i n 180, th e b a n d o f n e g a t i v e p a r i t y s t a t e s a s s o c i a t e d w it h th e r e f l e c t i o n a s y m m e t r ic n a t u r e o f a lp h a p a r t i c l e c l u s t e r i n g ( J ^ l ", 3, 5 ", 7 " w it h <0 2> = 5 0 % ) ct c a n be e a s i l y i d e n t i f i e d. In d e e d, e v e n th e g r o u n d s t a t e o f 20Ne, w it h a r e d u c e d w id t h o f o n l y 5% o f th e W ig n e r L i m i t, i s c o n s i d e r e d m o le c u la r i n th e s e n s e t h a t i t s w a v e f u n c t io n c a n be w r i t t e n a s Y = *0*i6oXa_i60(ra_i60)' w he re a n d $ 16q r e p r e s e n t th e i n t r i n s i c s t a t e s o f th e a lp h a p a r t i c l e a n d 160 c o r e, r e s p e c t i v e l y, a n d x _ i 6q c h a r a c t e r i z e s t h e i r r e l a t i v e p o s i t i o n. The d i f f e r e n c e i s t h a t t h o s e b a n d s w it h r e d u c e d w id t h s o f 40% a n d 50% o f th e W ig n e r L i m i t h a v e r e l a t i v e l y w e l l s e p a r a t e d c l u s t e r s, w h e r e a s i n th e g r o u n d s t a t e th e y a c t u a l l y i n t e r p e n e t r a t e to some d e g r e e. One e s s e n t i a l d i f f e r e n c e b e tw e e n 180 a n d 20Ne m u st be m e n tio n e d, h o w e v e r. I n 180, th e a l p h a p a r t i c l e c l u s t e r c o n f i g u r a t i o n g i v e s r i s e t o a n e l e c t r i c d i p o l e m om ent. T h i s i s n o t th e c a s e i n 20N e, s i n c e a i s
46 F i g u r e 2.1 P a r t i a l l e v e l s p e c tr u m o f 20Ne (A j 8 3 ).
47 CO CO + V + CO + V I in OJ O o' O J + fo H O A h" wo * ^ + + oj o = o + ^ m ii A *= oj a 05 = V + + z o' + 0 J o O O J ii ii f= A o j a 05 V o co + a = v i o o in II M k A cma z 05 V -25- <02>=5% 20 N e
48 -26- a s e l f - c o n j u g a t e s y s t e m s o t h a t E l t r a n s i t i o n s v a n i s h i n l e a d i n g o r d e r. We s h o u ld e m p h a s iz e t h a t w h i le 4H e + 14C s t a t e s p l a y an e s s e n t i a l r o l e i n 180 th e y a r e n e v e r t h e l e s s h e a v i l y m ix e d w it h n a n d 20N e + 2 ( p r o t o n h o l e ) c o n f i g u r a t i o n s a s w e l l a s th e q u a d r u p o le d e fo r m e d s t a t e s o f 180. C o n s e q u e n t ly, t h i s c o n f i g u r a t i o n m i x i n g m u st be ta k e n i n t o a c c o u n t when i n t e r p r e t i n g th e s p e c t r o s c o p i c i n f o r m a t i o n f o r t h i s n u c le u s. A s show n i n f i g u r e 1.4, th e s u g g e s t i o n t h a t l a r g e r e d u c e d a lp h a p a r t i c l e w id t h s a r e s i g n a t u r e s f o r a lp h a c l u s t e r i n g i n Ra a n d Th a p p l i e s t o c e r t a i n n u c l e i i n th e A =50 a n d 150 r e g i o n s a s w e l l. A s m e n tio n e d i n th e i n t r o d u c t i o n, th e SGA a p p r o a c h p r o v i d e s a u s e f u l p h e n o m e n o lo g ic a l fr a m e w o rk f o r th e a n a l y s i s o f d a t a on q u a d r u p o le c o l l e c t i v i t y a n d m o le c u la r b e h a v i o r. I n th e I n t e r a c t i n g B o s o n A p p r o x i m a t io n ( I B A ), th e SGA m o d e l f o r q u a d r u p o le c o l l e c t i v i t y i n e v e n -e v e n h e a v y n u c l e i, th e n u c le o n s o u t s i d e o f c l o s e d s h e l l s a r e p a i r e d, th e p a i r s b e i n g t r e a t e d a s b o s o n s. E a c h b o s o n c a n o c c u p y two q uan tu m s t a t e s : th e g r o u n d, o r s - b o s o n, s t a t e w it h a n g u l a r momentum z e r o ; a n d th e e x c i t e d, o r d - b o s o n, s t a t e w it h a n g u l a r momentum o f 2FT a n d e x c i t a t i o n e n e r g y e ^. I n a d d i t i o n, th e d - b o s o n c a n o c c u p y a n y o f th e f i v e m a g n e t ic s u b s t a t e s o f a s p i n 2 p a r t i c l e. C o n s e q u e n t ly, t h e r e a r e s i x t y p e s o f b o s o n s p r e s e n t i n t h i s m o d e l. The t o t a l b o s o n num ber, N = n + n,, s d (w h ere n a n d n, a r e th e n u m b e rs o f s - a n d d - b o s o n s, r e s p e c t i v e l y ) i s s d h e l d c o n s t a n t a t a v a l u e e q u a l t o h a l f th e num ber o f v a le n c e n u c le o n s (the total of valence protons and neutrons). For example, 220Ra has 6
49 F i g u r e 2. 2 C l a s s i c a l r e p r e s e n t a t i o n o f q u a d r u p o le a n d d i n u c l e a r m o l e c u l a r d e g r e e s o f fr e e d o m a n d th e c o r r e s p o n d i n g sym m e try g r o u p s U ( n ) (E n 8 4 ).
50 (a) z ii (b) Z i YNT 70 A 1 / / / ] // M / I 'a/ * / / x y ~ ^y 0, y, d\, 8 Z, 6 Z r,0,< U(6) U(4)
51 -28- v a le n c e p r o t o n s o v e r th e s h e l l c l o s u r e Z = 8 2, a n d 6 v a le n c e n e u t r o n s o v e r th e m a g ic num ber 12 6, y i e l d i n g N =6. When th e c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s f o r t h e s e b o s o n s a r e u s e d t o c o n s t r u c t a b o s o n n u m b e r - c o n s e r v i n g h a m i l t o n i a n, th e y a c t a s g e n e r a t o r s f o r th e sy m m e try g r o u p o f th e q u a d r u p o le c o l l e c t i v e s y ste m, U ( 6 ). A s i m p l y s t a t e d r u l e r e l a t i n g th e p o s s i b l e b o s o n s t a t e s ( i n c l u d i n g m a g n e t ic s u b s t a t e s ) t o th e sy m m e try g r o u p f o r th e s y s t e m i s t h a t i f t h e r e a r e n b o s o n s t a t e s p r e s e n t, th e n th e sy m m e try g r o u p i s U ( n ). The e i g e n s t a t e s o f a S p e c tr u m G e n e r a t i n g A l g e b r a c a n be d e s c r i b e d i n te rm s o f a s e t o f p h y s i c a l b e n c h m a r k s c a l l e d d y n a m ic a l l i m i t s. I n th e s t u d y o f n u c l e a r c o l l e c t i v e ph e nom en a, a s u b g r o u p A o f th e SGA sym m e t r y g r o u p i s a p h y s i c a l l y m e a n in g f u l d y n a m ic a l l i m i t o f th e SGA i f a n d o n l y i f th e sy m m e try g r o u p f o r a n g u l a r momentum, 0 ( 3 ), i s a s u b g r o u p o f A ( l a 8 0 ). O n ly th e n do th e m o d e l s t a t e s h a v e a n g u l a r momentum a s a g o o d q u a n tu m n um be r. P h y s i c a l l y, a d y n a m ic a l l i m i t i s a l i m i t i n g c a s e o f th e p h y s i c a l m o t io n d e s c r ib e d. F o r i n s t a n c e, th e IB A h a s th r e e d y n a m ic a l l i m i t s : S U ( 3 ), th e d e fo rm e d s y m m e tr ic r o t o r ; U ( 5 ), th e a n h a r - m o n ic v i b r a t o r ; a n d 0 ( 6 ), th e a x i a l l y a s y m m e t r ic, g a m m a - u n s t a b le r o t o r. A s i m p l i f i e d v e r s i o n o f th e IB A h a m i l t o n i a n c a n i l l u s t r a t e how t h i s m o d e l t r e a t s th e q u a d r u p o le v i b r a t i o n a l t o d e fo r m e d ( S U ( 5 ) t o S U ( 3 ), i n th e la n g u a g e o f th e IB A ) t r a n s i t i o n. T h i s h a m i l t o n i a n i s H = Ednd - KdQd'Qd. (2.2.1) w here n ^ a n d a r e d e f i n e d a s a b o v e, r e p r e s e n t s th e q u a d r u p o le - q u a d r u p o le i n t e r a c t i o n b e tw e e n b o s o n s, a n d i s a p a r a m e t e r r e p r e s e n t i n g th e s t r e n g t h o f t h i s te rm. I n a d d i t i o n, i s t r e a t e d a s a p a ra m e - S
52 -29- t e r. When cj> >K d ' t5 e e x c i t a t 4 o n s p e c tr u m c o n s i s t s o f d e g e n e r a t e m u l t i p l e t s o f s t a t e s (e a c h m u l t i p l e t c o r r e s p o n d i n g t o a g i v e n v a l u e o f n ^ ) w h ic h a r e s p a c e d a p a r t. T h i s i s th e b e h a v i o r we e x p e c t i n v i b r a t i o n a l n u c l e i, a n d c o r r e s p o n d s t o th e U ( 5 ) d y n a m ic a l l i m i t. We c a n b r e a k th e d e g e n e r a c i e s o f th e m u l t i p l e t s b y i n c l u d i n g a term [k' + (3/8)Kd]Ld-L^ {2 2 2) w he re L d g i v e s th e a n g u l a r momentum o f a b o s o n, i n th e h a m i l t o n i a n. The o p e r a t o r Qd c a n be w r i t t e n i n te rm s o f b o s o n c r e a t i o n ( s + a n d d+ ) a n d a n n i h i l a t i o n ( s a n d d ) o p e r a t o r s a s Qd = ( s + d + d+ s ) + ( l / 5 ) 1 / 2 x ( d + d ) ( 2 ), w here th e s u p e r s c r i p t (2 ) d e n o t e s a c o u p l i n g to two u n i t s o f a n g u l a r momentum, a n d x i s a p a r a m e t e r. I f <cj> > d an<4 X= 0, th e n th e h a m i l t o n i a n c o r r e s p o n d s to th e 0 ( 6 ) g a m m a -u n s ta b le l i m i t ; a v a l u e x = ~ y i e l d s th e S U ( 3 ) d e fo r m e d r o t o r l i m i t. The v i b r a t i o n a l t o d e fo r m e d t r a n s i t i o n i s r e p r o d u c e d b y i n c r e a s i n g th e im p o r t a n c e o f th e te rm r e l a t i v e t o th e d n d te rm ( i. e. b y i n c r e a s i n g T h i s a l g e b r a i c p r e s c r i p t i o n c a n a l s o be a p p l i e d t o s im p le m o le c u la r b e h a v i o r. The r e s u l t, c a l l e d th e v i b r o n m o d e l, u s e s s - b o s o n s (1 = 0 ) a n d p - b o s o n s ( 1 = 1 ), a n d h a s th e sy m m e try g r o u p U ( 4 ). No b o s o n c o u n t i n g r u l e s, s u c h a s t h o s e u s e d f o r th e IB A, h a ve b e e n d e d u c e d f o r th e v i b r o n m o d e l. T h i s m o d e l h a s two d y n a m ic a l l i m i t s : 0 ( 4 ), th e r i g i d r o t a t i n g m o le c u le w it h s t i f f p a r t i c i p a n t n u c l e i ; a n d U ( 3 ), a m ore c o m p le x mode w h ic h i s p r i m a r i l y o f o s c i l l a t o r y c h a r a c t e r ( l a 8 2 a ). T h e se two l i m i t s r e s u l t i n e n e r g y l e v e l s p e c t r a a n d s e l e c t i o n r u l e s f o r ga m m a -ra y t r a n
53 s i t i o n s t h a t a r e i l l u s t r a t e d i n f i g u r e The s im p le v i b r o n m o d e l h a s b e e n a p p l i e d t o a v a r i e t y o f n u c l e a r s y s t e m s, i n c l u d i n g 12C + 12C ( E r 81, C s 8 5 ) a n d i 2C (R u 8 4 ). I n th e p r e s e n t s t u d y, we w i l l be i n t e r e s t e d o n l y i n th e U ( 3 ) l i m i t o f th e U ( 4 ) sy m m e try g r o u p ; th e h a m i l t o n i a n f o r t h i s l i m i t i s H = s n + a n (n - 1 ) + k 'L *L, ( ) P P P P P P P P w he re i s a p a r a m e t e r r e p r e s e n t i n g th e e n e r g y r e q u i r e d t o p ro m o te a b o s o n fro m an s s t a t e t o a p s t a t e, a n d k 1 a n d a a r e th e s t r e n g t h P p a r a m e t e r s f o r t h e i r r e s p e c t i v e te r m s. The p h e n o m e n o lo g y o f t h i s m o d e l i s n o t a s w e l l u n d e r s t o o d a s t h a t o f th e IB A ; h o w e v e r, we w o u ld e x p e c t t h a t e w o u ld c o n t r o l th e e x c i t a t i o n e n e r q y o f th e b a n d h e a d o f th e P " d i p o l e v i b r a t i o n " b a n d, th e b a n d o f s t a t e s h a v i n g n = 1. F u r t h e r, th e P L L term c l e a r l y a c t s t o b r e a k v i b r a t i o n a l m u l t i p l e t s. P P When th e i n t r i n s i c b e h a v i o r o f one o r b o t h o f th e m o l e c u l a r c o n s t i t u e n t s i s i m p o r t a n t, th e s i t u a t i o n b e co m e s m ore c o m p le x. F o r th e c a s e o f a lp h a p a r t i c l e c l u s t e r i n g i n h e a v y n u c l e i, th e q u a d r u p o le c o l l e c t i v e e x c i t a t i o n s p e c tr u m o f th e c o r e m u st be ta k e n i n t o a c c o u n t. T h i s i s done i n th e H y b r i d M o d e l, a m a r r i a g e b e tw e e n th e IB A a n d th e V i b r o n M o d e l (D a 8 3 ). Two s e t s o f b o s o n s a r e u s e d, th e t o t a l b o s o n numb e r i n e a c h s e t b e i n g c o n s e r v e d. The f i r s t s e t c o n s i s t s o f th e s a n d d b o s o n s u s e d t o a c c o u n t f o r th e q u a d r u p o le m o t io n o f th e c o r e ; th e m o le c u l a r a s p e c t s o f th e c o r e - c l u s t e r s t r u c t u r e a r e d e s c r i b e d b y th e s a n d p b o s o n s o f th e U ( 4 ) SGA. F o r c l a r i t y, we d e n o te th e b o s o n s o f th e l a t t e r s e t b y s * a n d p *. We a l s o u se th e n o t a t i o n
54 F i g u r e 2. 3 A s c h e m a t ic r e p r e s e n t a t i o n o f th e e n e r g y s p e c t r a f o r th e tw o d y n a m ic a l l i m i t s o f th e V i b r o n M o d e l. The a r r o w s r e p r e s e n t f i r s t o r d e r a llo w e d E l t r a n s i t i o n s (E n 8 4 ).
55 E (MeV) _ YNT 75 _ YNT 77 -JC 0+ 0 _ > 3 " ; 2+ r ^ o + i i ( 4 )
56 -32- N = n. + n. p s * p * f o r th e c o n s e r v e d b o s o n n u m b e rs f o r e a c h s e t. The sy m m e try g r o u p f o r t h i s m o d e l i s s i m p l y U ( 6 ) $ U ( 4 ). B o s o n s fro m th e two s e t s i n t e r a c t v i a th e h a m i l t o n i a n % - >=«d'8p + 2* v y <2-2-4) I t h a s b e e n fo u n d i n p r e v i o u s s t u d i e s u s i n g t h i s m o d e l t h a t th e U ( 3 ) v i b r o n h a m i l t o n i a n i s m o st a p p r o p r i a t e f o r th e d e s c r i p t i o n o f Ra a n d Th i s o t o p e s. C o n s e q u e n t ly, th e h a m i l t o n i a n t h a t we u se i n o u r s t u d y i s th e sum o f t h o s e fo u n d i n ( ), ( ), ( ) a n d ( ). The H y b r i d M o d e l c a l c u l a t i o n i t s e l f a s su m e s th e p r e s e n c e o f two c o n f i g u r a t i o n s, one w it h o u t a lp h a p a r t i c l e c l u s t e r i n g ( t h e z e r o a lp h a c o n f i g u r a t i o n ), f o r w h ic h N =0 a n d N, i s d e t e r m in e d b y th e u s u a l IB A P Q c o u n t i n g r u l e ; a n d a n o t h e r w it h a s i n g l e v a le n c e a lp h a p a r t i c l e c l u s t e r ( t h e one a lp h a c o n f i g u r a t i o n ), i n w h ic h N^=2 a n d i s two l e s s t h a n i n th e z e r o a lp h a c o n f i g u r a t i o n. The z e r o a lp h a c o n f i g u r a t i o n h a s o n l y th e q u a d r u p o le d e g r e e o f fr e e d o m ; c o n s e q u e n t ly, o n l y p o s i t i v e p a r i t y e v e n s p i n s t a t e s a r e p r e s e n t i n t h i s c o n f i g u r a t i o n. B o t h p o s i t i v e a n d n e g a t i v e p a r i t y s t a t e s a r i s e fro m th e r e f l e c t i o n a s y m m e t r ic one a lp h a c o n f i g u r a t i o n. S t a t e s fr o m b o t h c o n f i g u r a t i o n s a r e p u t i n t o a s i n g l e s p a c e, i n w h ic h th e y a r e m ix e d t h r o u g h th e u s e o f a m i x i n g h a m i l t o n i a n a n d f i r s t o r d e r p e r t u r b a t i o n t h e o r y. V. = y ( s + 2 s * 2 + s * + 2 s 2 ), m ix (w h e re y i s a p a r a m e t e r ) a n d f i r s t o r d e r p e r t u r b a t i o n t h e o r y. The e n e r g y d e n o m in a t o r u s e d i n th e e x p r e s s i o n f o r th e p e r t u r b e d w a v e f u n c t io n
57 -33- i s a l s o a p a r a m e t e r, <f>. C o n f i g u r a t i o n m i x i n g t a k e s p l a c e am ong th e a p o s i t i v e p a r i t y s t a t e s, b u t th e one a lp h a n e g a t i v e p a r i t y s t a t e s h a v e no z e r o a lp h a s t a t e s w i t h w h ic h t o i n t e r a c t a n d, t h u s, re m a in p u r e. An e x a m p le i s s k e t c h e d i n f i g u r e The H y b r i d M o d e l h a s b e e n a p p l i e d to n u c l e i i n th e r a r e e a r t h, l i g h t a c t i n i d e a n d h e a v y a c t i n i d e r e g i o n s (D a 8 3, Da 8 4, Da 8 6 a, Da 8 6 b ). I n p a r t i c u l a r, tw o s e p a r a t e s e t s o f p a r a m e t e r s h a v e b e e n u s e d to d e s c r i b e Ra a n d Th i s o t o p e s (D a 83, Da 8 4 ) O C T U P O L E V I B R A T I O N A L B E H A V I O R I N E V E N - E V E N N U C L E I I n th e i n t r o d u c t i o n - we b r i e f l y d i s c u s s e d how th e c o u p l i n g o f an o c t u p o le p h o n o n t o th e g r o u n d s t a t e q u a d r u p o le c o l l e c t i v e b a n d c a n a c c o u n t f o r th e p r e s e n c e o f l o w - l y i n g n e g a t i v e p a r i t y s t a t e s. I t a p p e a r s t h a t s u c h a c o u p l i n g p r o d u c e s an a l t e r n a t i n g p a r i t y b a n d a t s p i n s o f 3H a n d a b o v e i n th e s p h e r i c a l n u c le u s 150G d. F u r t h e r, i n th e d e fo rm e d r a r e e a r t h n u c l e i th e lo w e s t o c t u p o le v i b r a t i o n a l s t a t e o f t e n a r i s e s fro m th e c o u p l i n g o f an o c t u p o le p h o n o n w hose a n g u l a r momentum v e c t o r l i e s, i n th e i n t r i n s i c fra m e, n o r m a l t o th e sy m m e try a x i s o f th e n u c le u s. The a n g u l a r momentum p r o j e c t i o n on th e sy m m e try a x i s i s a g o o d q uantum num ber w h ic h i s d e n o te d b y K. C o n s e q u e n t ly, th e o c t u p o le p h o n o n we h a v e j u s t d e s c r i b e d h a s K ^ O ". T h i s a l i g n e d o c t u p o l e p h o n o n c a n g i v e th e s p i n - p a r i t y s e q u e n c e 1 ", 3 ", 5 ", e t c., s u c h a s t h a t i n 162E r ( f i g u r e 2. 5 ). The octupole vibrational interpretation of low-lying negative-par-
58 F ig u r e 2. 4 A s c h e m a t ic r e p r e s e n t a t i o n o f th e H y b r i d M o d e l f o r 218Ra - th e m i x i n g o f th e c o n f i g u r a t i o n w it h no a lp h a p a r t i c l e s w it h t h a t h a v i n g one a lp h a p a r t i c l e (E n 8 4 ).
59 THE LOW LYING QUANTUM LEVELS OF RADIUM-218 SHOWING BOTH QUADRUPOLE AND DIPOLE COLLECTIVITY AND THEIR INTERACTION EXCITATION ENERGY IN M e V \ \ \ \ \.6-7 " ' ' \._ ^ \ \ \ \ \ \. \ - 7 ' ' ' 2' r O 0 P U R E QUADRUPOLE ACTUAL N U C L E A R L E V E L DIAGRAM FOR RADIUM 218 PURE DIPOLE A. W. W R I G H T N U C L E A R S T R U C T U R E L A B O R A T O R Y - Y A L E
60 F i g u r e 2. 5 A p a r t i a l l e v e l s p e c trum o f 162E r (He 8 5 ). The n e g a t i v e p a r i t y s t a t e s show n h a v e b e e n i n t e r p r e t e d i n (He 8 5 ) t o be th e r e s u l t o f th e c o u p l i n g o f an o c t u p o le p h o n o n h a v i n g K ^ O " t o th e q u a d r u p o le d e fo r m e d c o r e.
61 I l62er 68 94
62 -36- i t y s t a t e s i n th e t r a n s i t i o n a l r a r e - e a r t h n u c l e i i s s u p p o r t e d i n s e v e r a l w a y s. F i r s t, s t r o n g E3 t r a n s i t i o n s d e e x c i t e th e lo w e s t 3 ' s t a t e s i n t h e s e n u c l e i (B e 7 1 ). N e x t, two o c t u p o le p h o n o n s t a t e s h a v e b e e n o b s e r v e d i n 147Gd a n d 148Gd (K 1 82, Lu 8 4 ) ; t h e s e s t a t e s w ere r e c o g n i z e d p r i m a r i l y b e c a u s e o f t h e i r d e e x c i t a t i o n b y a c a s c a d e o f two E3 t r a n s i t i o n s. An i n t e r p r e t a t i o n o f th e c o r r e s p o n d i n g n e g a t i v e p a r i t y b a n d s o f t h e s e n u c l e i i n te rm s o f th e c o u p l i n g o f an o c t u p o le ph o n o n t o s u c c e s s i v e g r o u n d s t a t e b a n d m em bers i s s u g g e s t e d b y th e s i m i l a r i t y o f th e s t r u c t u r e s o f th e g r o u n d s t a t e a n d n e g a t i v e p a r i t y b a n d s a s w e l l a s th e e n h a n c e d E l t r a n s i t i o n s c o n n e c t i n g th e two b a n d s. F u r t h e r, a num ber o f m o d e ls i n c o r p o r a t i n g o c t u p o le p h o n o n s h a v e s u c c e s s f u l l y r e p r o d u c e d o b s e r v e d e x c i t a t i o n s p e c t r a ( s e e, f o r i n s t a n c e, Ha 77 a n d Su 7 7 ). We s h o u l d a l s o n o t e th e r e s u l t s o f (V o 7 6 ), w h ic h s u g g e s t t h a t th e c h a r a c t e r o f n e g a t i v e p a r i t y s t a t e s i n d e fo r m e d r a r e e a r t h n u c l e i c h a n g e s fro m o c t u p o le v i b r a t i o n a l to two q u a s i p a r t i c l e a s th e s p i n i n c r e a s e s fr o m 9R t o 15fi. S e v e r a l a u t h o r s (H a 7 7, Ha 79, Su 7 7 ) h a ve n o t e d t h a t t h i s s u g g e s t i o n may a p p ly t o t r a n s i t i o n a l la n t h a n id e n u c l e i a s w e l l. F u r t h e r, i t i s a l s o p r e d i c t e d i n (V o 7 6 ) t h a t an e q u i v a l e n t t r a n s f o r m a t i o n o c c u r s i n th e h e a v y a c t i n i d e s ( f o r e x a m p le, 238U ) a t s p i n s n e a r 25R. An o c t u p o le v i b r a t i o n a l b e h a v i o r c a n p r o d u c e E l t r a n s i t i o n s c o n s i d e r a b l y s t r o n g e r th a n t h o s e a r i s i n g fro m s i n g l e p a r t i c l e t r a n s i t i o n s. By s i m p l y c o n s i d e r i n g th e e x p r e s s i o n f o r th e d i p o l e moment r e s u l t i n g fro m an o c t u p o le s h a p e, an e x p r e s s i o n w h ic h i s common t o b o t h th e S t r u t i n s k i ( S t 5 7 ) a n d B o h r - M o t t e l s o n (B o 5 7 ) t r e a t m e n t s d i s c u s s e d e a r l i e r, n a m e ly :
63 -37- D = cazep p (2.2.1) a n d th e e x p r e s s i o n f o r th e f i r s t o r d e r r e d u c e d E l m a t r i x e le m e n t i n a q u a d r u p o le d e fo r m e d n u c le u s, n a m e ly : B ( E l : I. - > I tf) = ( 3 / 4 -it) < D 2>< I I. 1 I. 0> ( ) i f l i f we se e t h a t B ( E 1 : I. + I f ) cc 02 2 < 8 3 2> ( ) T h u s, an o c t u p o l e p h o n o n i s s u f f i c i e n t t o g i v e a s t r o n g E l t r a n s i t i o n when e v e n a s m a l l q u a d r u p o le d e f o r m a t io n e x i s t s. I t b e a r s e m p h a s is, h o w e v e r, t h a t w h i le r e d u c e d m a t r i x e le m e n t s o f E l t r a n s i t i o n s a s s o c i a t e d w it h o c t u p o le v i b r a t i o n s i n th e l a n t h a n i d e r e g i o n a r e e n h a n c e d r e l a t i v e to s i n g l e p a r t i c l e E l d e e x c i t a t i o n s i n h e a v y n u c l e i, th e y a r e s t i l l a f a c t o r o f te n lo w e r th a n t h o s e m e a s u r e d i n 218Ra (G a 8 6 ). One a p p a r e n t s h o r t c o m in g i n t h e s e m o d e ls, h o w e v e r, c o n c e r n s th e p r e d i c t e d d e p e n d e n c e o f th e s t r e n g t h o f th e E l t r a n s i t i o n on th e p a r i t y o f th e i n i t i a l s t a t e. S e v e r a l c a l c u l a t i o n s, i n c l u d i n g t h o s e i n (S u 7 7 ) a n d (H a 7 7 ), p r e d i c t t h a t th e E l t r a n s i t i o n s o r i g i n a t i n g fro m p o s i t i v e p a r i t y s t a t e s s h o u l d be s i g n i f i c a n t l y lo w e r th a n t h o s e o r i g i n a t i n g fr o m n e g a t i v e p a r i t y s t a t e s i n s p h e r i c a l a n d t r a n s i t i o n a l n u c l e i. A d i f f e r en ce o f a f a c t o r o f t h r e e i s p r e d i c t e d i n (H a 7 9 ) f o r 148Sm; a c a l c u l a t i o n i n (H a 7 7 ) f o r 150Gd u s i n g a d i f f e r e n t t e c h n iq u e p r e d i c t s a f a c t o r o f H o w e v e r, m o st n u c l e i i n t h i s n e ig h b o r h o o d show no e v id e n c e f o r s u c h a d i f f e r e n c e ( s e e f i g u r e 2. 6 ). The c a l c u l a t i o n i n ( S u 7 7 ) o v e r com es t h i s d i f f i c u l t y b y i n c l u d i n g an a d d i t i o n a l tw o -b o d y te rm i n th e
64 F i g u r e 2.6 The r a t i o o f the r e d u c e d m a t r i x e le m e n t s o f th e E l and E2 d e e x c i t a t i o n s, d e n o te d b y B ( E 1 : J -» J - 1 ) / B ( E 2 : J-+ J-2 ), betw e en s t a t e s b e l o n g i n g to th e g r o u n d s t a t e a n d o c t u p o le b a n d s i n 150Gd, 148Sm a n d 150Sm. No o d d - e v e n p a r i t y s t a g g e r i n g o f l a r g e m a g n it u d e i s e v id e n t. D a t a a r e ta k e n fr o m r e f e r e n c e s (S u 77, Ha 77, Pe 8 4 b ).
65 B(El: J J -D /B (E 2 : J -J -2 ) (W.u.) -38-
66 -39- quantum operator for the El transition. The p r i m a r y o b j e c t i o n t o th e i n t e r p r e t a t i o n o f Ra a n d Th b e h a v i o r i n te rm s o f o c t u p o l e v i b r a t i o n s h a s b e e n t h a t two o c t u p o le p h o n o n s t a t e s o f s p i n - p a r i t y 0+ (w h ic h i n th e c a s e o f h a r m o n ic o s c i l l a t i o n w o u ld be fo u n d a t t w ic e th e e x c i t a t i o n e n e r g y o f th e n e g a t i v e p a r i t y b a n d h e a d ) h a v e n o t b e e n o b s e r v e d i n e v e n - e v e n Ra a n d Th i s o t o p e s n e a r m a ss 22 5, w here e x t e n s i v e s t u d i e s o f s t a t e s o f lo w s p i n a n d e x c i t a t i o n e n e r g y h a v e b e e n p e r f o r m e d t h r o u g h e x a m in a t io n o f a l p h a - a n d b e t a - p a r t i c l e d e c a y. T h i s a r g u m e n t c a n be a n s w e r e d i n two w a y s. F i r s t, P i e p e n b r i n g ( P i 8 2 ) h a s p e r fo r m e d c a l c u l a t i o n s t h a t show t h a t a n h a r m o n i c i t i e s i n th e v i b r a t i o n a l b e h a v i o r o f Ra a n d Th c a n d r i v e th e tw o - ph o n o n s t a t e s t o much h i g h e r e n e r g i e s, p o s s i b l y a s h i g h a s f i v e t im e s th e o n e -p h o n o n e n e r g y. S e c o n d, i t i s c l e a r t h a t a n y s t a t i c o c t u p o le d e fo rm e d b e h a v i o r m u st be th e end r e s u l t o f, i n a n a l o g y t o th e q u a d r u p o le c a s e, a sm o o th t r a n s i t i o n fr o m o c t u p o l e v i b r a t i o n t o o c t u p o le d e f o r m a t io n t a k i n g p l a c e o v e r a s e r i e s o f n u c l e i. T h i s c o u l d m ean, f o r i n s t a n c e, t h a t 22 4 Ra h a s a s t a t i c o c t u p o l e d e f o r m a t i o n, w h i le R a s i m p l y e x h i b i t s an o c t u p o le v i b r a t i o n a l b e h a v i o r (C h 7 9 ) S T A T I C O C T U P O L E D E F O R M A T I O N S I N E V E N - E V E N N U C L E I J u s t a s s t a t i c a l l y d e fo rm e d q u a d r u p o le s h a p e s e x i s t i n n u c l e i, we m ig h t e x p e c t t h a t s t a t i c o c t u p o le s h a p e s c a n be fo u n d a s w e l l. U n l i k e v i b r a t i o n a l b e h a v i o r, a s t a t i c s h a p e c o m p o n e n t p r e s e n t i n th e g r o u n d s t a t e a f f e c t s g r o u n d s t a t e p r o p e r t i e s, s u c h a s th e n u c l e a r m a s s. I n
67 , M t f lle r a n d N i x p u b l i s h e d a c a l c u l a t i o n o f n u c l e a r m a s s e s t a k i n g b o t h q u a d r u p o le a n d h e x a d e c a p o le d e f o r m a t i o n s i n t o a c c o u n t (Mo 8 1 b ). T h e i r c a l c u l a t i o n i s q u i t e s u c c e s s f u l o v e r th e e n t i r e r a n g e o f h e a v y (A > 8 0 ) n u c l e i ; h o w e v e r, r e l a t i v e l y l a r g e d i s c r e p a n c i e s a r e fo u n d b e tw e e n t h e o r e t i c a l r e s u l t s a n d e x p e r im e n t a l v a l u e s a r o u n d m a ss 220 (s e e f i g u r e 2. 7 ). M o l l e r a n d N i x com m ented t h a t t h i s p r o b le m may a r i s e fr o m th e p r e s e n c e o f a s t a t i c o c t u p o le co m p o n e n t i n th e g r o u n d s t a t e s h a p e o f Ra a n d Th i s o t o p e s (Mo 8 1 a ). T h i s o b s e r v a t i o n i n s p i r e d s e v e r a l t h e o r e t i c a l i n v e s t i g a t i o n s o f g r o u n d s t a t e p o t e n t i a l e n e r g y s u r f a c e s o f Ra a n d T h. The m o st p r o m in e n t o f t h e s e a r e s t u d i e s b y L e a n d e r, e t a l., (L e 8 2 a ) a n d b y N a z a r e w ic z, e t a l. (N a 8 4 b ). B o t h s t u d i e s i n c l u d e d q u a d r u p o le, o c t u p o le a n d h e x a d e c a p o l e d e g r e e s o f fre e d o m a n d c o n c lu d e d t h a t s t a b l e o c t u p o le d e fo r m e d m in im a do o c c u r i n th e p o t e n t i a l e n e r g y s u r f a c e s f o r s e v e r a l Ra a n d T h i s o t o p e s. I n p a r t i c u l a r, th e r e s u l t s o f N a z a r e w ic z, e t a l. ( s e e f i g u r e 2. 8 ) p r e d i c t t h a t ' 224R a a n d ^ ^ h a v e s t a t i c s h a p e s w h ic h i n c l u d e b o t h q u a d r u p o le a n d o c t u p o le m om ents. N a z a r e w ic z, e t a l. a l s o e x t e n d e d th e c a l c u l a t i o n s t o o t h e r r e g i o n s o f th e p e r i o d i c t a b l e a n d fo u n d t h a t h i s m e th o d p r e d i c t e d a s h a l lo w s t a t i c o c t u p o le d e fo r m e d m i n i mum i n th e p o t e n t i a l e n e r g y s u r f a c e o f th e 146Ba g r o u n d s t a t e a s w e l l. We s h o u l d n o t e h e r e t h a t a c o m p le te a n a l y s i s o f g r o u n d s t a t e p o t e n t i a l e n e r g i e s w o u ld i n c l u d e a s f r e e p a r a m e t e r s a l a r g e num ber o f m u l t i p o l e s o f o r d e r h i g h e r t h a n h e x a d e c a p o le a s w e l l a s th e lo w e r o r d e r s i n c l u d e d i n (N a 8 4 b ). H o w e v e r, s u c h c a l c u l a t i o n s a r e i m p r a c t i c a l on c o n v e n t i o n a l c o m p u t in g e q u ip m e n t. I n s t e a d, N a z a r e w ic z sh ow ed t h a t th e i n c l u s i o n o f a n d c o m p o n e n ts o f r e a s o n a b l e s i z e w o u ld a f f e c t th e
68 F i g u r e 2.7 A c o m p a r is o n o f g r o u n d s t a t e m i c r o s c o p i c e n e r g i e s, a s c a l c u l a t e d i n (Mo 8 1 b ), t o e x p e r im e n t a l v a l u e s f o r 1323 n u c l i d e s (Mo 8 1 b ). The b o tto m l i n e sh o w s th e d i f f e r e n c e b e tw e e n e x p e r im e n t a l a n d c a l c u l a t e d v a l u e s. I s o t o p e s a r e c o n n e c t e d b y l i n e s. The l a r g e d i s c r e p a n c y a r i s i n g i n Ra a n d T h i s l o c a t e d n e a r n e u t r o n num ber 134.
69 G r o u n d - S t a t e M i c r o s c o p i c E n e r g y ( M e V ) 10 i- 0 : r 0 : r i i i 1 m i i i [ i r i i i i i i i i m t i ir i i i i i it i \ i i i r i'i r~p "ii i r r i i i i i i i i i i i i > \ Ilf! Experimental Discrepancy (Expt. - Calc.) Microscopic zero-p oin t energies \il1lli Ill1iiii1i»t'1\*i*I\t*»Ii*t\I*i\\\i\it1*Ii'I1ii*1*ill1itii\,ti,I Neutron Number N
70 -42- p o t e n t i a l e n e r g y b y s m a l l a m o u n ts, l e s s t h a n.5 MeV, i n q u a d r u p o le s h a p e s r a n g i n g fro m s p h e r i c a l to e x tre m e e l o n g a t i o n ( 3 2 = 1. 0 ). T h i s r e s u l t l e n t c r e d i b i l i t y to h i s d e l e t i o n o f h i g h e r m om ents fr o m h i s c a l c u l a t i o n. A r o u g h d e s c r i p t i o n o f th e e n e r g y s p e c t r a a r i s i n g fro m d i f f e r e n t fo r m s o f o c t u p o le c o l l e c t i v i t y h a s b e e n g i v e n i n r e f e r e n c e (L e 8 2 a ). A s i l l u s t r a t e d i n f i g u r e 2. 8, th e p o t e n t i a l e n e r g y s u r f a c e s a r e s y m m e tr ic a b o u t C o n s e q u e n t ly, when th e p o t e n t i a l e n e r g y i s m in im iz e d w it h r e s p e c t t o 3 ^ anc^ 3 4 an<3 p l o t t e d a s a f u n c t i o n o f 3 ^, th e r e s u l t a p p e a r s a s i n f i g u r e I f th e p o t e n t i a l b a r r i e r b e tw e e n th e tw o m in im a i s i n f i n i t e, a p e r f e c t l y s t a t i c sh a p e r e s u l t s a n d th e e n e r g y l e v e l se q u e n c e i s 0 +, 1 ", 2+, 3 ", e t c. F o r a f i n i t e b a r r i e r, t u n n e l i n g b e tw e e n th e m in im a r e s u l t s i n th e d i s p la c e m e n t o f th e n e g a t i v e - p a r i t y s t a t e s w i t h r e s p e c t t o th e p o s i t i v e - p a r i t y s t a t e s. F i n a l l y, i n th e p u r e l y v i b r a t i o n a l s i t u a t i o n th e n e g a t i v e - p a r i t y s t a t e s a r e d i s p l a c e d r e l a t i v e t o th e g r o u n d s t a t e b y th e one ph o n o n e n e r g y, Tiw-j. The d e fo rm e d t o v i b r a t i o n a l t r a n s i t i o n d e p i c t e d h e r e i s p r e c i s e l y t h a t to w h ic h we r e f e r r e d i n th e p r e v i o u s s e c t i o n. The i d e a o f a s t a t i c o c t u p o le sh a p e h a s a l s o b e e n a p p l i e d t o e x p l a i n an a p p a r e n t a n o m a ly i n th e h i g h s p i n b e h a v i o r o f T h. D u d ek, e t a l. (D u 8 2 ) fo u n d t h a t th e c r a n k i n g m o d e l b a s e d o n l y on q u a d r u p o le a n d h e x a d e c a p o le d e f o r m a t io n s p r e d i c t e d a p r o n o u n c e d b a c k b e n d, a r o t a t i o n a l s i g n a t u r e o f th e b r e a k i n g o f a p a i r o f n u c le o n s a t a p a r t i c u l a r r o t a t i o n a l f r e q u e n c y ( b e c a u s e o f C o r i o l i s e f f e c t s ), on th e y r a s t l i n e i n 222Th ( s e e f i g u r e ). S u b s e q u e n t m e a su re m e n ts o f th e e x c i t a t i o n s p e c t r u m o f 222Th w ere n o t r e p r o d u c e d b y t h i s p r e d i c t i o n (B o 8 5, Wa 8 3 ).
71 F i g u r e 2. 8 P o t e n t i a l e n e r g y s u r f a c e s f o r th e g r o u n d s t a t e s o f B a, R a, 222Ra a n d 224Ra a s p r e s e n t e d i n (N a 8 4 b ). The e n e r g y i s m in im iz e d w i t h r e s p e c t t o 8 4 a t e a c h p o i n t, a n d th e 8 5 a n d 8 6 p a r a m e t e r s a r e s e t a s f u n c t i o n s o f 8 ^/ 8 3 a n i3 8 4 e x p r e s s i o n s s t a t e d i n th e r e f e r e n c e. The c o n t o u r l i n e s a r e 0.1 MeV a p a r t. The d a s h e d l i n e s i n th e 220Ra s u r f a c e t r a c e th e p a t h o f maximum s o f t n e s s ( t h e m o st l i k e l y p a t h f o r o s c i l l a t i o n s ) i n th e ( ) p l a n e.
72 I CO <T I 91*0 zsl 80*0 0*0 :S/ freo 91*0 80*0 0*0 80*0- '222
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