Louis Montague Martz 9 - Wright Laboratory...Louis Montague Martz Yale University 1978 We have...

ABSTRACT

THREE-NUCLEON TRANSFER REACTIONS AND CLUSTER STRUCTURE IN THE A=15 TO A=19 NUCLEI

Louis Montague Martz Yale Un i v e r s i t y 1978

6 6 3We have s t ud i e d the ( L i , t ) and ( Li , He) r e a c t i o n s on t a r g e t s of12C, 13C> ^ N , ^3 N and ^ 0 a t E ,-AA MeV and 0. =15°. A p r e f e r e n t i a l

Li lab

po p u l a t i o n of f i n a l s t a t e s is e x h i b i t e d in s p e c t r a for the A=15 to A=19 n u c l e i . The s t r ong forward peaking of angul a r d i s t r i b u t i o n s in the , 3 C(6 L i ( t ) l 6 0 and 13C(6 Li ,3 He) l 6 N r e a c t i o n s can be reproduced by DWBA c a l c u l a t i o n s but not by the Hauser -Feshbach model . Given such i n d i c a t i o n s of a p r i ma r i l y d i r e c t mechanism a t forward a n g l e s , we use t hese t h r e e - n u c l e o n t r a n s f e r r e a c t i o n s t o i d e n t i f y c and i da t e s f or 3p_nh s t a t e s .A compar i son wi t h o t h e r mu 11 i -nuc 1 eon t r a n s f e r d a t a , e . g . t he ( ^Li , a ) and ( ^ L i , t ) r e a c t i o n s on ^ C and ^ N t a r g e t s , f u r t h e r t e s t s dominantp a r t i c 1e - h o l e c o n f i g u r a t i o n s . The r e l a t i o n s h i p between ( ^ L i , t ) and 6 3( Li , He) s p e c t r a r e vea l s anal og s t a t e s , no t ab l y T=1,1^=0 l e ve l s a t

high e x c i t a t i o n in ^ 0 . Through a p p l i c a t i o n s of nuc l e a r t heor y , wei n v e s t i g a t e the r o l e of t r i t o n c l u s t e r i n g in such s t r u c t u r e . The 2N+L=6

18 15band p r e d i c t e d by a f o l d e d - p o t e n t i a l model of 0= N+t shows an under l ying cor respondence to the exper i ment a l l e v e l s in t r i t o n - t r a n s f e r da t a . T r i t o n s p e c t r o s c o p i c f a c t o r s c a l c u l a t e d from the SU(3) s h e l l model f u r t h e r sugges t t he broad i n f l uence of c l u s t e r i n g phenomena in t h i s mass r eg i on .We f i nd exper i ment a l ev i dence of s y s t e ma t i c behavi or in the t r i t o n bi nd-

— n 3ing e n e r g i e s of proposed p (sd) c o n f i g u r a t i o n s .

(c) Copyright by Louis Montague Martz 1979

ACKNOWLEDGEMENTS

I w o u l d l i k e t o t h a n k t h e many p e o p l e who h a v e made v i t a l c o n t r i b u

t i o n s t o t h i s r e s e a r c h . P r o f e s s o r P e t e r D. P a r k e r h a s b e e n a t h e s i s

a d v i s o r w i t h b o t h p a t i e n t f l e x i b i l i t y a nd h i g h s t a n d a r d s , as w e l l as a

c o l l a b o r a t o r w i t h g r e a t e x p e r i m e n t a l e x p e r t i s e . D r . S t e p h e n J . S a n d e r s

h e l p e d t o s e t up t h e s e e x p e r i m e n t s , k e p t t h e m r u n n i n g s m o o t h l y d u r i n g

t h e d a y a n d w r o t e s e v e r a l c o m p u t e r c o d e s w h i c h w e r e e x t e n s i v e l y u s e d

f o r t h e d a t a a n a l y s i s .

A t t h e B r o o k h a v e n N a t i o n a l L a b o r a t o r y , D r . C a r l B. D o v e r o f f e r e d

me a f i n e o p p o r t u n i t y t o w o r k i n n u c l e a r t h e o r y a nd i n s t r u c t e d me i n

t h e u s e o f h i s f o l d e d - p o t e n t i a l m o d e l . D r . D. J o h n M i l l e n e r i n t r o d u c e d

me t o t h e S U ( 3 ) s h e l l mod e l a nd a l l o w e d me t o p r e s e n t h i s i m p o r t a n t

c a l c u l a t i o n s f o r ^ 0 .

P r o f e s s o r D. A l l a n B r o m l e y p r o v i d e d v a l u a b l e c o m m e n t a r y on b o t h t h e

o u t l i n e d a n d w r i t t e n f o r m s o f t h i s t h e s i s a n d , s p e c i f i c a l l y , p r o p o s e d

t h e i n v e s t i g a t i o n o f b i n d i n g - e n e r g y s y s t e m a t i c s .

When q u e s t i o n s a r o s e c o n c e r n i n g t h e c o m p u t e r s y s t e m o r t a r g e t p r e

p a r a t i o n , a n s w e r s w e r e a l w a y s a v a i l a b l e f r o m D r . M a r t i n W. Sa c h s and

D r . E d i t h F e h r r e s p e c t i v e l y .

I w o u l d a l s o l i k e t o t h a n k member s o f t h e s t a f f a t t h e 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 . The a c c e l e r a t o r was m a i n t a i n e d and

o p e r a t e d t h r o u g h t h e p e r s i s t e n t e f f o r t s o f K e n z o S a t o , J o h n B e n j a m i n ,

R i c h a r d D ' A l e x a n d e r , P h i l l i p C l a r k i n , T h e o d o r e Duda a nd R o b e r t H e r r i n g t o n .

C h a r l e s G i n g e l l a nd h i s s t a f f h a n d l e d a n y e l e c t r o n i c s p r o b l e m s . I n t h e

m a c h i n e s h o p w e r e Edmond Comeau , G e o r g e S a p o r t i n a nd J o s e p h C i m i n o .

TABLE OF CONTENTS

A b s t r a c t

A c k n o w l e d g e m e n t s .............................................................................................................................................. 1 M

T a b l e o f C o n t e n t s ........................................................................................................................................ v

L i s t o f F i g u r e s ............................................................................................................................................ v i i i

L i s t o f Tab 1 e s ....................................................................................................................................................... x '

I n t r o d u c t i o n

1 . 1 Mo t i v a t i o n .................................................................................................................................. 1

1 . 2 Re s u i t s ........................................................................................................................................ 3

Expe r i m e n t

2 . 1 Appa r a t u s .........................................................................................................................................7

2 . 2 E n e r g y R e s o l u t i o n ............................................................................................................ 10

2 . 3 E l e c t r o n i c I n s t r u m e n t a t i o n ....................................................................................... 12

2 . £ D a t a Ana l y s i s ............................................................................................................................ 13

R e a c t i o n M e c h a n i s m

3 . 1 E x c i t a t i o n F u n c t i o n s ........................................................... 16

3 . 2 A n g u l a r D i s t r i b u t i o n s ................................................................................................. 17

3 - 3 Se 1 ec t i v i t y ..................................................................................................................................19

3 . £ C 1 us t e r i n g .......................................................................................................................................23

T h e o r y

£ . 1 F o l d e d - P o t e n t i a l C l u s t e r Mo d e l ............................................................................ 26

£ . 2 SU ( 3 ) S h e l l Mod e l ................................................................................................................. 31

Section Page

5 . 1 1 2 C (6 L i , t ) 1 5 0 a nd 1 2 C (6 L i , 3 H e ) 1 5 N ............................................................. 3^

5 - 2 O t h e r T r a n s f e r R e a c t i o n s ......................................................................................... 3^

5 . 3 Mod e l P r e d i c t i o n s .......................................................................................................... 39

6 . 1 1 6 0 (6 L i , t ) 1 9 Ne a nd 1 6 0 (6 L i , 3 H e ) 1 9 F ...................................................................... A5

6 . 2 O t h e r T r a n s f e r R e a c t i o n s ................................................................................................ k~J

6 . 3 Mo d e l P r e d i c t i o n s ....................................................................................................... 52

7 . 1 1 5 n (6 L i , t ) l 8 F and 1 9 N ( 6 L i , 3 H e ) l 8 0 ...................................................................56

7 . 2 O t h e r T r a n s f e r R e a c t i o n s ................................... 58

7 . 3 Mod e l P r e d i c t i o n s .......................................................................................................... 62

8 . 1 1 ( 6 L i , t ) 1 7 F a nd 1 (6 L i , 3 H e ) 1 7 0 ...................................................................68

8 . 2 O t h e r T r a n s f e r R e a c t i o n s .......................................................................................... 70

9 . 1 1 3 C ( 6 L i , t ) l 6 0 a nd 13 C (6 L i , 3 H e ) l 6 N ............................................................ 77

9 . 2 O t h e r T r a n s f e r R e a c t i o n s ............. ............................................................................. 79

9 . 3 A n g u l a r D i s t r i b u t i o n s ............................................................................................ 83

Cone 1 us i on

10 . 1 Sys t e m a t i c s ............................................................................................................................... 93

1 0 . 2 S u m m a r y ........................................................................................................................................100

A p p e n d i x A Z R D W B A ........................................................................................................................................ 103

A p p e n d i x B

R e f e r e n c e s

LIST OF FIGURES

2 . 1 S c a t t e r i n g Cha mb e r ............................................................................................................... 9

2 . 2 E l e c t r o n i c s .............................................................................................................................. 14

2 . 3 P a r t i c l e I d e n t i f i c a t i o n .............................................................................................. 14

3 . 1 DWBA v s . H a u s e r - F e s h b a c h ............................................................................................... 18

4 . 1 C o o r d i n a t e s ...................................................................................................................................... 27

4 . 2 P o t e n t i a 1 s ........................................................................................................................ 27

4 . 3 R e s o n a n c e s ..................................................................................................................................... 30

4 . 4 Vs ( r ) a n d \y\ 30

5 . 1 1 2 C ( 6 L i , t ) 1 5 0 .......................................................................................................................... 35

5 . 2 12 C (6 L i , 3 H e ) 1 5 N ............................................................................................................................35

5 . 3 1 2 C ( 7 L i ,ot ) 1 5 n ................................................................................................................................. 37

5 . 4 C o m p a r i s o n ............................................................................................................................................37

5 . 5 1 5 N = 1 2 C + t ............................................................................................................................................ 40

6 . 1 , 6 0 ( 6 L i , t ) l 9 N e ................................................................................................................................. 46

6 . 2 16 0 (6 L i , 3 H e ) 1 9 F ............................................................................................................................ 46

6 . 3 1 5 N ( 7 L i , t ) 1 9 F ................................................................................................................................. 48

6 . 4 C o m p a r i s o n ............................................................................................................................................ 48

6 . 5 ( p l / 2 ) " 1® 2 0 N e .................................................................................................................. . 51

6 . 6 19 F = l 6 0 + t .............................................................................................................................................53

7 . 1 1 5 N ( 6 L i , t ) l 8 F ..................................................................................................................................57

7 . 2 1 5 N ( 6 L i , 3 He) 180 ............................................................................................................................ 57

Number Title Page

v i i i

C o m p a r i s o n ........................... .....

,fW V t ...............1 8

S h e l l Mod e l o f 0 (tt = — 1)

15N ( 7L i , a ) ,80 ............

' V u , t ) ' 7F.

' V 6U , 3H e ) , 7 0

, 3 c ( 7u , t ) 17o .

C o m p a r i s o n . .

p 1 / 2 ® , 6 ° • . .

13C ( 6L i , t ) ' 60 .

' 3C ( 6 L i , 3He) ' 6N

, 3 C ( 7L i , a ) , 6 N .

C o m p a r i s o n • •

0 , a b = * 5 ° ’ ' ’

do / do, • • •c . m.H a u s e r - F e s h b a c h

FRDWBA . . . .

3( s d ) T r a n s f e r ............................

B i n d i n g E n e r g i e s ......................

2 2 p ( s d ) a n d ( s d ) f p T r a n s f e r

ZRDWBA . . .

ZR v s . FR . .

,70 ( 6Li>t) 20Ne

B. 2 , 7 0 ( 6 L i , 3 H e ) 2 0 F .................................................................................................................... 107

B . 3 ' 8 0 ( 6 L i , 3 H e ) 2 l F .................................................................................................................... 107

LIST OF TABLES

1 . 1 Mass R e g i o n .............................................................................................................................. £

5 . 1 A= 1 5 ........................................................................................................................................ 38

6 . 1 A= 1 9 ........................................................................................................................................................... £9

7 . 1 A = 1 8 ........................................................................................................................................................... 60

8 . 1 A = 1 7 .................................................................................................................................................... 73

9 . 1 A = 1 6 ........................................................................................................................................................... 82

1 0 . 1 p “ n ( s d ) n ‘ ............................................................. 97

A . l O p t i c a l P o t e n t i a l s .............................................................................................................. 105

B. 1 2 0 F, 21 F .............................................................................................................................................. 108

Number Title Page

CHAPTER 1 INTRODUCTION

T h e s t r u c t u r e o f l i g h t n u c l e i c a n be s t u d i e d v i a t h e s e l e c t i v i t y

o f m u l t i - n u c l e o n t r a n s f e r r e a c t i o n s . I n c o n j u n c t i o n w i t h t h e s p e c i a l

i z e d p r e d i c t i o n s o f n u c l e a r m o d e l s , e x p e r i m e n t a l s p e c t r a r e v e a l p a r t i c l e -

h o l e c o n f i g u r a t i o n s and c l u s t e r i n g p h e n o me n a i n f i n a l s t a t e s . Such

m i c r o s c o p i c s h e l l s t r u c t u r e a n d m a c r o s c o p i c c o l l e c t i v e b e h a v i o r a r e

o f i n t e r e s t as an i n d i c a t i o n o f s i m p l i f i e d p r o p e r t i e s and as a c h a l l e n g e

t o u n i f i e d t h e o r i e s f o r t h e f i n i t e , m a n y - b o d y s y s t e m . A l t h o u g h t h e s e

s i m p l e f e a t u r e s o f e x c i t e d s t a t e s a r e g e n e r a l l y b l u r r e d by a d e g r e e o f

i m p u r i t y i n t h e c o n f i g u r a t i o n o r a m b i g u i t y i n t h e p a r e n t a g e , t h e f i r s t

s t e p t o w a r d a p r e c i s e u n d e r s t a n d i n g o f n u c l e a r s t r u c t u r e l i e s i n an

i d e n t i f i c a t i o n and i n t e r p r e t a t i o n o f t h e p r i m a r y a s p e c t s .

1 . 1 M o t i v a t i on

I m p o r t a n t p r o g r e s s i s b e i n g made t h r o u g h i n v e s t i g a t i o n s o f f o u r -

p a r t i c l e s t r u c t u r e . T r a n s f e r r e a c t i o n s s u c h as ( 7 | _ i , t ) ( e . g . Co76)

d e m o n s t r a t e a h i g h l y p r e f e r e n t i a l p o p u l a t i o n o f f i n a l s t a t e s i n e v e n -

e v e n , 4N n u c l e i s u c h as ^ 0 a nd ^ N e . C l u s t e r m o d e l s , e . g . t h e f o l d e d -

p o t e n t i a l mo d e l ( B u 7 5 ) , p r e d i c t e n e r g y l e v e l s i n s u b s t a n t i a l a g r e e m e n t

w i t h t h e s e d a t a . The r e s u l t i s a g r o w i n g k n o w l e d g e o f A p - n h c o n f i g u r a

t i o n s a nd a 1p h a - p a r t i c 1e c l u s t e r i n g .

T h r e e - p a r t i c l e s t r u c t u r e i s l e s s e r k nown b u t g e n e r a t i n g i n t e r e s t .

I n e a r l y s t u d i e s o f t h e ( ^ L i , t ) a n d ( ^ L i , ^ H e ) r e a c t i o n s , f o r i n s t a n c e ,

a p r e d o m i n a n t l y d i r e c t m e c h a n i s m i s i n d i c a t e d by e x c i t a t i o n f u n c t i o n s

( Bi 7 3 b , B a 7 0 ) , a n g u l a r d i s t r i b u t i o n s ( e . g . B i 7 5 ) a nd s t r u c t u r a l

s e l e c t i v i t y ( e . g . L i 7 2 ) . T h i s p r e v i o u s w o r k , h o w e v e r , d o e s n o t t r e a t

many a s p e c t s s u c h as T=1 s t a t e s , a c c e s s i b l e f r o m o d d - A t a r g e t s a n d r e

l a t i n g Tz =0 t o Tz = l n u c l e i . I n i t i a l p r e d i c t i o n s f o r t r i t o n - c 1 us t e r

s t a t e s i n f r o m a f o l d e d - p o t e n t i a l mo d e l ( B u 7 5 ) e n c o u r a g e f u r t h e r

a p p l i c a t i o n s t o t h e i n t e r p r e t a t i o n o f t h r e e - n u c l e o n t r a n s f e r d a t a . The

c e n t r a l g o a l o f t h e p r e s e n t r e s e a r c h , t h e r e f o r e , i s t w o f o l d : t o i d e n t i

f y e x p e r i m e n t a l l y f i n a l s t a t e s w i t h d o m i n a n t 3 p _ nh c o n f i g u r a t i o n s and

t o i n d i c a t e t h e o r e t i c a l l y t h e r o l e o f t h r e e - n u c l e o n c l u s t e r i n g i n t h e

A = 15 t o A = 19 n u c l e i .

A f i r s t q u e s t i o n c o n c e r n s t h e c h o i c e o f a t h r e e - n u c l e o n t r a n s f e r

r e a c t i o n . T h e m e c h a n i s m o f t h e ( o t , p ) r e a c t i o n i s r e l a t i v e l y c o m p l e x

b e l o w an i n c i d e n t e n e r g y o f AO MeV, s i n c e b a c k w a r d p e a k i n g i n t h e

a n g u l a r d i s t r i b u t i o n s n e a r Ea - 3 0 MeV r e f l e c t s a c o n t r i b u t i o n f r o m c om-

p o u n d - n u c l e u s f o r m a t i o n , h e a v y - p a r t i c l e s t r i p p i n g a n d / o r a k n o c k - o u t

p r o c e s s ( H i 6 6 ) . W i t h o u t n e u t r o n d e t e c t i o n , m o r e o v e r , t h e ( a , n) r e a c

t i o n i n t o N=Z a nd N=Z- 1 n u c l e i c a n n o t be o b s e r v e d . I n h e a v y - i o n - i n d u c e d ,

t h r e e - n u c l e o n t r a n s f e r , e . g . by ^ B , o r ( H a 7 6 , S c 7 2 , P i 7 7) »

h i g h - r e s o l u t i o n s p e c t r a a r e d i f f i c u l t t o o b t a i n . T h e i n t e r m e d i a t e

c h o i c e i s a l i t h i u m p r o j e c t i l e . The ( ^ L i , a ) r e a c t i o n h a s t w o d i s a d v a n

t a g e s o f i t s o w n , a r i s i n g f r o m Q - v a l u e e f f e c t s . A l a r g e c o n t i n u u m i n

t h e a l p h a - p a r t i c l e s p e c t r u m i s g e n e r a t e d by C o u l o m b d i s s o c i a t i o n , w h i c h

h a s a t h r e s h o l d o f o n l y 2 MeV, a n d a r e d u c t i o n i n s e l e c t i v i t y i s c a u s e d

by w e l l - m a t c h e d a n g u l a r m o m e n t a , w h i c h a l l o w t h e s t r o n g p o p u l a t i o n o f

l o w - s p i n s t a t e s . T h r e e - n u c l e o n t r a n s f e r i n d u c e d by ^ L i , h o w e v e r , f e a

t u r e s l o w b a c k g r o u n d and h i g h - s p i n s e l e c t i v i t y , b e c a u s e t h e Q - v a l u e f o r

b r e a k - u p e q u a l s - 1 6 MeV a n d t h e m i s m a t c h A L i s t y p i c a l l y 6 f i . A l t h o u g h

6 7t h e p a r e n t a g e o f L i i s l e s s e v i d e n t t h a n L i = a + t , a s p e c t r o s c o p i c

f a c t o r f o r H e + t i s f o u n d t o be l a r g e and c o m p a r a b l e t o t h a t f o r a+d

3( Ro 7 6 ) . A d i r e c t t r a n s f e r o f e i t h e r He o r a t r i t o n c a n be o b s e r v e d

6f r o m L i , w i t h g o o d e n e r g y r e s o l u t i o n . Ou r e x p e r i m e n t a l w o r k , t h e r e -

6 6 3f o r e , c e n t e r s on t h e ( L i , t ) and ( L i , He) r e a c t i o n s , w h i c h a r e mea-

12 13 l £ 15 16s u r e d o n t a r g e t s o f C, C, N, N a n d 0 a t E ^ . = £ 0 , ££ o r £6 MeV

a nd 0 l a b = l O ° o r 1 5 ° .

1 . 2 R e s u 1 t s

A p r e f e r e n t i a l p o p u l a t i o n o f f i n a l s t a t e s i n t h e A = 15 t o A = 19

n u c l e i l e a d s t o t h e i d e n t i f i c a t i o n o f new c a n d i d a t e s f o r 3 p ~ n h c o n -

16 18 . . f i g u r a t i o n s . I n T “ 1 n u c l e i , n a m e l y N a nd 0 ( T a b l e 1 . 1 ) , h i g h

s e l e c t i v i t y c h a r a c t e r i z e s p r e v i o u s l y u n o b s e r v e d , t r i t o n - t r a n s f e r r e a c -

3t i o n s . A c o m p a r i s o n w i t h He t r a n s f e r i n t o T z =0 n u c l e i r e v e a l s d i s t i n c t

16 18 a n a l o g s t a t e s i n 0 b u t a h i g h e r l e v e l d e n s i t y i n F. I n m i r r o r

s p e c t r a f o r T z = ± l / 2 n u c l e i ( T a b l e 1 . 1 ) , t h e l a r g e s t c r o s s s e c t i o n s

o c c u r a t h i g h e x c i t a t i o n e n e r g y , i . e . l £ . 9 MeV i n ^ F / ^ 0 and £ 9 MeV i n

19 19N e / F. T h e d o m i n a n t s t r u c t u r e o f s u c h s t a t e s i s c h e c k e d whe n t h e

6 6 3( L i , t ) a nd ( L i , He) r e a c t i o n s a r e p l a c e d i n t h e c o n t e x t o f o t h e r m u l t i -

/? x ,7 13n u c l e o n t r a n s f e r d a t a , s u c h as ( L i , a ) a nd ( L 1 , t ) s p e c t r a f r o m C and

15 " n 3N t a r g e t s . I n p ( s d ) c o n f i g u r a t i o n s , t h e r e e x i s t s t e n t a t i v e e v i

d e n c e o f s y s t e m a t i c b e h a v i o r w i t h r e s p e c t t o a n g u 1a r - m o m e n t u m c o u p l i n g

a n d t r i t o n b i n d i n g e n e r g y .

13 6 16 13 6 A n g u l a r d i s t r i b u t i o n s , m e a s u r e d f o r t h e C( L i , t ) 0 and C( L i ,

3 16He) N r e a c t i o n s a n d c o m p a r e d w i t h DWBA a nd H a u s e r - F e s h b a c h p r e d i c t i o n s ,

c o n f i r m p r e v i o u s i n d i c a t i o n s ( B a 7 1 a , B i 7 3 b ) o f a p r e d o m i n a n t l y d i r e c t

m e c h a n i s m a t f o r w a r d a n g l e s . A l t h o u g h r e l e v a n t t o a n a l o g a s s i g n m e n t s ,

t h e s e a n g u l a r d i s t r i b u t i o n s do n o t u n i q u e l y d e t e r m i n e t r a n s f e r r e d a n g u l a r

T A B L E 1 .1 MASS REGION

TARGET TRANSFER FINAL STATES

leo 0+ L ® i+

14 +N 1

12 +C 0

3tt 19. _ ^ iHe Ne T = 4

3 18He F T = 1, T = 0

t 18c T = 1

3tt 17^ ,He F T = 417o T = 4

3 16He O T = 1, T = 0

t 16n T = 1

3 15 _He O T = |

t 15n t = i

m o m e n t a ( s e e a l s o B i 7 5 ) • T h e i r s t r u c t u r e l e s s b e h a v i o r i m p l i e s t h a t a p

p r o x i m a t e , r e l a t i v 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 i s c o n t a i n e d i n f o r w a r d -

a n g l e s p e c t r a . F r o m t h e s e l e c t i v i t y e x h i b i t e d i n t h e s e s p e c t r a , we

3d r a w s u p p o r t f o r a o n e - s t e p r e a c t i o n m e c h a n i s m ; t h e i n f l u e n c e o f He-

o r t r i t o n - c l u s t e r t r a n s f e r i s s u g g e s t e d by a t h e o r e t i c a l s t u d y o f t h e

f i n a l - s t a t e p a r e n t a g e .

A p p l i c a t i o n o f a f o l d e d - p o t e n t i a l m o d e l t o t h e p r e d i c t i o n o f

6t r i t o n - c 1u s t e r s t r u c t u r e p r o v i d e s a s i m p l i f i e d i n t e r p r e t a t i o n o f ( L i ,

3 19 3He) d a t a . As i n F (Bu77a) , c a l c u l a t e d ( s d ) e x c i t a t i o n s s how a s i g

n i f i c a n t , t h o u g h a p p r o x i m a t e , c o r r e s p o n d e n c e t o s t r o n g l y p o p u l a t e d

^ 15 18 18 15s t a t e s o f N a nd 0 . A w e a k - c o u p 1 i n g r e l a t i o n s h i p b e t w e e n 0 s N + t

19 16a n d F= 0 + t ( T a b l e 1 . 1 ) a p p e a r s t o h a v e s u b s t a n t i a l v a l i d i t y . S i n c e

13 •t h e i n t e r a c t i o n o f a t r i t o n w i t h t h e v a l e n c e n e u t r o n o f C o r w i t h t h e

s p i n 1+ o f ( T a b l e 1 . 1 ) h as e x p e r i m e n t a l l y u n k n o w n a nd p o t e n t i a l l y

16 17c o m p l e x e f f e c t s , N a n d 0 a r e a t p r e s e n t b e y o n d d e s c r i p t i o n b y s u c h

a c l u s t e r m o d e l . M o r e s o p h i s t i c a t e d c a l c u l a t i o n s a r e o b t a i n e d f r o m a

s h e l l m o d e l w i t h an S U ( 3 ) b a s i s ( M i 7 7 ) - S p e c t r o s c o p i c f a c t o r s p r e d i c t e d

18f o r a t r i t o n c l u s t e r i n 0 p r o v e t o be w e l l c o r r e l a t e d w i t h c r o s s

15 .6 3 . 1 8s e c t i o n s m e a s u r e d f o r t h e N( L i , He) 0 r e a c t i o n . I n a d d i t i o n t o

s u g g e s t i n g s p i n v a l u e s , t h e r e f o r e , t h e S U ( 3 ) s h e l l m o d e l s u p p o r t s e v i

d e n c e f r o m t h e f o l d e d - p o t e n t i a l m o d e l t h a t t r i t o n c l u s t e r i n g p l a y s an

i m p o r t a n t r o l e i n t h e s t r u c t u r e o f l i g h t n u c l e i .

T h e f o r m a l i s m o f t h e s e n u c l e a r m o d e l s i s d i s c u s s e d i n C h a p t e r A,

f o l l o w i n g a d e s c r i p t i o n o f e x p e r i m e n t a l a p p a r a t u s a nd p r o c e d u r e ( C h a p t e r

62 ) a n d a s u r v e y o f e x i s t i n g e v i d e n c e o n t h e m e c h a n i s m o f t h e ( L i , t ) a nd

6 3( Li, He) reactions (Chapter 3)- The presentation of results is intro-

d u c e d i n C h a p t e r 5 b y t h e A= 15 n u c l e i , w h i c h r e p r e s e n t t h e f o c u s o f

e a r l i e r s t u d y , b u t t h e d e s c r i p t i o n a nd i n t e r p r e t a t i o n o f new d a t a b e g i n

w i t h t h e A= 19 n u c l e i ( C h a p t e r 6 ) , w h i c h p r o v i d e an e q u a l l y f a v o r a b l e

c a s e f o r t h r e e - n u c l e o n c l u s t e r s t r u c t u r e . We p r o c e e d d o w n w a r d i n t h e

s d s h e l l ( C h a p t e r s 7 " 9 ) a nd e n d w i t h an a n a l y s i s o f a n g u l a r d i s t r i b u

t i o n s f o r t h e A = 1 6 n u c l e i ( S e c t i o n 9 . 3 ) . I n C h a p t e r 10, a c o m p a r i s o n

d e m o n s t r a t e s t h e common f e a t u r e s o f n u c l e i i n t h i s mass r e g i o n , a n d a

s u m m a r y i n t e g r a t e s t h e new c o n t r i b u t i o n s o u t l i n e d a b o v e w i t h t h e b o d y

o f p r e v i o u s w o r k . A f t e r a d d i t i o n a l DWBA c a l c u l a t i o n s ( A p p e n d i x A ) , we

20 20 21i n c l u d e d a t a f o r Ne , F a nd F ( A p p e n d i x B ) , i n w h i c h t h r e e - n u c l e o n

t r a n s f e r r e a c t i o n s v e n t u r e b e y o n d p - s h e l l t a r g e t s .

CHAPTER 2 EXPERIMENT

M e a s u r e m e n t s on t h e ( ^ L i , t ) , ( ^ L i , ^ H e ) , ( ^ L i , a ) a nd ( ^ L i , t )

r e a c t i o n s a r e p e r f o r m e d u s i n g t h e MP-1 Tandem v a n de G r a a f f a c c e l e r a t o r

o f t h e 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 . I n t h e E x t r i o n s p u t

t e r i n g s o u r c e ( M i 7 ^ ) » a + 4 0 0 nA beam o f L i i o n s i s g e n e r a t e d by bom

b a r d i n g an a n n u l a r , c o n e - s h a p e d s a m p l e o f l i t h i u m w i t h p o s i t i v e c e s i u m

i o n s . The l i t h i u m i s p r e s s e d i n t o a s t e p p e d c o p p e r c y l i n d e r , t o i n c r e a s e

t h e r m a l c o n d u c t i v i t y , a nd i s p l a c e d q u i c k l y i n v a c u u m t o a v o i d o x i d a

t i o n . Common s o u r c e p a r a m e t e r s a r e an e x t r a c t i o n v o l t a g e o f + 3 0 kV and

an e x t r a c t i o n c u r r e n t o f +1 mA. A f l o w o f o x y g e n o n t o t h e l i t h i u m s u r

f a c e e n h a n c e s t h e c u r r e n t o u t p u t b y + 5 0 % , a f t e r t h e c o n e h as b e e n i n

u s e f o r a f e w h o u r s . T h e o x y g e n l e a k r a t e i s c r i t i c a l and v e r y s m a l l ,

- 7 - 6f o r t h e s o u r c e p r e s s u r e r e m a i n s i n t h e 10 - 1 0 r a n g e . A f t e r n e g a t i v e

i o n s a r e i n j e c t e d i n t o t h e a c c e l e r a t o r a nd f o i 1- s t r i p p e d a t i t s t e r m i -

3+n a l , L i n u c l e i a r e a n a l y z e d b y t h e 9 0 ° m a g n e t . T h e e m e r g i n g beam

t y p i c a l l y h a s an e n e r g y o f 44 MeV, a s p r e a d o f + 7 keV and a c u r r e n t o f

+ 4 0 0 n A , w h i c h i s r e d u c e d t o + 2 0 0 nA by c o l l i m a t o r s i n f r o n t o f t h e

t a r g e t ( s e e S e c t i o n 2 . 2 ) .

B o t h s o l i d a nd g a s t a r g e t s a r e u s e d i n t h i s s e r i e s o f e x p e r i m e n t s .

1 21/ 2 " - d i a m e t e r t a r g e t f r a m e s s u p p o r t f o i l s o f C ( Y i s s u m C o r p o r a t i o n ,

1 3I s r a e l ) o r C ( AECL, C h a l k R i v e r , C a n a d a ) , t h e l a t t e r b e i n g i s o t o p i c a l l y

, ^ 15 16 I 8 rt ,e n r i c h e d t o ^ 9 6 % . A g a s c e l l ( Co 7* 0 c o n t a i n s N, N, 0 , 0 ( Mon

s a n t o C o r p o r a t i o n ) o r ^ 0 ( M i l e s L a b o r a t o r i e s ) , f o r w h i c h t h e o n l y

o b s e r v a b l e i m p u r i t y i s "°4% ^ 0 i n t h e ^ 0 g a s . T h e c e l l h a s a 1 / 4 " -

d i a m e t e r e n t r a n c e w i n d o w o f 20 p i n c h n i c k e l a nd a w i d e e x i t w i n d o w ( s e e

2.1 Apparatus

F i g . 2 . 1 ) o f 100 y i n c h H a v a r . T a r g e t c h a r a c t e r i s t i c s a r e f u r t h e r d i s

c u s s e d i n S e c t i o n 2 . 2 .

R e a c t i o n p r o d u c t s a r e o b s e r v e d w i t h t w o S i ( S B ) d e t e c t o r t e l e s c o p e s ,

3one d e s i g n e d f o r t r i t o n s , t h e o t h e r f o r He n u c l e i a nd a l p h a p a r t i c l e s .

E n e r g y d e t e r m i n a t i o n r e q u i r e s a t o t a l d e t e c t o r t h i c k n e s s c a p a b l e o f

3 3 4s t o p p i n g t h e r e l e v a n t n u c l e u s ; 4 2 0 0 y f o r H a nd 2 0 0 0 y f o r H e / He a r e

s u f f i c i e n t i n o u r c a s e . P a r t i c l e i d e n t i f i c a t i o n d e p e n d s u p o n t h e f i r s t

d e t e c t o r i n e a c h t e l e s c o p e , w h i c h s h o u l d p r o v i d e an a d e q u a t e b u t n o t

e x c e s s i v e e n e r g y - l o s s s i g n a l . T h e c r i t e r i a AE>1 MeV and A E / E < l / 2 a r e

s a t i s f i e d by a t h i c k n e s s o f 2 00 y f o r t r i t o n s w i t h 3<E<52 and by 64 y

f o r He n u c l e i w i t h 11< E < 7 4 . T h e s e t w o d e t e c t o r s , f o r e x a m p l e , a l l o w

m e a s u r e m e n t o f t h e ^ C ( 8 L i , t ) ^ 8 0 a nd ( ^ L i , ^ H e ) r e a c t i o n s f r o m a

l a b o r a t o r y a n g l e o f 10° t o 7 0 ° f o r e x c i t a t i o n e n e r g i e s up t o 28 MeV i n

160 a n d 15 MeV i n l 6 N.

T h e a p p a r a t u s i n s i d e a 3 0 " - d i a m e t e r , O r t e c s c a t t e r i n g c h a m b e r i s

d e p i c t e d i n F i g . 2 . 1 . T h e d i r e c t i o n a nd s i z e o f a l i t h i u m beam f r o m

t h e a c c e l e r a t o r a r e c o n s t r a i n e d b y t w o t a n t a l u m c o l l i m a t o r s ( S e c t i o n

2 . 2 ) , a n d t h e r e s u l t i n g s c a t t e r i n g e f f e c t s a r e r e d u c e d by an a d d i t i o n a l

a p e r t u r e c l o s e t o t h e t a r g e t . A f t e r p a s s i n g t h r o u g h w i n d o w s o f t h e

gas c e l l , t h e beam i s c o l l e c t e d b y a m a g n e t i c a l l y s h i e l d e d F a r a d a y c u p .

M a g n e t s a r e a l s o a t t a c h e d t o e a c h s n o u t i n o r d e r t o d e f l e c t e l e c t r o n s

away f r o m t h e d e t e c t o r s . The t w o d e t e c t o r t e l e s c o p e s c a n be moved

i n d e p e n d e n t l y b u t a r e l i m i t e d t o Gj ^ 1 2 . 5 ° * T h i s a n g l e b e t w e e n e a c h

p a i r o f s l i t s a nd t h e beam d i r e c t i o n i s c a l i b r a t e d t o ± 0 . 1 ° b y o p t i c a l

a l i g n m e n t w i t h t h e c o l l i m a t o r s . I n t h e c a s e o f a s o l i d t a r g e t , we u s e

r e c t a n g u l a r r a t h e r t h a n c i r c u l a r beam c o l l i m a t o r s a n d r e m o v e t h e s l i t

F i g u r e 2 . 1 Ch a mb e r s e t - u p f o r a g a s t a r g e t

FROM ACCELERATOR

TO FARADAY CUP

a t t h e t a r g e t e n d o f e a c h d e t e c t o r s n o u t . I n o r d e r t o b l o c k t h e a n t i

s c a t t e r i n g a p e r t u r e f r o m t h e v i e w o f t h e d e t e c t o r s , a n a d d i t i o n a l s n o u t

i s i n t r o d u c e d b e t w e e n t h i s a p e r t u r e a nd t h e s o l i d t a r g e t . A 8 " - s q u a r e

1 3c o p p e r p l a t e c o o l e d by l i q u i d n i t r o g e n i s p o s i t i o n e d n e a r a C t a r g e t

1 2t o r e d u c e C b u i l d - u p .

2 . 2 E n e r g y R e s o l u t i o n

T h e beam c o l l i m a t i o n , t a r g e t t h i c k n e s s and d e t e c t o r s o l i d

a n g l e a r e d e s i g n e d t o b a l a n c e r e s o l u t i o n w i t h y i e l d . T h e s e t w o c o m p e t i n g

c h a r a c t e r i s t i c s a r e d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n s f o r a s o l i d

t a r g e t n o r m a l t o t h e beam;

iEi ■ (f ) Ae* Qn° - “ 1 / 2 R + b s i nGX - 1 / 2 R - b s i n 0 \ , nA0 = 180 - t a n (---------— r ---------- ) - t a n (------——----------- - ) ( 2 . 1 )

a + b c o s 0 a + b c o s 0

- 2 t a n 1 ( a ^ - 0 a t s m a l l © ( P i 73 )

AE. . = ( 4 ^ ) , . • t - AE . a t s m a l l 0 ( 2 . 2 )L i d x L i 2

Y S n ( p t ) & J A « l a b . ( 2 . 3 )c . m .

T h e k i n e m a t i c b r o a d e n i n g AE] i s d u e t o a v a r i a t i o n i n t h e e n e r g y E o f

t h e o u t g o i n g n u c l e u s as a f u n c t i o n o f t h e l a b o r a t o r y a n g l e o f o b s e r v a

t i o n 0 . I n t h e e x p r e s s i o n f o r A 0, r e l e v a n t g e o m e t r i c a l p a r a m e t e r s a r e

t h e w i d t h a o f t h e d e t e c t o r s l i t , t h e w i d t h b o f t h e beam s p o t on t h e

t a r g e t a nd t h e d i s t a n c e R b e t w e e n t h e t w o . T h e a d d i t i o n a l b r o a d e n i n g

AE2 i n t h e o u t g o i n g e n e r g y a r i s e s f r o m a c h a n g e A E | j o f t h e i n c i d e n t

e n e r g y as t h e beam p a s s e s t h r o u g h a t a r g e t o f g i v e n s t o p p i n g p o w e r

( d E / d x ) j j a n d t h i c k n e s s t . I n t h e d e f i n i t i o n o f t h e d i f f e r e n t i a l c r o s s

s e c t i o n — , Y i s t h e y i e l d o f d e t e c t e d n u c l e i , n i s t h e t o t a l n u m b e rd f i c . m .

o f L i ^ + i o n s c o l l e c t e d , p t i s t h e a r e a l d e n s i t y o f t h e t a r g e t , Af t . i s1 ab

t h e l a b o r a t o r y s o l i d a n g l e o f t h e d e t e c t o r s l i t , a nd J i s t h e J a c o b i a n

c o n v e r t i n g i t i n t o t h e c e n t e r - o f - m a s s s y s t e m . T h e i n t e r d e p e n d e n c e o f

t h e s e t h r e e e q u a t i o n s i s f i r s t e v i d e n t i n a p r o p o r t i o n a l i t y o f b o t h

A E ^ . a n d Y t o t h e t a r g e t t h i c k n e s s . The d e t e c t i o n g e o m e t r y , m o r e o v e r ,

f i x e s Aft a n d a f f e c t s A 0 ; t h e beam c o l l i m a t i o n n o t o n l y d e t e r m i n e s bI 3 D

b u t a l s o i n f l u e n c e s n . A l i t h i u m beam c a n be a d e q u a t e l y f o c u s e d t h r o u g h

t w o c o l l i m a t o r s w i t h d i m e n s i o n s w x h = 0 . 0 4 0 " * 0 . 1 2 0 " a n d p o s i t i o n s 5 0 " a nd

2 0 " f r o m t h e t a r g e t . T h e r e s u l t i n g b e a m - s p o t h e i g h t o f 0 . 2 8 0 " i s s a f e

l y w i t h i n t h e 1/ 2 " - d i a m e t e r s o l i d t a r g e t , a nd t h e v a l u e o f b = 0 . 0 9 3 "

i s e f f i c i e n t l y c l o s e t o t h a t o f a = 0 . 0 6 2 " ( s e e Eq . 2 . 1 ) . A l / l 6 " x l / A "

- Ad e t e c t o r s l i t a t R = 1 0 " a n d 0 * 1 0 ° l e a d s t o A 0 = O . 8 8 ° a nd Af t . = 1 . 5 6 * 1 0 s r .

W i t h a 100 y g / c m c a r b o n t a r g e t , t h e e n e r g y r e s o l u t i o n i s ' v l l O keV f o r

t h i s e x p e r i m e n t a l c o n f i g u r a t i o n .

A g a s t a r g e t i n t r o d u c e s f u r t h e r i n t e r r e l a t i o n b e t w e e n t h e a b o v e

e f f e c t s , b e c a u s e a d o u b l e - s l i t d e t e c t i o n g e o m e t r y c o m b i n e s w i t h t h e

beam c o l l i m a t i o n t o d e f i n e an a c t i v e t a r g e t v o l u m e . Two 1/ I 6 " - d i a m e t e r

c o l l i m a t o r s p l a c e d 5 0 " a nd 2 0 " f r o m t h e t a r g e t a s s u r e t h a t t h e beam

p a s s e s c l e a n l y t h r o u g h t h e 1/ A " - d i a m e t e r e n t r a n c e w i n d o w o f t h e g a s c e l l .

S i n c e t h e a n g l e o f o b s e r v a t i o n © d e p e n d s o n t w o d e t e c t o r s l i t s and

s i n c e t h e o b s e r v e d g a s v o l u m e h a s s i g n i f i c a n t l e n g t h a l o n g t h e beam

d i r e c t i o n z , b o t h 0 a nd A0 v a r y w i t h z , as d o e s Ej_ j . K i n e m a t i c b r o a d

e n i n g t h e r e f o r e b ec o me s i n t e r w o v e n w i t h t h e l o s s o f i n c i d e n t e n e r g y ,

a n d t h e t w o e f f e c t s m u s t be t r e a t e d t o g e t h e r ( s e e C o 7 ^ ) . The y i e l d o f

d e t e c t e d n u c l e i i s c a l c u l a t e d f r o m t h e f o l l o w i n g e x p r e s s i o n s (S i 59 )

I n a d d i t i o n t o s y m b o l s f o u n d i n E q s . 2 . 1 a nd 2 . 3 , N i s t h e n u m b e r o f

t a r g e t n u c l e i p e r u n i t v o l u m e , and 1 i s t h e h e i g h t o f t h e b a c k d e t e c

t o r s l i t . G_ = 5 . A x 10 ^ cm- s r i s i m p l i e d b y s l i t s w i t h a . x l . = 1 / I 6 " x l / A " 00 i ia t 1 . 6 5 " a n d 7 . 6 5 " f r o m t h e t a r g e t . T o g e t h e r w i t h a g a s p r e s s u r e o f

1 / 8 a t m , t h i s c o n f i g u r a t i o n l e a d s t o an e n e r g y r e s o l u t i o n o f M 10 k e V .

F u r t h e r c o n t r i b u t i o n s , h o w e v e r , a r i s e f r o m s t r a g g l i n g i n t h e g a s a n d i n

t h e w i n d o w s o f t h e c e l l . T h i s b r o a d e n i n g n ( k e V ) c a n be e s t i m a t e d f r o m

t h e r e l a t i o n ( B o 15 , Co66)

n 2 = 4.35.1 z 2 | p t , ( 2 . 5 )

w h e r e z i s t h e a t o m i c n u m b e r o f t h e m o v i n g n u c l e u s , Z and A a r e t h e

a t o m i c n u m b e r a n d a t o m i c w e i g h t o f t h e m e d i u m , a nd p t i s t h e a r e a l

2d e n s i t y o f t h e m e d i u m ( mg / c m ) . F o r l i t h i u m p a s s i n g t h r o u g h t h e e n

t r a n c e w i n d o w a nd t h e g a s t o t h e c e n t e r o f t h e c e l l , a common r e s u l t

i s h ^ j ~ A 2 keV a n d n ^ . ^ 2 5 keV r e s p e c t i v e l y . E n e r g y s t r a g g l i n g o f t h e

o u t g o i n g n u c l e i i s n e g l i g i b l e i n t h e g a s b u t i s t y p i c a l l y n ^ = 2 n t - 6 2 keV

i n t h e t h i c k e x i t w i n d o w .

2 . 3 E l e c t r o n i c I n s t r u m e n t a t i o n

T h e e n e r g y s i g n a l s f r o m t h e t w o d e t e c t o r t e l e s c o p e s a r e d i f f e r e n

t i a t e d , a m p l i f i e d , s h a p e d , s y n c h r o n i z e d a nd g a t e d , i n t h e way d e s c r i b e d

b y F i g . 2 . 2 f o r t h e h e l i u m t e l e s c o p e . M u t u a l g a t i n g o c c u r s b e t w e e n t h e

e n e r g y l o s s AE a nd t h e t o t a l e n e r g y E= AE+ E^ . I n o r d e r t o r e d u c e t h e

c o u n t r a t e a t t h e c o m p u t e r i n t e r f a c e , t h e g a t e on E by a AE w i n d o w r e

mov e s e l a s t i c a l l y s c a t t e r e d l i t h i u m a n d some o f t h e h y d r o g e n s i g n a l s .

I n o r d e r t o a s s u r e t h a t v a l i d AE - E p a i r s a r e t r a n s f e r r e d t o t h e c o m p u t e r ,

t h e s e c o n d g a t e r e m o v e s AE s i g n a l s w h i c h c o r r e s p o n d t o E s i g n a l s b e l o w

t h e d i s c r i m i n a t o r l e v e l o f t h e b i a s e d a m p l i f i e r . A f t e r AE a nd E r e a c h

t h e a n a l o g - t o - d i g i t a l c o n v e r t e r s o n t h e i n t e r f a c e , EVENT 1 t r i g g e r s t h e i r

t r a n s f e r t o t h e c o m p u t e r . T h e i n t e g r a t e d beam c u r r e n t , t h e n u m b e r o f

e v e n t s a n d t h e d e a d t i m e a r e a l s o s t o r e d . D u p l i c a t e h a r d w a r e f o r t h e

o b s e r v a t i o n o f t r i t o n s p r o v i d e s EVENT 2 , A E 1, and E 1= A E ' + E j ' + E 2 1 s i g n a l s

f r e e f r o m h e l i u m a nd l i t h i u m c o n t r i b u t i o n s . By means o f d a t a a c q u i s i

t i o n a n d a n a l y s i s s o f t w a r e on an IBM 3 6 0 / A A , t h e i n f o r m a t i o n i s s e n t

b o t h t o m a g n e t i c t a p e f o r l a t e r a n a l y s i s a n d t o c o m p u t e r memor y f o r o n

l i n e m o n i t o r i n g o f t h e e x p e r i m e n t .

2 . £ D a t a A n a l y s i s

A t w o - p a r a m e t e r d i s p l a y o f AE a n d E ( F i g . 2 . 3 ) a l l o w s p r e c i s e i s o

t o p e i d e n t i f i c a t i o n v i a a s o f t w a r e g a t e w h i c h d e f i n e s M i n ( E ) < A E < M a x ( E ) .

D a t a t a p e s a r e r e p l a y e d a f t e r t h e g a t e has b e e n r e f i n e d f r o m i t s p r e l i m

i n a r y f o r m d u r i n g t h e e x p e r i m e n t . A l t h o u g h t h e s e p a r a t i o n o f t r i t o n s

3f r o m d e u t e r o n s i s r a t h e r c l e a r - c u t , an i d e n t i f i c a t i o n o f He r e q u i r e s

t h i s l e n g t h y p r o c e d u r e . I n F i g . 2 . 3 , t h e L i , cc) r e a c t i o n g e n e r a t e s a

h u g e c o n t i n u u m t h r o u g h C o u l o m b d i s s o c i a t i o n b u t a n e g l i g i b l e t a i l w i t h i n

t h e He g a t e . No e v i d e n c e o f a l p h a p a r t i c l e s a p p e a r s i n o u r ( L i , He)

s p e c t r a .

6 6 3T h e e n e r g y c a l i b r a t i o n o f s p e c t r a f r o m t h e ( L i , t ) , ( L i , He) and

( 7 L i , a ) r e a c t i o n s i s d e t e r m i n e d f r o m known e x c i t a t i o n e n e r g i e s i n t h e

A= 1 5 a nd A= 1 9 n u c l e i ( s e e f o o t n o t e s t o T a b l e s 5 - 1 , 6 . 1 ) . C a l i b r a t i o n

12 16d a t a a r e c o l l e c t e d f r o m C a n d / o r 0 t a r g e t s i m m e d i a t e l y b e f o r e o r

after the study of a different target under the same experimental conditions.

Figure 2.2

F i g u r e 2 . 3

PRE: P r e - a m p ) i f i e r

AMP: M a i n A m p l i f i e r

TSCA: T i m i n g S i n g l e C h a n n e l A n a l y z e r

LG: L i n e a r G a t e a nd S t r e t c h e r

B I A S : B i a s e d A m p l i f i e r

I d e n t i f i c a t i o n o f He

C o u n t s w i t h i n t h e d a s h e d l i n e s

3a r e a t t r i b u t e d t o He n u c l e i .

Electronics for helium observation

E V E N T I

Ex ( ,80) (MeV)IT) O tt

Eh« (bin)

F o r a g a s t a r g e t , t h e e n e r g y l o s s o f an o u t g o i n g p a r t i c l e i n t h e g a s and

i n t h e e x i t w i n d o w i s g i v e n t o s u f f i c i e n t a c c u r a c y by

^ - ( c o n s t a n t ) (-!■— - ) , ( 2 . 6 )

w h e r e t h e c o n s t a n t i s c h o s e n t o f i t t a b u l a t e d v a l u e s o f t h e s t o p p i n g

p o w e r ( N o 7 0 ) . A f t e r t h i s c o r r e c t i o n o f 2 0 0 - 4 0 0 keV f o r ^He and 5 0 - 1 0 0

keV f o r t r i t o n s , t h e o u t g o i n g e n e r g i e s a r e r e l a t e d t o o b s e r v e d c h a n n e l s

by a l i n e a r f i t ( B e 6 9 ) , as i n t h e c a s e o f a s o l i d t a r g e t . Two s e t s o f

e x c i t a t i o n e n e r g i e s , b a s e d on t h e ( ^ L i , ^ He ) a nd ^ 0 ( ^ L i , ^ He ) ^ F

r e a c t i o n s r e s p e c t i v e l y , a r e c h e c k e d f o r c o n s i s t e n c y and a r e f o u n d t o

15 6 3 18a g r e e w i t h i n 5 keV f o r f i n a l s t a t e s i n t h e N( L i , He) 0 r e a c t i o n .

,6 »I n t h e ( L i , t ) r e a c t i o n , t h e t w o s t a n d a r d s a p p l y t o l a r g e l y c o m p l e m e n

t a r y r e g i o n s o f Q - v a l u e . W i t h i n t h e r a n g e o f i n t e r p o l a t i o n b e t w e e n

k n own l e v e l s , a c a l i b r a t i o n g e n e r a l l y has t h e e s t i m a t e d u n c e r t a i n t y o f

AE = ± 2 0 keV ( s e e T a b l e s 5 - l " 9 * l ) -

A f i n a l s t e p i n t h i s a n a l y s i s i s t h e e x t r a c t i o n o f d i f f e r e n t i a l

c r o s s s e c t i o n s . A G a u s s i a n d i s t r i b u t i o n i s f i t t e d ( Be 6 9 ) t o t h e c o n

t i n u u m a r i s i n g f r o m C o u l o m b d i s s o c i a t i o n and i s s u b t r a c t e d f r o m t h e

s p e c t r u m . I n f i t t i n g i n d i v i d u a l p e a k s , we r e p r e s e n t s t r o n g l y o v e r

l a p p i n g s t a t e s b y a d o u b l e o r t r i p l e G a u s s i a n . I n a d d i t i o n t o a r e

f i n e m e n t o f t h e p e a k p o s i t i o n a nd w i d t h , t h i s p r o c e d u r e p r o v i d e s an

a r e a Y u s e d i n t h e c a l c u l a t i o n o f d o / d ^ c m f o r a g i v e n f i n a l s t a t e

( E q s . 2 . 3 , 2 . 4 ) . S t a t i s t i c a l u n c e r t a i n t y i n t h e y i e l d and s y s t e m a t i c

e r r o r i n t h e t a r g e t t h i c k n e s s and s l i t w i d t h o f t e n a dd up t o ^20% o f

t h e a b s o l u t e c r o s s s e c t i o n . T h e e r r o r c o n t r i b u t i o n a r i s i n g f r o m b a c k

g r o u n d s u b t r a c t i o n v a r i e s w i d e l y b u t , i n t h e c a s e o f a n g u l a r d i s t r i b u

t i o n s , c o n s i s t e n c y i n t h e f i t t i n g p r o c e d u r e as a f u n c t i o n o f a n g l e i s

c h e c k e d .

6 6 3I f t h e ( L i , t ) and ( L i , He) r e a c t i o n s a r e t o be u s e f u l i n t h e

s t u d y o f 3p**nh c o n f i g u r a t i o n s , t h e n t h e y m u s t p r o c e e d p r i m a r i l y v i a

a d i r e c t m e c h a n i s m . T h i s c h a p t e r p r e s e n t s a s u r v e y o f t h e e x i s t i n g

e x p e r i m e n t a l e v i d e n c e , as i t r e l a t e s t o t h e o r e t i c a l e x p e c t a t i o n s .

We i n v e s t i g a t e t h e o n e - s t e p n a t u r e o f t h i s d i r e c t t r a n s f e r and e nc

w i t h a d i s c u s s i o n o f t h e m o s t r e s t r i c t i v e p r o c e s s , n a m e l y c l u s t e r t r a n s -

f e r o f He o r a t r i t o n .

3 . 1 E x c i t a t i o n F u n c t i o n s

T h e b e h a v i o r o f c r o s s s e c t i o n s as a f u n c t i o n o f i n c i d e n t e n e r g y

p r o v i d e s a f i r s t t e s t o f t h e r e a c t i o n m e c h a n i s m . F o r m a t i o n o f a com

p o u n d n u c l e u s , i n i s o l a t e d e n e r g y l e v e l s , w o u l d g e n e r a t e s h a r p r e s o

n a n c e s . Ev e n i n t h e c a s e o f s t r o n g l y o v e r l a p p i n g l e v e l s , n a r r o w f l u c

t u a t i o n s c a n be p r o d u c e d by r a n d o m p h a s e v a r i a t i o n s i n t h e c o n t r i b u t i n g

a m p l i t u d e s , a n d b r o a d s t r u c t u r e i s o f t e n o b s e r v e d i n e x c i t a t i o n f u n c

t i o n s . A s m o o t h e n e r g y d e p e n d e n c e i s e x p e c t e d o f a d i r e c t r e a c t i o n ,

a l t h o u g h i t i s n o t c o n c l u s i v e e v i d e n c e .

S e v e r a l m e a s u r e m e n t s o f e x c i t a t i o n f u n c t i o n s h a v e b e e n r e p o r t e d

f o r 1 i t h i u r n - i n d u c e d , t h r e e - n u c l e o n t r a n s f e r r e a c t i o n s . The c r o s s

s e c t i o n s o f t h e 5 / 2 + a n d 9 / 2 + s t a t e s o f ^ N e i n t h e ( 8 L i , t ) r e a c t i o n

( s e e F i g . 6 . 1 ) d e m o n s t r a t e a f l a t e n e r g y d e p e n d e n c e f o r E = 2 2 . 0 , 2 2 . 1 ,

. . . , 2 4 . 6 MeV a nd 0 , = 7 - 5 ° ( B i 7 3 b ) . I n a s t u d y o f 13 C ( 6 L i , t ) 16 0 * ( A l 1 . 0 9 )laba t E . = 2 0 , 2 1 , . . . , 3 2 MeV a nd 0 , = 1 5 ° ( B a 7 0 ) , o n l y t h e b a r e l y o b s e r v e d ,

L i l a b

g r o u n d s t a t e h a s a s t r u c t u r e d e x c i t a t i o n f u n c t i o n a nd t h e s t r o n g l y p o p u

l a t e d , 1 1 . 0 9 MeV d o u b l e t ( s e e F i g . 9 - 1 ) h as an e s p e c i a l l y s m o o t h o n e .

CHAPTER 3 REACTION MECHANISM

e n e r g i e s o f 32 MeV ( B a 7 0 ) , A6 MeV ( t h i s w o r k ) and 60 MeV ( B i 7 5 ) - E x p e r i -

7 12m e n t a l r e s u l t s a r e s i m i l a r f o r t h e ( L i , a ) r e a c t i o n on a C t a r g e t f r o m

an e n e r g y o f 28 MeV t o 38 MeV ( T s 7 3 ) • F o r f i n a l s t a t e s w i t h l a r g e c r o s s

s e c t i o n s , t h e r e f o r e , t h e s e d a t a s h ow no e f f e c t s o f a c o m p o u n d - n u c 1e u s

c o m p o n e n t i n t h e r e a c t i o n m e c h a n i s m .

3 . 2 A n g u l a r D i s t r i b u t i o n s

T h e r e l a t i v e c o n t r i b u t i o n s o f d i r e c t a nd c o m p o u n d - n u c l e u s p r o c e s s e s

c a n be e s t i m a t e d by c o m p a r i n g DWBA and H a u s e r - F e s h b a c h p r e d i c t i o n s w i t h

m e a s u r e d a n g u l a r d i s t r i b u t i o n s . We h a v e o b s e r v e d t h e ^ C ( ^ L i , t ) ^ 0 and

( ^ L i , ^ H e ) r e a c t i o n s f r o m 0 = 1 5 ° t o 8 0 ° a t E. . =AA MeV. E x p e r i -c . m . L i

m e n t a l a n g u l a r d i s t r i b u t i o n s a nd t h e o r e t i c a l c u r v e s a r e p r e s e n t e d i n

F i g s . 9 - 6 - 9 . 8 a n d a r e d i s c u s s e d i n S e c t i o n 9 . 3 . The e x a m p l e o f F i g . 3 - 1

i l l u s t r a t e s t h e g e n e r a l c o n c l u s i o n t h a t a s t r o n g f o r w a r d p e a k i n g i n t h e

d a t a c a n be r e p r o d u c e d by f i n i t e - r a n g e DWBA c a l c u l a t i o n s b u t n o t by t h e

H a u s e r - F e s h b a c h m o d e l . The m a g n i t u d e o f t h e DWBA c u r v e i n F i g . 3 . 1 i s

n o r m a l i z e d t o t h e m e a s u r e d c r o s s s e c t i o n a t 0 . , = 1 0 ° , w h e r e a s an u p p e rl ab

l i m i t i s p l a c e d on t h e o v e r a l l m a g n i t u d e o f t h e H a u s e r - F e s h b a c h p r e

d i c t i o n s b y t h e 1 0 . 3 5 3 MeV s t a t e o f ^ 0 ( s e e F i g . 9 - 7 ) . The r e s u l t i n g

d i f f e r e n c e b e t w e e n t h e a b s o l u t e c r o s s s e c t i o n s f r o m s t a t i s t i c a l t h e o r y

a nd f r o m e x p e r i m e n t i n d i c a t e s a n e g l i g i b l e r o l e f o r t h e c o m p o u n d - n u c 1e us

m e c h a n i s m a t s m a l l a n g l e s .

A n o t h e r d i f f e r e n c e l i e s i n t h e s y m m e t r y w i t h r e s p e c t t o 0 = 9 0 °c . m .e x p e c t e d o f an e n e r g y - a v e r a g e d a n g u l a r d i s t r i b u t i o n f r o m t h e d e c a y o f a

c o m p o u n d n u c l e u s . When t h e 1 1 . 0 9 MeV s t a t e o f ^ 0 i s o b s e r v e d o u t t o

©c m = 1 5 0 ° i n t h e ( ^ L i , t ) r e a c t i o n a t Ej_ j = 28 MeV ( B a 7 1 a ) , t h e a n g u l a r

The selectivity of the ^ C ( ^ L i , t ) ^ 0 reaction is consistent for incident

F i g u r e 3 . 1 An e x a m p l e o f ( ^ L i , t ) a n g u l a r d i s t r i b u t i o n s

D a t a f o r t h e 1 6 . 8 1 MeV, ( 3 + ) s t a t e o f ^ 0 a r e c o m p a r e d

w i t h t h e a n g u l a r d i s t r i b u t i o n s p r e d i c t e d by f i n i t e - r a n g e

DWBA a nd H a u s e r - F e s h b a c h t h e o r y .

(JS/qr/) UJ'°

^ c . m .

d i s t r i b u t i o n p r o v e s t o be a s y m m e t r i c , i n c o n t r a s t t o t h e H a u s e r - F e s h b a c h

c u r v e . O t h e r r e p o r t e d w o r k s u p p l i e s f u r t h e r i n d i c a t i o n o f a d i r e c t

r e a c t i o n . F o r a ^ 0 t a r g e t ( Ga7 2 ) , t h e ( ^ L i , ^ H e ) r e a c t i o n a t

El j = 2 A MeV p r o d u c e s a d i f f e r e n t i a l c r o s s s e c t i o n f o r t h e 9 / 2 + s t a t e a t

^ F ” ( 2 . 7 8 0 ) w h i c h f a l l s e v e n f a s t e r w i t h a n g l e t h a n t h e z e r o - r a n g e

DWBA p r e d i c t i o n . A t E ^ . = 6 0 MeV ( B i 7 5 ) » a n g u l a r d i s t r i b u t i o n s f o r t h e

t h r e e s t a t e s m o s t s t r o n g l y p o p u l a t e d by t h e ^ C ( ^ L i , t ) ^ 0 r e a c t i o n c a n

be f i t t e d b y f i n i t e - r a n g e DWBA c a l c u l a t i o n s . I n t h e ( ^ L i , a ) r e a c t i o n

( T s 7 3 , 7 A ) , s t r o n g f o r w a r d p e a k i n g a n d a s y m m e t r y a r e a g a i n o b s e r v e d .

A n g u l a r d i s t r i b u t i o n s , t h e r e f o r e , t o g e t h e r w i t h e x c i t a t i o n f u n c

t i o n s , p r o v i d e s t r o n g e v i d e n c e t h a t t h e d o m i n a n t m e c h a n i s m o f t h e

( ^ L i , t ) a nd ( ^ L i , ^ H e ) r e a c t i o n s i s d i r e c t a t f o r w a r d a n g l e s . S i n c e o u r

s p e c t r a a r e m e a s u r e d a t E . ^ A 0 MeV and 0 , . £ 1 5 ° , i . e . a t i n c i d e n tL i l a b

e n e r g i e s h i g h e r t h a n i n a l l o f t h e d a t a r e f e r e n c e d a b o v e ( e x c e p t B i 7 5 ) ,

e v e n l e s s c o m p o u n d - n u c 1e u s f o r m a t i o n i s e x p e c t e d . T h e f r a c t i o n o f t h e

e x i t - c h a n n e l f l u x f r o m a c o mp o u n d n u c l e u s , m o r e o v e r , i s s m a l l e r f o r t h e

( ^ L i , t ) a n d ( ^ L i , ^ H e ) r e a c t i o n s t h a n f o r ( ^ L i , a ) . I n o u r H a u s e r - F e s h b a c h

c a l c u l a t i o n o f t h e d e c a y o f ^ F , AA% o f t h e f l u x g o e s i n t o t h e ^ N + a

c h a n n e l b u t o n l y b% i n t o ^ 0 + t .

3 . 3 S e l e c t i v i t y

A o n e - s t e p , d i r e c t r e a c t i o n t r a n s f e r r i n g t h r e e n u c l e o n s as a g r o u p

w o u l d l e a d t o s t r o n g p o p u l a t i o n o f 3 p _ nh c o n f i g u r a t i o n s i n t h e A = 15 t o

A = 19 n u c l e i . Known f i n a l s t a t e s i n w h i c h 3 p - n h s t r u c t u r e i s p r o b a b l e

c a n t e s t t h i s a s p e c t o f t h e ( ^ L i , t ) a nd ( ^ L i , ^ H e ) r e a c t i o n s . I n ^ F

( F i g . 6 . 2 , B i 7 1 , W e 7 2 ) , a l l h i g h - s p i n membe r s o f t h e p o s i t i v e - p a r i t y ,

g r o u n d - s t a t e b a n d a r e s t r o n g l y p o p u l a t e d . I n c o n t r a s t , t h e 7 / 2 + s t a t e

a t £ . 3 7 7 MeV a n d t h e l l / 2 + s t a t e s a t 6 . 5 0 0 , 7 - 9 3 7 a n d 9 - 2 6 7 MeV ( A j 7 8 ,

S y 7 7 ) a r e o b s e r v e d w e a k l y i f a t a l l , d e s p i t e t h e a n g u 1a r - m o m e n t u m m i s

m a t c h A L = 6 o f t h e ^ 0 ( ^ L i , 3 He) ^ F r e a c t i o n . T h e s e l e c t i v i t y , t h e r e f o r e ,

i s n o t m e r e l y o f a h i g h - s p i n n a t u r e c h a r a c t e r i s t i c o f c o m p o u n d - n u c 1e us

r e a c t i o n s . S i m i l a r l y , i n t h e ^ 7 0 ( ^ L i , t ) 2 ^ N e s p e c t r u m ( F i g . B . l ) , t h e

7T — ■» — 7T ■»J =5 , 6 a nd 7 membe r s o f t h e K =2 b a n d a r e c l e a r l y w e a k o r a b s e n t

a t E = 8 . £ £ 9 , 1 0 . 6 0 9 a nd 1 3 • 3 3£ MeV r e s p e c t i v e l y ( A j 7 8 ) . When o n e h o l e

i n t h e p s h e l l i s p r o v i d e d by t h e t a r g e t n u c l e u s , n a m e l y ^ N , l o w - l y i n g

18 18n e g a t i v e - p a r i t y s t a t e s o f F and 0 a r e p r e f e r e n t i a l l y p o p u l a t e d

( F i g s . 7 . 1 , 7 . 2 , L i 7 2 ) . T h e (6 L i , 3 He) r e a c t i o n a l s o p r o d u c e s l a r g e

p e a k s c o r r e s p o n d i n g t o ^ 7 CT ( 8 . £ 7 £ , 7 / 2 + ) a nd ( 1 0 . 6 9 3 , 9 / 2 + ) ( F i g s . 8 . 2 ,

5 . 2 ) . T h e s e t w o s t a t e s h a v e t h e p o s i t i v e p a r i t y and h i g h e x c i t a t i o n

e n e r g y e x p e c t e d o f 3 p ~ 2 h a nd 3 p _ £h c o n f i g u r a t i o n s , as w e l l as s p i n

a s s i g n m e n t s w h i c h a r e t o o h i g h f o r ( s d ) a nd c o n f i g u r a t i o n s

r e s p e c t i v e l y . O v e r a l l , kn own c a n d i d a t e s f o r 3 p - n h s t a t e s p r o v e t o be

6 6 3p r o m i n e n t i n ( L i , t ) a nd ( L i , He) s p e c t r a , a f e a t u r e c o n s i s t e n t w i t h

a o n e - s t e p p r o c e s s o f d i r e c t t r a n s f e r .

I n t h r e e - n u c l e o n t r a n s f e r d a t a f r o m a t a r g e t w i t h n h o l e s , an

a b s e n c e o f £ p - ( n + l ) h s t a t e s w o u l d be e v i d e n c e c o n t r a r y t o a t w o - s t e p

p r o c e s s i n v o l v i n g b o t h s i n g l e - n u c l e o n p i c k - u p and a l p h a - p a r t i c l e t r a n s

f e r . A ( ^ L i , d ) ( d , t ) m e c h a n i s m h as b e e n s u g g e s t e d as a p o s s i b l e e x p l a n a

t i o n f o r t h e p o p u l a t i o n o f a p r o p o s e d f ^ ^ n e u t r o n - h o l e s t a t e i n t h e

^ Fe ( ^ L i , t ) i r e a c t i o n (W0 7 8 ) . I n a b s o l u t e c r o s s s e c t i o n , h o w e v e r ,

t h e ( ^ L i , d ) r e a c t i o n on p - s h e l l t a r g e t s i s c o m p a r a b l e t o t h e ( ^ L i , t )

r e a c t i o n ( S e c t i o n 3 - £ ) and w e a k e r t h a n t h e ( ^ L i , t ) r e a c t i o n ( C o 7 6 ) . We

t h e r e f o r e c o n s i d e r a ( ^ L i , ^ L i ) ( ^ L i , t ) p r o c e s s , w h i c h w o u l d a l s o p o p u l a t e

f i n a l s t a t e s w i t h A p - ( n + l ) h c o n f i g u r a t i o n s and a l p h a - p a r t i c l e c l u s t e r

i n g . T h e l a c k o f a c o n s i s t e n t c o r r e l a t i o n i n s e l e c t i v i t y b e t w e e n t h e

r e a c t i o n a nd t h e ( ^ L i , t ) o r ( ^ L i , ^ H e ) r e a c t i o n i s d o c u m e n t e d

19b y t h e f o l l o w i n g c a s e s . T h e A p - l h , n e g a t i v e - p a r i t y b a n d o f F ( F i g .

6 . 3 , M i 7 0 ) h a s a 1 3 / 2 member a t 8 . 2 8 8 MeV w h i c h i s q u i t e w e a k l y p o p u

l a t e d i n t h r e e - n u c l e o n t r a n s f e r ( F i g s . 6 . 2 , 6 . A ) . The b+ s t a t e a t

I 8 *** 7TF ( 5 . 2 9 8 ) , a member o f t h e A p - 2 h , K7r= l + b an d ( R o 7 3 b , Co77) , i s a l m o s t

1 5 6 1 8 6 3 a b s e n t i n N( L i , t ) F d a t a ( F i g . 7 . 1 ) . The ( L i , He) r e a c t i o n i g n o r e s

t h e s t a t e s a t ( 1 8 . 1 5 , 1 9 . 2 4 ) w h i c h a r e s t r o n g l y p o p u l a t e d i n a l p h a -

p a r t i c l e t r a n s f e r ( F i g . 8 . A) a nd a p p e a r t o h a v e 4 p - 3 h c o n f i g u r a t i o n s .

L a s t l y , i n t h e ^ C ( ^ L i , t ) ^ 0 r e a c t i o n ( F i g . 9 - 1 ) , t h e r e i s o n l y m i n o r

p o p u l a t i o n o f t h e b s t a t e a t 1 0 . 3 5 3 MeV a n d t h e 6+ s t a t e a t 1 6 . 2 9 MeV,

b o t h b e l o n g i n g t o t h e A p - A h , K7T= 0 + b a n d o f ^ 0 ( e . g . Co7 6 ) . T h e s e

c o u n t e r e x a m p l e s f o r e a c h t a r g e t n u c l e u s a r g u e a g a i n s t t h e s y s t e m a t i c

e f f e c t e x p e c t e d o f s u c h a t w o - s t e p m e c h a n i s m . S p e c i a l c a s e s a p p e a r ,

19 *h o w e v e r , i n w h i c h a s t a t e s u c h as F ( 8 . 9 5 3 , 1 1 / 2 ) i s s t r o n g l y p o p u -

7 6 6 3l a t e d b o t h i n ( L i , t ) d a t a a nd i n t h e ( L i , t ) o r ( L i , He) r e a c t i o n .

G i v e n t h e a b o v e r e s u l t , we a t t r i b u t e s u c h e x c e p t i o n s t o m i x e d s t r u c t u r e

i n t h e f i n a l s t a t e , s i n c e i m p u r i t y i n t h e g r o u n d - s t a t e c o n f i g u r a t i o n

o f t h e t a r g e t i s g e n e r a l l y s m a l l .

T h e c o n s i d e r a t i o n o f a s e q u e n t i a l t r a n s f e r o f t h r e e i n d i v i d u a l

n u c l e o n s i n v o l v e s 2 p - ( n - l ) h a nd l p - ( n - 2 ) h s t a t e s , w h e r e n i s t h e n u m b e r

o f h o l e s i n t h e t a r g e t n u c l e u s . A l t h o u g h t h e y a r e a c c e s s i b l e t o a o n e -

s t e p , d i r e c t r e a c t i o n , w e a k p o p u l a t i o n o f t h e s e c o n f i g u r a t i o n s c o u l d

p l a c e an u p p e r l i m i t on t h e r o l e o f a s e q u e n t i a l p r o c e s s . I n a ( ^ L i , t )

s p e c t r u m f o r ^ 0 ( £ 1 6 . 2 ) , some e v i d e n c e h a s b e e n p o i n t e d o u t ( B a 7 1 a )

f o r a m e c h a n i s m o f s i n g l e - n u c l e o n t r a n s f e r i n t o t h e p s h e l l f o l l o w e d

b y t r a n s f e r o f a p r o t o n - n e u t r o n p a i r e q u i v a l e n t t o t h e ( a , d ) r e a c t i o n .

A c o n t r a s t i n s e l e c t i v i t y b e t w e e n t h e s e t w o r e a c t i o n s , h o w e v e r , i s

18d e m o n s t r a t e d by o t h e r n u c l e i . T h e 6 . 8 1 MeV s t a t e o f F i s p r o m i n e n t

i n ( a , d ) d a t a ( Ma 6 8 , R i 6 6 ) b u t n o t i n ( ^ L i , t ) d a t a ( F i g . 7 * 1 ) . S i m i l a r

c o u n t e r e x a m p l e s a t ^ 0 ( 9 - 1 * 0 a nd ( 1 1 . 9 5 ) ( F i g s . 8 . 2 , 5 - 2 and L u 6 9 )

s h o w t h a t t h i s m e c h a n i s m p l a y s a t m o s t a m i n o r r o l e . T h e c a s e s o f

c o r r e s p o n d i n g s e l e c t i v i t y i n t h e ( ^ L i , t ) a nd ( a , d ) r e a c t i o n s , e . g .

2 p - ( n - l ) h s t a t e s a t 1 5 n ' ( 1 3 - 0 0 , 1 1 / 2 ~ ) a nd n '" ' ( 5 . 7 3 , 5+ ) ( F i g s . 5 . 2 , 9 . 2

a n d L u 6 9 ) , c a n be a d e q u a t e l y a c c o u n t e d f o r by a s i m p l e o n e - s t e p p r o c e s s

6 24 28( s e e S e c t i o n 5 - 3 ) . I n ( L i , t ) s p e c t r a f r o m Mg a nd Si t a r g e t s , e v i

d e n c e e x i s t s o f a s e q u e n t i a l m e c h a n i s m p o p u l a t i n g l p - ( n - 2 ) h s t a t e s ( W o 7 8 ) .

31T h e r e l a t i v e l y l a r g e c r o s s s e c t i o n o f t h e 4 . 4 5 MeV s t a t e o f S i s t e n

t a t i v e l y i n t e r p r e t e d i n t e r m s o f t r a n s f e r o f a p r o t o n p a i r i n t o t h e l o w

e s t a v a i l a b l e l e v e l p l u s n e u t r o n t r a n s f e r i n t o t h e f y / 2 s h e l l . T h e o n l y

t a r g e t s we h a v e s t u d i e d w h i c h c o n t a i n t w o p r o t o n h o l e s a r e and ^ C .

I n t h e ^ C ( ^ L i , t ) ^ 0 r e a c t i o n ( F i g . 5 - 1 ) , an i n h i b i t e d p o p u l a t i o n o f t h e

l p - 2 h c o n f i g u r a t i o n ( L i 70 ) a t ^ 0 ( 7 - 2 7 6 , 7 / 2 + ) a r g u e s a g a i n s t t h i s

m e c h a n i s m . I n t h e ^ C ( ^ L i , t ) ^ 0 r e a c t i o n ( F i g . 9 - 1 ) , a m o r e p r o m i n e n t

p e a k f o r t h e p r i m a r i l y l p - l h s t a t e a t ^ 0 ( 6 . 1 3 0 , 3 ) may t h e r e f o r e

a r i s e f r o m an a d d i t i o n a l 3 p " 3 h c o m p o n e n t ( s e e D e 7 1 ) . I n g e n e r a l , f e a

t u r e s o f t h e ( ^ L i , t ) r e a c t i o n m e c h a n i s m s u g g e s t e d i n t h e c a s e o f h e a v y

n u c l e i (Wo78) d o n o t a p p e a r t o a p p l y t o t h e s e l i g h t n u c l e i .

6 6 3 I n s u m m a r y , t h e s e l e c t i v i t y o f t h e ( L i , t ) a nd ( L i , He) r e a c t i o n s

c o n t a i n s much s p e c i f i c , e x p e r i m e n t a l e v i d e n c e i n s u p p o r t o f a o n e - s t e p

p r o c e s s o f d i r e c t t r a n s f e r . F i n a l s t a t e s w h i c h a r e e x p e c t e d a p r i o r ?

t o h a v e 3 p _ nh c o n f i g u r a t i o n s a r e i n d e e d s t r o n g l y p o p u l a t e d by t h e s e

t h r e e - n u c l e o n t r a n s f e r r e a c t i o n s , w h e r e a s s e v e r a l o t h e r k n own h i g h -

s p i n s t a t e s a r e p o p u l a t e d w e a k l y i f a t a l l . I n t h e c a s e o f A p - ( n + l ) h

s t a t e s as w e l l as t w o - p a r t i c l e o r o n e - p a r t i c l e s t r u c t u r e , t h e r e e x i s t

c o u n t e r e x a m p l e s t o a t w o - s t e p m e c h a n i s m i n v o l v i n g a l p h a - p a r t i c l e ,

d e u t e r o n o r n e u t r o n t r a n s f e r r e s p e c t i v e l y .

3>.h C l u s t e r i n g

T h e s i m p l e s t a nd m o s t r e s t r i c t i v e r e a c t i o n m e c h a n i s m f o r t h e

6 6 3( L i , t ) o r ( L i , He) r e a c t i o n w o u l d be t h e d i r e c t t r a n s f e r o f a t r i t o n

3o r He c l u s t e r , n a m e l y a g r o u p o f t h r e e n u c l e o n s c o u p l e d t o T = S = l / 2

a n d b o u n d i n t h e i r g r o u n d s t a t e . I n t h e f o u r - n u c l e o n c a s e , t h e ( 2 L i , t )

r e a c t i o n b e h a v e s p r i m a r i l y as t h e c l u s t e r t r a n s f e r o f an a l p h a p a r t i c l e .

T h e a n a l o g y t o t h e t h r e e - n u c l e o n s y s t e m i s n o t i m m e d i a t e b e c a u s e t h e

3t r i t o n o r He h a s a b i n d i n g e n e r g y o f 8 MeV, s m a l l c o m p a r e d w i t h t h e

a l p h a - p a r t i c l e v a l u e o f 28 MeV t h o u g h l a r g e r t h a n t h e d e u t e r o n b i n d i n g

o f 2 MeV. T h e r o l e o f c l u s t e r t r a n s f e r i s i n f l u e n c e d by t h e e x t e n t

o f t h r e e - n u c l e o n c l u s t e r i n g i n t h e i n i t i a l s t a t e o f ^ L i and i s r e

f l e c t e d b y t h a t i n t h e f i n a l s t a t e s o f t h e r e s i d u a l n u c l e u s .

T h e c l u s t e r s t r u c t u r e o f ^ L i has b e e n i n v e s t i g a t e d i n a w i d e

3 6v a r i e t y o f e x p e r i m e n t s ( s e e H a 7 7 ) • T h e t ( H e . y ^ ) L i r e a c t i o n y i e l d s

a s p e c t r o s c o p i c f a c t o r S ( t ) = 0 . 7 ' n o n e a n a l y s i s ( Y o 7 0 ) . T h e k n o c k - o u t

6 3 6r e a c t i o n s L i ( p , p H e ) t and L i ( p , p a ) d g i v e S ( t ) = 0 . 3 3 a nd S ( a ) = 0 . A 5

f o r c l u s t e r w a v e f u n c t i o n s , o r S ( t ) = 0 . 7 8 a n d S ( a ) = 0 . 5 8 f o r W o o d s - S a x o n

£w a v e f u n c t i o n s ( R o 7 6 ) . A l t h o u g h t h e p a r e n t a g e o f L i i s n o t u n i q u e ,

3s u c h r e s u l t s i n d i c a t e t h a t H e + t s t r u c t u r e i s m a j o r a nd c o m p a r a b l e t o

a + d s t r u c t u r e . N o n o r t h o g o n a l i t y o f t h e c l u s t e r w a v e f u n c t i o n s ( C 1 7 ^ )

may a c c o u n t f o r t h e d u a l i t y o f s p e c t r o s c o p i c s t r e n g t h . F u r t h e r e v i d e n c e

c a n be f o u n d i n t h e c o m p a r a b l e c r o s s s e c t i o n s o f t h e (8 L i , d ) and

( ^ L i , ^ H e ) r e a c t i o n s . F o r ^ C ( 8 L i , d ) ^ 0 ( 1 0 . 3 5 3 , 4 " * " ) a t . —40 MeV and

0 l a b = 1O° a n d f o r 1 6 ° ( 6 l ' >d ) 2 ° Ne* ( 1 0 . 2 6 1 , 5 ’ ) a t Eu = 4 6 MeV a nd © | a b = 1 5 ° ,

t h e d i f f e r e n t i a l c r o s s s e c t i o n i s ^ 6 0 0 y b / s r . M a g n i t u d e s o f ^ 9 0 0 y b / s r

a nd ^ 3 0 0 y b / s r a r e o b t a i n e d f o r ^ C ( 8 L i , ^ H e ) ^ N ( 1 0 . 6 9 3 , 9 / ) and

^ 0 (^ L i , ^ H e ) ^ F ( 8 . 9 5 3 , 1 1 / 2 ) r e s p e c t i v e l y u n d e r t h e same e x p e r i m e n t a l

c o n d i t i o n s ( T a b l e s 5 . 1 , 6 . 1 ) . L a s t l y , b e c a u s e t r i t o n c l u s t e r i n g i n 7 L i

i s m o r e d o m i n a n t t h a n i n 8 L i , t h e s e l e c t i v i t y o f t h e (7 L i , a ) r e a c t i o n

s h o u l d be c o m p a r e d t o t h a t o f (8 L i , ^ H e ) . A s u b s t a n t i a l o v e r l a p d o e s

e x i s t ( F i g s . 5 . 4 , 7 . 4 , 9 . 4 ) d e s p i t e t h e l a r g e d i f f e r e n c e b e t w e e n t h e s e

r e a c t i o n s i n a n g u 1a r - m o m e n t u m m i s m a t c h , e . g . A L - 3 and A L - 6 r e s p e c t i v e l y .

6 3 6I n v i e w o f t h e s e i n d i c a t i o n s o f L i = H e + t p a r e n t a g e , t h e ( L i , t ) a nd

6 3( L i , He) r e a c t i o n s c o u l d o f t e n p r o c e e d v i a c l u s t e r t r a n s f e r .

T h e r e m a i n i n g q u e s t i o n o f f i n a l - s t a t e p a r e n t a g e i s s t u d i e d h e r e by

an a p p l i c a t i o n o f n u c l e a r m o d e l s t o t h e r e s i d u a l n u c l e i . I n ^ F , ^ 8 0

1 r 6 3a n d N, s t a t e s s t r o n g l y p o p u l a t e d b y t h e ( L i , He) r e a c t i o n h a v e a

s i g n i f i c a n t c o r r e s p o n d e n c e w i t h t r i t o n - c 1 us t e r s t a t e s p r e d i c t e d by t h e

f o l d e d - p o t e n t i a l m o d e l a n d a m o r e p r e c i s e c o r r e l a t i o n w i t h c o n c e n t r a t i o n s

o f t r i t o n - c l u s t e r s p e c t r o s c o p i c s t r e n g t h p r e d i c t e d b y t h e S U ( 3 ) s h e l l

18 * — 6 3 m o d e l . F o r e x a m p l e , t h e p r o m i n e n c e o f 0 ( 8 . 1 0 , 5 ) i n t h e ( L i , He)

s p e c t r u m ( F i g . 7 . 2 ) i s i n a g r e e m e n t w i t h t h e s u b s t a n t i a l c l u s t e r i n g

e x p e c t e d i n t h i s s t a t e ( F i g . 7 . 6 ) . I n c o n t r a s t , no l a r g e p e a k a p p e a r s

f o r t h e 5^ s h e l l - m o d e l l e v e l p r e d i c t e d a t 9 . 0 MeV, w h i c h has a d o m i n a n t

S U ( 3 ) c o m p o n e n t ( X p ) = ( 0 4 ) —( 0 1 ) * ( 0 3 ) i n v o l v i n g t h r e e t o t a l l y a n t i s y m

m e t r i c n u c l e o n s . T h e o r e t i c a l e v i d e n c e on t h e s t r u c t u r e o f f i n a l s t a t e s

t h e r e f o r e s u g g e s t s t h a t c l u s t e r t r a n s f e r p l a y s a c o n s i d e r a b l e p a r t i n

t h e m e c h a n i s m o f t h e ( ^ L i , t ) a nd ( ^ L i , ^ H e ) r e a c t i o n s . T h e f o r m a l i s m o f

t h e t w o m o d e l s i s i n t r o d u c e d i n t h e n e x t c h a p t e r .

CHAPTER A THEORY

A . l F o l d e d - P o t e n t i a l C l u s t e r Mo d e l

C l u s t e r i n g p h e n o me n a i n t h e s t r u c t u r e o f l i g h t n u c l e i c a n be

c a l c u l a t e d t o f i r s t o r d e r f r o m a m a c r o s c o p i c mo d e l h a v i n g m i c r o s c o p i c

o r i g i n s . A t r i t o n c l u s t e r i s a s s u m e d t o e x i s t i n a p o t e n t i a l d e r i v e d

f r o m n u c l e a r d e n s i t i e s . T h e s i m p l i f i e d a p p r o a c h o f t h i s t h e o r y l e a d s

t o i n t u i t i v e p h y s i c a l c o n t e n t a nd c o n v e n i e n t n u m e r i c a l c a l c u l a t i o n s .

T h e l i m i t e d s c o p e o f a t r i t o n - c l u s t e r mo d e l i m p l i e s t h a t p r e d i c t e d

e n e r g y l e v e l s a r e t o be c o m p a r e d w i t h a s p e c i a l c l a s s o f n u c l e a r s t a t e s ,

6 3e . g . t h o s e s e l e c t e d b y t h e ( L i , He) r e a c t i o n .

T h e i n t e r a c t i o n b e t w e e n t h e c l u s t e r a nd t h e c o r e n u c l e u s i s r e p r e

s e n t e d by t h e p o t e n t i a l s o f F i g . A . 2 u s i n g t h e c o o r d i n a t e s o f F i g . A . I .

T h e f o l d e d p o t e n t i a l V ^ ( r ) ' s a c o n v o l u t i o n o f t h e d e n s i t i e s o f t h e

c l u s t e r a n d c o r e w i t h an e f f e c t i v e , n u c 1e o n - n u c 1 e o n a m p l i t u d e (Do 7A,

Va 7 A , B u 7 5 ) . T h e d e f i n i t i o n o f r e l a t e s t h e a d j u s t a b l e s t r e n g t h f o f

t h e f i n i t e - r a n g e i n t e r a c t i o n t o t h e n u c l e o n - n u c l e o n , f o r w a r d s c a t t e r i n g

a m p l i t u d e . A p p l i c a t i o n s t o h e a v y - i o n s c a t t e r i n g h a v e t e s t e d t h e f o r m o f

V ^ ( r ) , w i t h f a v o r a b l e r e s u l t s f o r e l a s t i c , i n e l a s t i c a nd o n e - n u c l e o n -

t r a n s f e r p r o c e s s e s ( V a 7 3 c , D o 7 5 , M o 7 7 ) . T h e c l u s t e r d e n s i t y i s d e t e r

m i n e d by e l e c t r o n s c a t t e r i n g ( C o 6 7 ) and n o r m a l i z e d t o A = 3 :

P 1 ( r ) = p 0 e x p [ j ( ^ - ) 2 ]

( A . l )

_ , /_L,o3/2p 0 ^ v i r a ’

■3w h e r e a = 1 . 6 A f o r a t r i t o n a nd a = 1 . 7 7 f o r ^ H e . I n t h e m o s t i n f l u e n t i a l

f a c t o r o f t h e f o l d e d - p o t e n t i a l i n t e g r a n d , t h e c o r e d e n s i t y , we u s e a

t h e o r e t i c a l mass d e n s i t y r a t h e r t h a n an e x p e r i m e n t a l c h a r g e d e n s i t y ,

F i g u r e 4 .

1 C o o r d i n a t e s o f t h e f o l d e d p o t e n t i a l

2 P o t e n t i a l s o f t h e c l u s t e r - c o r e i n t e r a c t i o n

1 = c l u s t e r

2 = c o r e

p = d e n s i t y

M = n u c l e o n mass

y = 1 fm

Z = a t o m i c n u m b e r

A = a t o m i c w e i g h t

( t i / m ^ c ) 2 * 2 . 0 f m 2

"£ = r e l a t i v e o r b i t a l a n g u l a r moment um

1 • • • •Ho = i n t r i n s i c s p i n

—►j = t o t a l a n g u l a r momen t um o f t h e c l u s t e r

f , v SOi » Vs o 2 = s t r e n 9 t h P a r a m e t e r s

CLUSTER COORDINATES

CLUSTER POTENTIALS

FOLDED Vs(r) = /d r , / dr2 p\(r{) vs (ir +T|-"r20 pz (r2 )

STRONG V f27rfc __________

M (7r / 2)3/2exp

COULOMB Z , * 2 2Vrs = — 2— — e Z

SPIN-ORBIT v (r) )2 v , -±-soi Vm^c/ soi j

r+ r,-r2

L r dr L • CT,

HYPERFINE v ^ r ) ; - ^ ) 2 vso2 I Lx dr i

b e c a u s e H a r t r e e - F o c k c a l c u l a t i o n s a r e mo r e a c c u r a t e i n t h e i m p o r t a n t

t a i l r e g i o n a n d a r e c o n s i s t e n t w i t h e l e c t r o n s c a t t e r i n g ( N e 7 0 ) .

I n an a l t e r n a t e p r e s c r i p t i o n f r o m t h e s h e l l m o d e l ,

P2(r) = I Wj U; I2, (+2)i

w h e r e Wj i s an o c c u p a t i o n w e i g h t a nd <J>j i s a s i n g l e - p a r t i c l e w a v e f u n c -

t i o n r e s u l t i n g f r o m w e l 1 p a r a m e t e r s f i t t o e x p e r i m e n t a l b i n d i n g e n e r g i e s

(M ? 7 3 ) - T h e o r e t i c a l d e n s i t i e s f r o m t h e s e t w o s o u r c e s g e n e r a t e s i m i l a r

p r e d i c t i o n s w h i c h , f o r e x a m p l e , d i f f e r by o n l y 0 . 0 5 MeV f o r t h e s p a c i n g

19b e t w e e n t h e L=2 a nd L= 4 l e v e l s o f Ne ( s e e F i g . 6 . 6 ) .

F o l l o w i n g t h e f o l d e d p o t e n t i a l o f t h e s t r o n g i n t e r a c t i o n i n F i g .

A . 2 a r e s t a n d a r d e x p r e s s i o n s f o r t h e C o u l o m b i n t e g r a l ^ ( r ) a n d f o r t h e

Thomas t e r m Vjjqj ( r ) , w h i c h t r e a t s t h e s p i n - o r b i t c o u p l i n g o f a t r i t o n

c l u s t e r . I f t h e c o r e h a s n o n - z e r o s p i n , we i n t r o d u c e a p o t e n t i a l

^ S 0 2 ^ P r o p o r t i o n a l t o t h e t o t a l a n g u l a r momen t um o f t h e c l u s t e r c o u p l e d

t o t h e s p i n o f t h e c o r e . As i n t h e s p i n - o r b i t c a s e , an a n a l o g y e x i s t s

t o t h e e l e c t r o m a g n e t i c c o u p l i n g o f an e l e c t r o n , s i n c e t h e h y p e r f i n e i n t e r

a c t i o n w i t h a n u c l e a r d i p o l e momen t i s a l s o p r o p o r t i o n a l t o

t h e s p i n - o r b i t i n t e r a c t i o n o f t h e c o r e a n d t h e s p i n - s p i n i n t e r a c t i o n a r e

, + + + + i + + i . . . . . . i •c o n t a i n e d i n j • o 2=1-' o 2+%° ] ' ° 2 » b u t t w o a s s u m p t i o n s a r e i m p l i c i t i n t h i s

p r e s c r i p t i o n . T h e e q u a l w e i g h t a s s u m e d f o r L * S 2 a nd t e r n s i s

n e c e s s a r y f o r t h e c o m m u t a t i v i t y o f ( r ) and VgQ2 ( r ) . "*"^e w e a ^ c o u p -

l i n g i m p l i e d by J = j +$2 i s v a l i d o n l y w h e n t h e s p i n - s p i n i n t e r a c t i o n i s

s m a l l c o m p a r e d w i t h t h e t r i t o n s p i n - o r b i t i n t e r a c t i o n . S t r o n g c o u p l i n g ,

i . e . S = S | + S 2 a nd J = L + S , w o u l d y i e l d c o m m u t i n g o p e r a t o r s L ' S a nd a j * a 2 »

b u t a s p i n - o r b i t p o t e n t i a l p r o p o r t i o n a l t o + t *?2 w o u l d r e q u i r e

e q u a l s t r e n g t h f o r t h e c l u s t e r a nd c o r e t e r m s , i n c o n t r a d i c t i o n t o

W i t h V ( r ) = V ^ ( p ) + V ^ ( r ) + V $ o i ^ + ^ S 0 2 ^ ’ s o ^ u t ' o n s a r e o b t a i n e d t o

t h e s i n g l e - p a r t i c l e S c h r b d i n g e r e q u a t i o n . W h i l e c o n v e r g e n c e u p o n b o u n d

s t a t e s p r o c e e d s v i a t h e m a t c h i n g c o n d i t i o n , a s e a r c h o c c u r s f o r u n b o u n d

s t a t e s i n t h e c a l c u l a t e d e l a s t i c s c a t t e r i n g o f a c l u s t e r p r o j e c t i l e on a

c o r e t a r g e t ( A u 7 6 ) . A r e s o n a n c e i n t h e c r o s s s e c t i o n i s i d e n t i f i e d

w h e r e I mS^ b e c o me s n e g a t i v e ( F i g . 4 . 3 ) , s i n c e t h e s c a t t e r i n g m a t r i x

i s g i v e n by S ^ = e x p (2 i <5^) a nd t h e a m p l i t u d e by f ^ 0" ( S ^ - l ) . B e c a u s e o f a

B r e i t - W i g n e r s h a p e ( B u 7 5 ) , t h e r e s o n a n c e h as a w i d t h r = Ec m ^ m8|_= ” ^ ”

E ( I mS = 1 ) . T h e c a l c u l a t e d e n e r g y l e v e l s a r e c l a s s i f i e d a c c o r d i n g t o c . m. L o

t h e c o n f i g u r a t i o n o f t h e t r i t o n c l u s t e r . A ( s d ) c o n f i g u r a t i o n c o r r e

s p o n d s t o a 2N+ L= 6 b a n d , w h e r e

32N + L = I 2 n j + l j . ( I t . 3)

T h i s p a r t i c u l a r r e s t r i c t i o n o n t h e o r b i t a l s o p e n t o t h e t h r e e n u c l e o n s

o f t h e c l u s t e r t r i v i a l l y s a t i s f i e s t h e P a u l i p r i n c i p l e f o r a n y p - s h e l l

c o r e . E x c h a n g e e f f e c t s , m o r e o v e r , s h o u l d be s m a l l o w i n g t o t h e l a r g e

rms r a d i u s o f t h e r e l a t i v e w a v e f u n c t i o n s ( F i g . 4 . 4 ) . W h i l e t h e s p a t i a l

l o c a l i z a t i o n o f t h e c l u s t e r i n c r e a s e s w i t h t h e o r b i t a l a n g u l a r momen

t u m L , t h e rms s e p a r a t i o n f r o m t h e c o r e r e m a i n s ^ 3 f m , i l l u s t r a t i n g t h a t

c e n t r i f u g a l s t r e t c h i n g i s b a l a n c e d by t h e d e c r e a s i n g n u m b e r N o f r a d i a l

n o d e s . A l t h o u g h t h e f o l d e d p o t e n t i a l i s t o o d e e p a t s m a l l r a d i i , i t i s

e x p e c t e d t o be m o r e v a l i d t h a n a W o o d s - S a x o n w e l l i n t h e s e n s i t i v e t a i l

r e g i o n , w h e r e m o s t o f t h e c l u s t e r p r o b a b i l i t y d e n s i t y i s l o c a t e d .

T h e f r e e p a r a m e t e r s o f t h i s t r i t o n - c l u s t e r m o d e l , w h i c h p a r t i a l l y

a b s o r b m a n y - b o d y e f f e c t s , a r e t h e s t r e n g t h s f , a nd ^ S 0 2 * * n c o n ”

2919experimental levels of F (see Bu77a ).

F i g u r e 4 . 3 S i n g l e - p a r t i c l e r e s o n a n c e s

T h e s e p r e d i c t i o n s r e p r e s e n t u n b o u n d , t r i t o n - c l u s t e r s t a t e s

o f , 9 F.

F i g u r e 4 . 4 F o l d e d p o t e n t i a l w i t h p r o b a b i l i t y d e n s i t i e s

T h e a r r o w s i n d i c a t e t h e rms s e p a r a t i o n o f a ( s d ) 3 t r i t o n

c l u s t e r f r o m a c o r e .

sin 28

t r a s t t o a W o o d s - S a x o n p o t e n t i a l , t h e g e o m e t r y o f a f o l d e d p o t e n t i a l i s

p r e d e t e r m i n e d by t h e c o r e d e n s i t y a n d , t o a l e s s e r e x t e n t , by t h e c l u s t e r

d e n s i t y a nd n u c 1e o n - n u c 1 e o n i n t e r a c t i o n . I n a d d i t i o n , a s i n g l e v a l u e o f

f g e n e r a t e s a n e n t i r e c l u s t e r b a n d , w h e r e a s a W o o d s - S a x o n d e p t h p a r a m e t e r

m u s t be r e a d j u s t e d t o e a c h e x p e r i m e n t a l l e v e l ( s e e B u 7 5 ) . A ^ 10 % r e n o r

m a l i z a t i o n o f f , a r i s i n g i n p a r t f r o m a s e n s i t i v i t y t o t h e c h o i c e o f

c o r e d e n s i t y , i s n e e d e d f o r 2N+ L=6 b a n d s i n d i f f e r e n t n u c l e i . T h e em

p i r i c a l r e s u l t f v l . 6 f m i s c o n s i s t e n t w i t h t h e t h e o r e t i c a l e s t i m a t e i n

a F e r m i - g a s a p p r o a c h ( V a 7 A ) . A s i n g l e v a l u e o f i s u s e d f o r a l l

2 N+ L= 6 c a l c u l a t i o n s , a nd e x c e p t f o r a c o r e o f n o n - z e r o s p i n ( s e e

S e c t i o n s 6 . 3 , 7 - 3 ) . B e c a u s e t h e p a r a m e t e r s a r e f i t t o e x p e r i m e n t a l

s t a t e s , w h i c h i n e v i t a b l y c o n t a i n some s t r u c t u r a l i m p u r i t y , t h e m o d e l

p r e d i c t s e n e r g y l e v e l s r e p r e s e n t i n g n o t r i g o r o u s c e n t r o i d s , b u t r a t h e r ,

l a r g e c o n c e n t r a t i o n s o f t r i t o n - c 1 us t e r , s p e c t r o s c o p i c s t r e n g t h .

A . 2 S U ( 3 ) S h e l l Mod e l

A m o r e s o p h i s t i c a t e d c a l c u l a t i o n o f c l u s t e r i n g p h e n o me n a c a n be

p e r f o r m e d f r o m a m i c r o s c o p i c mo d e l h a v i n g m a c r o s c o p i c c o n n e c t i o n s .

A s h e l l m o d e l l i n k e d t o SU ( 3 ) s y m m e t r y e v a l u a t e s s p e c t r o s c o p i c f a c t o r s ,

i n a d d i t i o n t o f a c i l i t a t i n g s p u r i o u s - s t a t e e l i m i n a t i o n a nd b a s i s t r u n c a

t i o n . G r e a t e r c o m p l e x i t y b r i n g s w i d e r a p p l i c a b i l i t y , r e l a t i v e t o t h e

f o l d e d - p o t e n t i a l m o d e l . C l u s t e r i n g i s now p r e d i c t e d , i n s t e a d o f

p o s t u l a t e d .

T h e b a s i s , i . e . t h e s e t o f e i g e n v e c t o r s f o r a o n e - b o d y c e n t r a l

i n t e r a c t i o n , i s l a b e l l e d a c c o r d i n g t o S U ( 3 ) s y m m e t r y ( s e e He6 A , Ha68)

a n d SU( A) q u a n t u m n u m b e r s . I n t h e sd s h e l l a l o n e , | [ f ] a ( A p ) k LM> and

| [ f ] $TMySM$> l e a d t o | [ f ] a 6 ( A y ) k L S J T > , w h e r e [ f ] r e p r e s e n t s t h e S U ( 6 )

o r b i t a l s y m m e t r y , ( Ay ) r e p r e s e n t s t h e S U ( 3 ) s y m m e t r y a n d a , $ a nd k

n u m b e r m u l t i p l i c i t i e s ; a c o u p l i n g w i t h t h e p s h e l l i n v o l v e s

I p " 1 ( X 1u , ) 6 l T 1S 1 , ( s d ) n 2 [ f 2 ] a 2 e 2 ( A 2 u 2 ) T 2 S2 ; (Ap ) k LSJT> ( M i 7 2 , 7 6 ) . I n t h e

” 1 3 18 p ( s d ) c o n f i g u r a t i o n o f 0 , f o r e x a m p l e , [ f ] ( A y ) = [ 3 ] ( 6 0 ) r e p r e s e n t s

max i mum s y m m e t r y f o r t h e t r i t o n a n d c o u p l e s w i t h [ £ £ £ 3 ] ( 0 1 ) o f t h e

g r o u n d s t a t e t o g i v e ( 0 1 ) x ( 6 0 ) - + ( 6 l ) , ( 5 0 ) . T r u n c a t i o n o f t h e m o d e l s p a c e

o c c u r s i n a n a t u r a l , s y s t e m a t i c way t h r o u g h t h e s e l e c t i o n o f h i g h o r b i t a l

s y m m e t r y a n d l a r g e v a l u e s o f t h e C a s i m i r o p e r a t o r , b o t h i m p l y i n g l o w

e x c i t a t i o n e n e r g y ( H a 6 8 ) . F o r i n s t a n c e , [ 2 1 ] ( 4 1 ) i s i n c l u d e d i n a ( s d ) ^

b a s i s , r e s u l t i n g i n (01 ) * ( £ l ) - + ( £ 2 ) , ( 5 0 ) , ( 3 1 ) f o r ^ 0 , w h e r e a s [ 1 1 1 ] ( 0 3 )

w o u l d be o m i t t e d f i r s t i f a c o m p l e t e b a s i s w e r e n o t u s e d . T h e mo d e l

s p a c e i s e x t e n d e d t o i n c l u d e c o n f i g u r a t i o n s h a v i n g o n e n u c l e o n i n t h e

18f p s h e l l (M i 7 7 ) , e . g . an ( s d ) ( f p ) c o n f i g u r a t i o n o f 0 w i t h ( 2 0 ) * ( 3 0 ) - > -

( 5 0 ) , ( 3 1 ) , ( 1 2 ) . T h e ( 5 0 ) s y m m e t r y , w h i c h o c c u r s i n t h e a b o v e e x a m p l e s ,

x 2c a n a l s o be c r e a t e d by A 1 ( 1 0 ) a c t i n g on t h e ( s d ) ( £ 0 ) c o n f i g u r a t i o n .

c . m .

Su c h s p u r i o u s , c e n t e r - o f - m a s s m o t i o n c a n be r e m o v e d r i g o r o u s l y t h r o u g h

t h e c o n s t r u c t i o n o f s p u r i o u s w a v e f u n c t i o n s f o l l o w e d by S c h m i d t o r t h o g

o n a l i z a t i o n ( H e 7 1 ) - D i a g o n a 1 i z a t i o n o f an e f f e c t i v e , t w o - b o d y r e s i d u a l

i n t e r a c t i o n (Mi 7 5 , Ku66 ) i s t h e n c a r r i e d o u t f o r an e n e r g y m a t r i x as

l a r g e as 2 0 0 x 2 0 0 f o r a g i v e n J 71.

V i a t h e S U ( 3 ) l a b e l l i n g , s p e c t r o s c o p i c f a c t o r s c a n be e x t r a c t e d

18f r o m t h e f i n a l s h e l l - m o d e l w a v e f u n c t i o n s . I n t h e 0 c a s e , a l a r g e ( 6 1 )

o r ( 5 0 ) c o m p o n e n t i n d i c a t e s p o s s i b l e t r i t o n - c l u s t e r s t r u c t u r e b u t d o e s

n o t g u a r a n t e e a l a r g e s p e c t r o s c o p i c f a c t o r , b e c a u s e a m p l i t u d e s f r o m

s e v e r a l c o m p o n e n t s may c a n c e l e a c h o t h e r o r ( 5 0 ) may r e s u l t f r o m a l e s s

s y m m e t r i c d e c o m p o s i t i o n n o t e d a b o v e . E x p r e s s i o n o f a c l u s t e r w a v e f u n c t i o n

i n t e r m s o f t h e S U ( 3 ) b a s i s { # . } ( l c 7 3 , He75) a l l o w s a c a l c u l a t i o n o f

i t s o v e r l a p w i t h t h e s h e l l - m o d e l w a v e f u n c t i o n T o f a g i v e n s t a t e :

w h e r e S i s t h e s p e c t r o s c o p i c f a c t o r a n d 6 t h e a m p l i t u d e . T h e g e n e r a l

f o r m o f t h e c o e f f i c i e n t s b j ( A n 7 4 ) b e c o me s much s i m p l e r f o r a s d - s h e l l

c l u s t e r a n d a p - s h e l l c o r e , o w i n g t o t h e a b s e n c e o f f r a c t i o n a 1- p a r e n t a g e

I n o r d e r t o t r a n s f o r m a s h e l l - m o d e l b a s i s f u n c t i o n i n t o p a r t o f a t r i t o n -

c l u s t e r w a v e f u n c t i o n , t h e f i r s t f a c t o r c o n v e r t s t o c e n t e r - o f - m a s s a nd

r e l a t i v e c o o r d i n a t e s f o r t h e c l u s t e r a nd c o r e , a nd t h e s e c o n d f a c t o r

d o e s t h e same f o r t h e t h r e e n u c l e o n s w i t h i n t h e c l u s t e r . T h e S U ( 3 )

C 1e b s c h - G o r d a n c o e f f i c i e n t t h e n d e c o m p o s e s ( 6 1 ) o r ( 5 0 ) i n t o ( 0 1 ) * ( 6 0 ) ,

a n d t h e W i g n e r 9j s y m b o l t r a n s f o r m s LS i n t o J 1 c o u p l i n g , w h e r e

( A ' y ' ) = ( 0 1 ) a n d | L 1 S 1 J 1 >= | 1 ^ > d e s c r i b e t h e g r o u n d s t a t e , a n d ( 6 0 )

a n d | L t h ^ t > r e p r e s e n t a t r i t o n c l u s t e r i n t h e sd s h e l l . An a p p l i c a t i o n

o f t h i s t h e o r e t i c a l f o r m a l i s m t o e x p e r i m e n t a l r e s u l t s b e g i n s w i t h t h e

n e x t c h a p t e r .

( 4 . A)

c 1 us t e r

18 15a nd r e c o u p l i n g c o e f f i c i e n t s . F o r 0 = N + t ( H I 7 7 ) .

( 4 . 5 )

5 . 1 12 C ( 6 L i , t ) 150 a nd 12 C ( 6 L i , 3 H e ) 15 N

T h e s e l e c t i v i t y e x h i b i t e d by t h r e e - n u c l e o n t r a n s f e r i n t o t h e

A= 15 n u c l e i has b e e n a s o u r c e o f b o t h e x p e r i m e n t a l a nd t h e o r e t i c a l

i n t e r e s t . T h e ( ^ L i , t ) a nd ( ^ L i , 3 He) r e a c t i o n s on a ^ 2 C t a r g e t a t

E ^ . = 4 0 MeV ( F i g s . 5 . 1 , 5 . 2 ) demons t r a t e a p r e f e r e n t i a l p o p u l a t i o n o f

t h r e e s t a t e s h a v i n g J Tr= 9 / 2 + , 1 1 / 2 a n d ( 1 3 / 2 + ) r e s p e c t i v e l y , a r e s u l t

c o n s i s t e n t w i t h m e a s u r e m e n t s a t E, . = 6 0 MeV ( B i 7 5 ) a nd w i t h e a r l y s t u d i e sL i

a t l o w i n c i d e n t e n e r g y ( B a 7 0 , O g 7 3 ) • O w i ng t o t h e m o n o t o n i c n a t u r e o f

a n g u l a r d i s t r i b u t i o n s i n t h e s e r e a c t i o n s ( B i 7 5 ) , f o r w a r d - a n g 1e s p e c t r a

c o n s t i t u t e a s o u r c e o f q u a l i t a t i v e i n f o r m a t i o n on r e l a t i v e s p e c t r o

s c o p i c s t r e n g t h s ( s e e a l s o S e c t i o n 9 * 3 ) . A o n e - t o - o n e c o r r e s p o n d e n c e

b e t w e e n T = ± 1 / 2 a n a l o g s t a t e s i n ^ 3 0 a n d i m p l i e s t h a t t h e d o m i n a n t

p e a k a t 1 2 . 8 4 MeV i n ^ 3 0 i s a n a r r o w d o u b l e t , c o r r e s p o n d i n g t o

15 * -N ( 1 3 . 0 0 , 1 1 / 2 ; 1 3 . 1 7 ) - B e f o r e c o n s i d e r i n g t h r e e - n u c l e o n c l u s t e r i n g

i n s u c h s t a t e s t h r o u g h an a p p l i c a t i o n o f n u c l e a r m o d e l s , we s t u d y t h e i r

6 6 3p r o p e r t i e s e x p e r i m e n t a l l y v i a a c o m p a r i s o n o f t h e ( L i , t ) and ( L i , He)

r e a c t i o n s w i t h d i v e r s e m u l t i - n u c l e o n t r a n s f e r d a t a .

5 . 2 O t h e r T r a n s f e r R e a c t i o n s

I n f o r w a r d - a n g l e s p e c t r a f o r ^ 3 N, t h e ( a , p ) r e a c t i o n a t E = 9 7 MeVa

6 3( F a 7 5 ) i s a l m o s t i d e n t i c a l t o t h e ( L i , He) r e a c t i o n a t h i g h i n c i d e n t

e n e r g y ( B i 7 5 ) - A l t h o u g h t h e ( 1 3 / 2 + ) s t a t e i s d o m i n a n t i n t h e s e d a t a ,

h i g h - s p i n s e l e c t i v i t y i s e v e n m o r e p r o n o u n c e d i n h e a v y - i o n - i n d u c e d , t h r e e -

n u c l e o n t r a n s f e r . A s e m i c 1 a s s i c a 1 c a l c u l a t i o n o f k i n e m a t i c p r o b a b i 1 i t y

f a v o r s L= 6 o v e r L= 4 by a f a c t o r o f t e n ( A n 7 4 ) , i n t h e c a s e o f t h e

1 2 1 2 9 i qC( C, Be) 0 r e a c t i o n a t an i n c i d e n t e n e r g y o f 114 MeV a n d a t an

CHAPTER 5 A=15

F i g u r e 5 * 1

F i g u r e 5 . 2

, 2 C ( 6 L . . t ) , 5 0

, 2 C ( 6 L i . 3 H e ) , 5 N

F i n a l s t a t e s o f t h e A= 15 n u c l e i , o b s e r v e d i n t r i t o n and

3He s p e c t r a a t E ^ j = A 0 MeV a nd 0 i a b ~ 1 5 ° , a r e g i v e n e x c i t a

t i o n e n e r g i e s f r o m i n t e r n a l c a l i b r a t i o n s a nd s p i n v a l u e s

f r o m r e f e r e n c e s i n T a b l e 5 . 1 . T h e s t a n d a r d l e v e l s a nd

e s t i m a t e d u n c e r t a i n t i e s o f t h e e n e r g y c a l i b r a t i o n s a r e

l i s t e d i n f o o t n o t e s t o T a b l e 5 . 1 . A s s i g n m e n t s o f

7T + " 1 5J = ( 9 / 2 ) a n d ( 1 1 / 2 ) i n 0 a r e b a s e d on t h e a n a l o g

r e l a t i o n s h i p w i t h ^ N .

EXCITATION

COUNTS

EXCITATION ENERGY

C O U N T S

as w e l l as t h e ( ^ B , ^ L i ) a n d ( ^ B , ^ B e ) r e a c t i o n s ( N a 7 3 ) , r e v e a l s an

a l m o s t e x c l u s i v e p o p u l a t i o n o f t h e 1 1 / 2 s t a t e n e a r 13 MeV a n d t h e ( 1 3 / 2 + )

s t a t e n e a r 15 MeV. T h e s e d i f f e r e n t t h r e e - n u c l e o n t r a n s f e r d a t a , t h e r e f o r e ,

p r o v i d e s u p p o r t i n g e v i d e n c e o f h i g h a n g u l a r - m o m e n t u m t r a n s f e r .

The ( ^ L i , a ) r e a c t i o n s e r v e s a n o p p o s i t e d y n a m i c a l f u n c t i o n . F o r

an e x c i t a t i o n e n e r g y o f 10 MeV i n ^ N , i n c o m i n g a nd o u t g o i n g o r b i t a l

a n g u l a r mo m e n t a a t t h e n u c l e a r s u r f a c e a r e w e l l m a t c h e d , i . e . A L - 2 i n

6 3c o n t r a s t t o A L = 5 o f t h e ( L i , He) r e a c t i o n . B e c a u s e l o w - s p i n s t a t e s a r e

t h u s m o r e a c c e s s i b l e t o t h e ( ^ L i , a ) r e a c t i o n , l e s s s e l e c t i v i t y i s o b

s e r v e d i n t h e s p e c t r u m f o r a t E ^ . = A 0 MeV ( F i g . 5 - 3 ) , w h i c h i s c o n

s i s t e n t w i t h p r e v i o u s m e a s u r e m e n t s a t E ^ . = A 8 MeV ( Z e 7 7 ) , 35 MeV ( T s 7 3 )

a n d 30 MeV ( 0 g 7 0 , 7 3 ) . W h i l e t h e 1 1 / 2 s t a t e b ec o me s s e c o n d a r y , l e v e l s

a t 1 2 . 5 5 MeV a n d 1 3 . 1 7 MeV r e c e i v e e n h a n c e d r e l a t i v e c r o s s s e c t i o n s i n

t h e ( ^ L i , a ) r e a c t i o n , s u g g e s t i n g L - A ( F i g . 5 - A ) . The Q - v a l u e , w h i c h i s

6 313 MeV m o r e f a v o r a b l e t h a n t h a t o f t h e ( L i , He) r e a c t i o n , a l l o w s a

s t r o n g p o p u l a t i o n o f a d d i t i o n a l s t a t e s a t h i g h e x c i t a t i o n e n e r g y , e . g .

I 5 *N ( 1 8 . 7 0 , 1 9 . 7 1 ) . O w i n g t o t h e o p p o s i n g i n f l u e n c e s o f l i n e a r - and

a n g u l a r - m o m e n t u m m a t c h i n g , t h e r e l a t i v e s t r e n g t h o f ( 1 0 . 6 9 3 , 9 / 2 + )

a n d ( 1 5 . A l , (1 3 / 2 + ) ) i n ( ^ L i , a ) d a t a i s a l m o s t u n c h a n g e d f r o m

6 3( L i , He) d a t a a t E ^ . = A 0 MeV. I n v i e w o f t h e u n a m b i g u o u s a + t p a r e n t a g e

o f ^ L i , t h e r e p e a t e d p r o m i n e n c e o f t h e s e t w o s t a t e s c o n f i r m s t h a t t h e y

p r o b a b l y h a v e 3 p~Ah c o n f i g u r a t i o n s .

I d e n t i f i c a t i o n o f 2 p - 3 h c o n f i g u r a t i o n s f o l l o w s f r o m a s t u d y o f t h e

13 15C ( a , d ) N r e a c t i o n a t ^ * ^ 0 MeV ( L u 6 9 ) • On t h e b a s i s o f i n t e g r a t e d

c r o s s s e c t i o n , a n g u l a r d i s t r i b u t i o n a n d Q - v a l u e , t h e d o m i n a n t s t a t e a t

excitation energy of 15 MeV. The observation of this reaction (Sc72),

E n e r g y c a l i b r a t i o n o f t h e a l p h a - p a r t i c l e s p e c t r u m a t

EL I - 4 0 MeV a nd i s i n d e p e n d e n t o f a nd c o n s i s t e n t

12 6 I ICw i t h C( L i , He) N r e s u l t s . A d d i t i o n a l e x c i t a t i o n

e n e r g i e s a nd s p i n v a l u e s a r e i n c l u d e d i n T a b l e 5 * 1 .

F i g u r e 5 . * * C o m p a r i s o n

T h e s e t h r e e - n u c l e o n t r a n s f e r s p e c t r a f r o m t h e ( 8 L i , t ) ,

6 3 7( L i , He) a n d ( ' L i , a ) r e a c t i o n s a r e m e a s u r e d a t t h e same

i n c i d e n t e n e r g y a nd l a b o r a t o r y a n g l e . P r o b a b l e a n a l o g

s t a t e s a n d r e l a t i v e c r o s s s e c t i o n s a r e a l s o c o m p a r e d i n

T a b l e 5 - 1 . The l a r g e b a c k g r o u n d i n t h e ( ^ L i , a ) r e a c t i o n

r e s u l t s f r o m a Q - v a l u e f a v o r a b l e t o C o u l o m b d i s s o c i a t i o n .

Figure 5-3 ^ C ( ^ L i , o t ) ^ N

EXCITATION

ENERGY

C O U N T S

u»COUNTS COUNTS COUNTS

CDe'en

rom o-■ <D" rO^ CD n> "rl < 01

T A B L E 5 . 1 A = 1 5

1 2 C ( 6 L i , t ) 1 5 0 l 2 C ( 6 L i , 3 H e ) 1 5 N 1 2 C ( 7 L 1 , o ) 1 5 n

1 5 0 E T = 4 0 M e V 9 . . = 1 5 * L i 1 f t b

D1 5 n

R e f . J f f E E ( 1 ) d * / d O ( 2 ) E « 31 d o ' d n 121 E <3 ) d a / d n ( 4 ) E J *X x c . m . x c m . x c . m . X

( M e V ) ( M e V ) ( r e l a t i v e ) ( M e V ) ( r e l a t i v e ) ( M e V ) ( r e l a t i v e ) ( M e V )

A J 7 6 1 / 2 " g . a . g . a . 1 / 2 -

l / 2 + 5 . 1 8 3 5 . 2 9 9 1 / 2 +

5 / 2 * 5 . 2 4 1 5 . 2 4 . 3 4 9 5 . 2 8 . 3 0 8 5 . 2 8 . 6 9 1 5 . 2 7 0 5 / 2 +

3 / 2 " 6 . 1 7 6 6 . 1 6 6 . 3 6 . 3 3 6 . 3 2 4 3 / 2 -

5 / 2 + 6 . 8 5 9 6 . 8 4 7 . 1 5 7 . 1 6 7 . 1 5 5 5 / 2 *

7 / 2 + 7 . 2 7 6 7 . 2 6 . 1 1 9 7 , 5 6 . 1 0 9 7 . 5 6 . 3 7 7 7 . 5 6 7 7 / 2 +

8 . 3 8 . 3 1 3 1 / 2 +

3 / 2 + 8 . 2 8 4 8 . 2 7 . 1 1 7 8 . 5 7 . 1 1 7 8 . 5 7 . 5 6 2 8 . 5 7 1 3 / 2 +

D r 7 7 5 / 2 + 8 . 9 2 2 8 . 9 1 9 . 1 6 . 1 6 5 9 . 1 5 . 7 8 3 9 . 1 5 5 5 / 2

3 / 2 - 8 . 9 8 2 9 . 1 5 2 3 / 2 "

A J 7 6 5 / 2 - 9 . 4 8 7 9 . 4 7 . 1 7 3 9 . 7 6 0 5 / 2 “

3 / 2 - 9 . 6 1 0 9 . 9 1 9 . 9 2 8 ( 3 / 2 “ )

( 7 / 2 , 9 / 2 ) - 9 . 6 6 2 9 . 6 4 . 4 9 7 9 . 8 2 . 6 9 6 9 . 7 8 9 . 8 2 9 7 / 2

K u 7 7 1 0 . 4 5 9 1 0 . 4 4 1 . 0 0 0 * 1 0 . 6 9 l . O O O t 1 0 . 6 9 2 . 2 7 7 1 0 . 6 9 3 9 / 2 +

1 1 . 2 3 1 1 . 2 3 5

1 1 , 4 3 1 1 . 4 4 . 7 3 6 1 1 . 4 3 8 l / 2 +

A J 7 6 1 1 . 7 1 9 1 1 . 7 1 1 1 , 9 6 1 1 . 9 4 1 1 . 9 5 ( 9 / 2 ~ )

" 1 1 . 9 6 5 1 / 2 "

5 / 2 - 1 1 . 9 8 1 2 . 0 1 2 . 3 2 1 2 . 3 3 . 5 2 0 1 2 . 3 2 7 5 / 2

1 2 . 2 9 5 1 2 . 2 9 . 1 4 2 1 2 . 5 6 . 2 3 0 1 2 . 5 5 2 . 4 3 7 1 2 . 5 5 9

1 2 . 8 3 5 1 2 . 8 4 1 . 3 1 9 1 3 . 0 2 1 . 1 7 8 1 2 . 9 9 1 . 2 6 4 1 3 . 0 0 4 1 1 / 2 -

1 3 . 1 7 . 4 8 8 1 3 . 1 7 2 . 6 8 1 1 3 . 1 7 3

1 3 . 8 4 1 3 . 8 5 1 3 . 8 4 3 / 2 *

1 4 . 1 1 1 4 . 1 0 1 4 . 0 9 / 1 0

1 4 . 0 1 4 . 8 6 / 9 2

( 1 3 / 2 + ) 1 5 . 0 5 1 5 . 0 6 . 6 2 0 1 5 . 4 1 . 9 0 7 1 5 . 4 0 2 . 6 0 5 1 5 . 4 0 ( 1 3 / 2 + )

1 5 . 5 4 1 5 . 5 5 1 5 . 8 3 . 1 6 7 1 5 . 8 1 1 . 0 6 3 e t c .

1 5 . 6 5 1 5 . 6 5 1 6 . 0 7 . 1 9 3 1 6 . 0 6 2 . 0 7 1

1 6 . 6 4

1 7 . 1 3

1 7 . 7 2 . 7 2 1

1 7 . 9 4 1 . 7 8 5

1 8 . 7 0 - 4 . 0

1 9 . 7 1

2 0 . 9 4

2 4 . 8

• d V d f 1 = 9 8 0 u b / 6 r c . m . r % E , . = 4 4 M e V , 9 , = 1 0 °

L i l a b

t d r / d n = 1 0 0 0 * b / s r c . m . “ * E . . - 4 4 M e V , e , . -

L i l a b1 0 °

c a l i b r a t e d f r o m 1 5 0 ( 5 . 2 4 1 , h . 2 8 4 , 1 0 . 4 5 , 1 2 . 8 3 5 , 1 5 . 0 5 )

A E ~ ± 2 0 k e V

to\* 2'r - 6°?, statistical

c a l i b r a t e d f r o m 1 5 N ( 5 . 2 7 0 , 7 . 5 6 7 , 8 . 5 7 1 , 1 0 . 6 9 3 )

A E ~ * 2 0 k e V , E < 1 6 M e V x

* 4 0 k e V , E > 1 6 M e V x(41* 3*J - 8%, statistical

R e f .

A J 7 6

M a 7 3

A J 7 6

L u 6 9

A J 7 6

F a 7 5

TT - - 3 21 3 . 0 3 MeV i s a s s i g n e d J = 1 1 / 2 a n d a ( P j ^ ^d 5 / 2 ^ c o n f i g u r a t i o n . As

1 2 6 3 1 5a r e s u l t , i t s p r o m i n e n c e i n C( L i , He) N d a t a ( F i g . 5 . 2 ) p r o b a b l y r e -

2 1 5 * _f l e e t s p ( s d ) t h r e e - n u c l e o n t r a n s f e r . T h e 2 p - 3 h s t a t e a t N ( 9 . 8 2 9 , 7 / 2 ” )

i s a l s o w e l l p o p u l a t e d ( F i g . 5 - 2 ) , b u t t h e p r o p o s e d 9 / 2 member o f t h e

" 3 2( p ) 1/ 2 ” ^^ ^ 5+ d o u b l e t has v e r y l i t t l e c r o s s s e c t i o n a t 1 1 . 9 6 MeV i n

6 3t h e ( L i , He) r e a c t i o n , w h i c h t h u s d e m o n s t r a t e s s t r u c t u r a l s e l e c t i v i t y

w i t h i n t h i s c o n f i g u r a t i o n . B e c a u s e t h e l e v e l s o f u n k n o w n s p i n a t 1 2 . 5 5

MeV a n d 1 3 - 1 7 MeV ( F i g . 5 . 3 ) a r e w e a k o r a b s e n t i n t h e ( a , d ) s p e c t r u m ,

t h e y e m e r g e as a d d i t i o n a l g o o d c a n d i d a t e s f o r 3 p - 4 h s t a t e s . O v e r a l l , a

.6 , 6 3c o m p a r i s o n o f ( L i , t ) a nd ( L i , He) d a t a w i t h o t h e r t r a n s f e r r e a c t i o n s

i n c r e a s e s t h e e x p e r i m e n t a l s e n s i t i v i t y t o t r a n s f e r r e d a n g u l a r momen t a

a n d f i n a l - s t a t e c o n f i g u r a t i o n s i n t h e A = 15 n u c l e i .

5 . 3 M o d e l P r e d i c t i o n s

We i n v e s t i g a t e t h e r o l e o f t r i t o n c l u s t e r i n g i n 3 p ~ 4 h s t a t e s o f

w i t h c a l c u l a t i o n s f r o m a f o l d e d - p o t e n t i a l m o d e l ( S e c t i o n 4 . 1 ) . T h i s

3p o t e n t i a l g e n e r a t e s a ( s d ) c l u s t e r b a n d w i t h an a p p r o x i m a t e l y L ( L + l )

s p a c i n g ( F i g . 5 . 5 ) , w h e r e a s a W o o d s - S a x o n p o t e n t i a l o f f i x e d d e p t h

w o u l d l e a d t o a l m o s t d e g e n e r a t e e n e r g y l e v e l s o r t o an i n v e r t e d s e q u e n c e

o f o r b i t a l a n g u l a r momen t a ( s e e B u 7 7 a ) . Each l e v e l o f g i v e n L i s s p l i t

by t h e t r i t o n s p i n - o r b i t i n t e r a c t i o n , w h e r e t h e s t r e n g t h p a r a m e t e r

19 3V$ q i ( F i g . 4 . 2 ) i s e q u a l t o t h e v a l u e o b t a i n e d i n F f o r t h e same ( s d )

c l u s t e r c o n f i g u r a t i o n ( F i g . 6 . 6 ) . T h e o n l y a d j u s t a b l e p a r a m e t e r i n t h e

p r e s e n t c a l c u l a t i o n , t h e s t r e n g t h f o f t h e f o l d e d p o t e n t i a l , i s f i t t e d

t o ( 1 0 . 6 9 3 , 9 / 2 + ) . I n a d d i t i o n t o h a v i n g kn own s p i n ( B e 7 5 ) , t h i s

n o r m a l i z a t i o n s t a t e i s p r e d i c t e d i n t h e s h e l l m o d e l t o h a v e a p u r e 3 p ~ 4 h

configuration (Li70) and the largest (sd) triton spectroscopic factor

F i g u r e 5 - 5 15N = 1 2 C + t

T he f o l d e d - p o t e n t i a l m o d e l p r e d i c t s a 2N+ L=6 t r i t o n - c 1 us t e r

b a n d o f ^ N , f o r c o m p a r i s o n w i t h f i n a l s t a t e s f r o m t h e

^ 2 C ( ^ L i , 3 H e ) r e a c t i o n a t E . = £ £ MeV a n d 0 * 1 0 ° . P e a k sL i l a b

c o r r e s p o n d i n g t o ( 9 . 8 2 9 , 7 / 2 ; 1 3 - 0 0 , 1 1 / 2 ) a r e d i s r e g a r d e d

i n t h e l i s t o f o b s e r v e d l e v e l s . The d o t t e d l i n e r e p r e s e n t s

e x p e r i m e n t a l s p l i t t i n g ; t h e s t a r i n d i c a t e s a n o r m a l i z a t i o n

s t a t e . W i t h s t r e n g t h p a r a m e t e r s o f f = 1 . 6 9 £ f m and

V ^ q ^ = 0 . 0 1 6 , t h e f o l l o w i n g e x c i t a t i o n e n e r g i e s (MeV) a r e

c a l c u l a t e d :

1 l / 2 + 21 .• 7£

1 3 / 2 + 17, . 28

7 / 2+ 13. . 57

9 / 2 + 10. . 69

3 / 2 + 8. . £5

5 / 2 + 6. . 90

l / 2 + 5,. 66

EXCITATION ENERGY (MeV)

in (An74). These expectations are supported by a large cross section

in triton-transfer reactions. Triton-cluster states calculated from a

normalized folded-potential model thus represent predicted positions of

concentrated spectroscopic strength.

The correspondence between this simple cluster theory and transfer

data in is limited but significant (Fig. 5.5). Given that 1p-2h

components affect the experimental, positive-parity levels of the low-

excitation region, we find the predictions with L=0 and L=2 to be quite

reasonable. The model places 3p~4h, triton-cluster states near

(5.299,1 /2+) and (8 . 571 > 3/2+) , but it suggests that 5/2+ spectroscopic strength is divided experimentally between the levels at 5.270 MeV

and 9-155 MeV. Both levels, as well as the 3/2+ state, are observed in

triton transfer via the (^Li,3 He) reaction, although high-spin selectivity

yields minor peaks. They all appear with more relative cross section in

the ( Li,a) reaction (Fig. 5*3), in contrast to a near absence of

(7.155,5/2+;7-301,3/2+;10.070,3/2+). At higher excitation energy,

the folded-potential model makes its most interesting prediction, a 7/2+

triton-c1 us ter state at 13.57 MeV. The only 7/2+ states known in

(Aj76) are the lp-2h state at 7.567 MeV (L i 70) and a IA. 3 8 MeV level

ignored by triton-transfer reactions. The only states of unknown spin

which are well populated in the 2C(^Li,3He)spectrum lie at 12.56 MeV

and 13.17 MeV (Fig. 5-5). Comparison to ( Li,a) and (a,d) data indicates

a 3p-4h configuration with L~A in this pair of levels (Section 5*2).

Angular distributions from the (2Li,a) reaction (Ts73) reflect a simi

larity to (10.693,9/2"4") . The 13.17 MeV state is the favored candi

date, with respect to excitation energy and (^Li,3 He) cross section, for

spin 7/2+ and substantial triton-cluster structure. Predicted well above

the 15*Al MeV level of ^N, the position of the 13/2+ cluster state in

dicates that additional spectroscopic strength at higher excitation

energy may be important, as the second 1 3/2+ state of 9F demonstrates

(Fig. 6 .6). For the ll/2+ prediction, (8Li,3He) data is not available

above E =20 MeV where, in any case, the cluster strength may fragment xamidst a higher level density. Overall, application of the folded-

potential model to suggests that triton clustering does influ

ence the structure of 3p_Ah configurations selected by the ( L i ,8He)

reaction. Useful predictive power, moreover, is illustrated by the case

of spin 7/2+ .

More detailed predictions, examining the degree of configuration

mixing and the distribution of spectroscopic strength, are found in

shell-model studies of the A=15 nuclei. Calculations from a SU (3) strong-

coupling basis are reported (An7A, see Section A.2), in addition to

results from a weak-coupling model which employs separate bases for the

p and sd shells (Li70,71,7 6a, An7A). Since the first l/2+ state and the

second 3/2+ state have larger 3p"Ah components than other shell-model

states with their respective spin values below E =10 MeV (L i 70). thexassociated experimental levels at (5 . 2 9 9*8 .5 7 1) are confirmed to be

reasonable positions for the 3p_Ah, triton-c1uster states predicted by

the folded-potential model (Fig. 5.5)- A primarily 1p-2h configuration,

however, is expected for 9N'' (5 .2 7 0 ,5/2+) and a large (sd) 3 spectroscopic

factor is calculated for ^N*‘ (9• 155,5/2+) (An7A) , suggesting that the

5/2+ cluster state is rather low. 3p**Ah triton clustering is most

highly developed in the 9/2* and 1 3/2* shell-model states, in agreement

with strong population of the corresponding experimental levels by the

(6Li ,3 He) reaction. For the weak-coupling prediction of a 7/2* state,

an excitation energy of 12.6 MeV and a spectroscopic factor of one-half

the 9/2* value prove consistent with evidence in Fig. 5.5. More frag

mentation of cluster strength occurs among the six shell-model states

of spin ll/2+ generated below E *18 MeV. Although the detailed distri-xbution of spectroscopic strength can be important in the general

outline of (sd) 3 triton-cluster structure indicated by the folded-

potential model receives considerable support from a comparison with

shell-model calculations.

The influence of triton clustering upon negative-parity states of

is a relevant question, since 2p-3h configurations at (9 .8 2 9,

7/2“;13.00,11/2”) are responsible for major peaks in the *2C(8Li ,3 He)^3Naspectrum (Fig. 5.2). A different cluster configuration of 'S

expected to entail a change in the strength of the triton spin-orbit

interaction, as well as a renormalization of the strength f of the

folded potential. Reliable determination of these parameters is pre

cluded at present by the lack of an experimental candidate for the upper

member of the L=3 or L=5 doublet, i.e. a 5/2” or 9/2” state populated by

three-nucleon transfer at high excitation energy in 3 N. If a calcula

tion is adjusted to the lower members of these doublets, the prediction

for a 3/2” level of the 2N+L=5 band is encouraging (Bu75), but if a

similar normalization procedure is followed for the 2N+L=6 band of ^N,

V is inconsistent with later results in ^F (Bu77a). In view of the SOIuncertainty implied for the 2N+L=5 case, we turn from the folded-potential

model to the shell model. Sizeable, three-nucleon spectroscopic factors

are indeed predicted for the 7 / 2 and 1 1 / 2 states of (An7£), account

ing for the large cross sections measured in the (^Li,3He) reaction. As2 12 a result, Pj/2 sc clustering appears to be viable, outside a C core.

6 6 3In summary, the ( Li,t) and ( Li, He) reactions, together with

other transfer reactions into the A=15 nuclei, identify probable p **(sd) 3

configurations at 15N*(10.693,9/2+;13.17;15.£l,(13/2+)). The folded-i c i opotential model for * N= C+t, through approximate correspondence with

experiment and general support from the shell model, suggests that tri

ton clustering plays a significant role in their structure.

16 12 The closed-shell target of 0, like the complete subshell of C,

should enhance the probability of clustering among transferred valence

nucleons. Since ^0 also has ^=0*, the spins and parities of triton-

cluster states in F are expected to be the same as those in 3N

(Table 1.1). In place of highly excited, 3p~4h states for A=15, however,

the (sd) 3 configuration implies a 3p"0h, ground-state band in the A=19nuclei.

6.1 160 (6L i , t)19Ne and 160(6Li ,3 He)19F

In mirror spectra for these T =±1/2 nuclei (Figs. 6.1, 6.2), three-znucleon transfer demonstrates a combination of structural and dynamic

selectivity. Final states with J7T=5/2+ , 9/2+ and 13/2+ have progressively

enhanced cross sections, in contrast to the minor peak for a 7 / 2 -9 / 2

19 *doublet at F (4.01) and to the absence of several known high-spin 19 * +states such as F (7.937,11/2 ) (see Section 3.3). Since negative-

parity levels become prominent above these positive-parity levels, the

pattern immediately suggests a presence of (sd) configurations followed 2by (sd) fp excitations. This choice of states in the A=19 nuclei by the

6 6 3( Li,t) and ( Li, He) reactions at E^.=46 MeV is consistent with previous

results, obtained at incident energies of 36 MeV (Pa72), 30 MeV (We72)

and 24 MeV (B i 71, see Section 3.1) for excitation energies below 9 MeV.1 9At higher excitation (Fig. 6.2), favored states of F are greater in

strength but not in number. Consequently, analog assignments are clear19 19 *in Ne (Fig. 6.1), although a shift in energy occurs at Ne (10.01)

and a more negative Q-value reduces ( Li,t) cross sections above ^^=14 MeV.

CHAPTER 6 A=19

Figure 6 .

1 l60(6 Li,t)19Ne

2 l60(6Li,3He)19F

For these spectra measured at E^.=A6 MeV and energy

calibrations are described and spin values are referenced in19Table 6.1. The extrapolation to high excitation in F is

generally consistent with a calibration of the (a,p) reac-IQ /Vtion (Va76), and the energy of Ne (8.9A) agrees with a pre

vious value from the ( Li,t) reaction (We72). Given addi

tional spins from the (a,y) reaction (e.g. Sy77), ^F is the

source of tentative spin assignments for analog states in

0 -------------------------------- 1-----w x - u20 15 10 5 0

high spin for high-lying levels. In view of the structural selectivity

illustrated at lower excitation, they can be identified as additional

candidates for 3p"0h states, perhaps involving several fp-shel1 excita-3 20tions. Evidence of (fp) structure in Ne is presented in Appendix B.

6.2 Other transfer reactions19 *High angu1ar-momentum transfer into F (12.71»14.10,15-00) is con

firmed by the large cross sections measured for these levels in the (a,p)

reaction at E =40 MeV (Va76) , where aL*9» and in the (^B,7 Be) reactionaat E =100 MeV, where AL^5 is expected semic1 assica11y (Ha76a,c). As inDthe A= 15 nuclei (Section 5.2), the known 13/2* states of 9F are most

prominent in these two spectra. Additional states at high excitation

energy are strongly populated by the (7 Li,a) reaction at j =35 MeV or

30 MeV, e.g. ^F (9 .6) (Ts7*0 and ^F (13-3) (We73) . Lower spin is

probable for such levels, since (7Li ,ot) is +3"h better matched than the 6 3( Li, He) reaction. If these dynamical differences between three-

nucleon transfer reactions are taken into account, however, there is an

underlying consistency in their selection of final states.

Candidates for 3p*0h configurations are more sensitively tested by

a comparison of three-nucleon transfer with alpha-particle transfer. The

(7 Li,t) reaction is predominantly direct at E^.=38 MeV and 0]ab = 15° and

highly selective in 1 60, l8F and 20Ne (Co7*t, 76,77) . The 1 5N (7Li , t) 19F

reaction of Fig. 6.3, similarly, is expected to favor 4p-lh configura

tions with "a 1pha-clus ter" structure, i.e. with large alpha-particle

spectroscopic factors. Since this reaction has the same angu1ar-momentum 16 6 3 19mismatch as 0( Li, He) F, dynamical effects should have little

6 3The angu1ar-momentum mismatch a L=7 of the ( Li, He) react ion ind icates

Figure 6 .

3 15N (7 Li,t)19F

19Final states of F are investigated here via alpha-particle

transfer at E, .=A0 MeV and 0 . ,=15°. Absolute differential Li 1 abcross sections are given in Table 6.1, together with rele-

19vant known levels of F.

A Comparison

6 6 3The ( Li,t) and ( Li, He) reactions indicate analog states in

the A=19 nuclei, which are listed adjacently in Table 6.1.

Below these three-nucleon transfer data is a contrasting alpha-19particle transfer spectrum for F, measured at a similar in

cident energy and at the same laboratory angle.

1000 -

COUNTS COUNTS COUNTS

TABLE 6.1

(AJ78)19Ne

J* Ex Ex(1)(MeV) (MeV)

1/2*5/2* 0.238 0.231/2- 0.2755/2- 1.5083/2* 1.536 1.543/2- 1.6169/2* 2.795 2.80(9/2-) 4.140(7/2-) 4.197 4.211/2* 4.379 4.38

13/2* 4.635 4.64(7/2*) 5.424 5.42

6.094 6.086.289 6.286.862 6. 85 7.218.06 8.088.44 8.45etc. 8.94

16o(6Li.t)l9Ne 160(6Li,3He)19F

9. 8110.0111.08-11.2411.40.

12.5613.113.2214.18 14.4414.78

®i v, = 15°lab(2)

182143

364246200

Ex(3> dVdp (2> * c. m.(MeV) (b/6r)0.20 691.321.542.784.014.37

4.645.45G. 106.526.927.25

8. 29 1.969.'

10.41 11.24 11.46 I 11.67 J

12.71-

13.7614.1015.0015.56

180147

521379221

15628087

15N(7Li,t)l9F Elj = 40 MeVe. . = is0lab

Ex(4) dVdO (5) c. m.(MeV) (jib/s r)0.19 1.341.462.784.02

5.466.106.326.94 7.548.298.95 9.35

263122162

495795

9.92 -1800 10.4011.5 11.7 12.01 12.30 1 12.57+J 1411

• or 12.63/12.77 t or 12.32/12.46/12.62(1) calibrated from l9Ne*(0.238, 2.795, 5.43)cons latent with 15o*(5.241, 7.276, 10.45, 12.835, 15.05)AE - ± 20 keV, Ex < 13 MeV

± 30 keV, Ex > 13 MeV(2) ±(1% - 4%), statistical ~ 4 10%, absolute(3) calibrated from 19F*(0.197, 2.780, 4.648, 6.925, 8.953, 10.411)

AE =- ± 15 keV, Ex < U MeV± 30 keV, Ex > U MeV

(4) calibrated from 19F*(2.780, 4.016, 8.953)AE - ± 15 keV, E* < 9 MeV± 30 keV, 9 MeV < Ex < 15 MeV± 50 keV, Ex > 15 MeV

13.7814.1214.5014.9216.0916.4517.418.218.719.93

7583366

Ex J"(MeV)g-s. 1/2*0,197 5/2+0.110 1/2-1.346 5/2-1. 554 3/2*1.459 3/2-2.780 9/2*4.032 9/2-3.999 7/2"4.377 1/2*4.550 3/2*4. 556 3/2-4.647 13/2*5.425 7/2-5. 465 1/2*5. 500 3/2*6.090 3/2-6.330 1/2*6.500 11/2*6.925 1/2-7.2657.56 1/2*8.288 13/2-8.953 11/2-9.3659.7109.819 5/2"9.8349.872 11/2-9.9010.411 13/2*11.217 11/2*etc.

Di77 A j7s

Fi77Sv 7 6AJ78Sy76Fi77Ko77Sy77

(5) ±(1% - 5%), statistical ~± 15%, absolute

bearing on the comparison. Since angular distributions have a structure

less behavior (Mi70, Ga72), forward-ang1e spectra contain approximate,

relative, spectroscopic information. Final states strongly populated in

the ( Li ,3 He) reaction but clearly inhibited in (7 Li,t),e.g. ^F'(2.78,

4.64,6.92,10.41,14.10) (Fig. 6.4), demonstrate probable 3p_0h configura

tions. Mixing between 3p"0h and 4p-lh structure is evident in the 11/2”

state at 8.953 MeV, where the two reactions yield similar relative cross

sections despite their different population of the 1 3/2” state at 8 . 2 8 8

MeV. Alternate known levels for the peaks at ^N (^Li,t)^F (5.46,9-92)

(Table 6.1) make their origin more uncertain. Near 12 MeV and 15 MeV

in excitation, multiplets further hinder a comparison with the 16 6 3 190 ( Li, He) F reaction, but primarily different states are suggested

6 3by the precise energies and widths (Table 6.1, Fig. 6.4). The ( Li, He)7 19and ( Li,t) reactions into F, therefore, produce generally contrasting

spectra but identify one major case of configuration mixing.

The resulting candidates for largely £p-lh configurations, e.g.19 *F (4.02,8.29,14.50), become of interest through their relation to 4p-0h

20configurations in Ne. We will study, in the next chapter, the relation

of 3p~lh to 3p_0h states by comparing ( Li,^He) spectra from ^N and ^0 targets. We first investigate, as a simpler and better known case, the

coupling of a Pj/ 2 to alpha-cluster structure by comparing ( Li,t)data from ^N and ^0 targets. Narrow, negati ve-pari ty doublets of ^F

(Fig. 6.5) exhibit a weak-coup 1 i ng relationship to the J7T=2+ , 4 and 6

members of the (sd)\ ground-state band of ^Ne (Mi70, Pi76, E170) .Because the 8+ member at 11.95 MeV is hardly observed in the ^0(^Li,t)^Ne

reaction, the 6+ state at 12.59 MeV may correspond to the pair at 12.57/

Figure 6.5 Weak coupling in 4p-lh configurations

The ^N(2Li,t)^F reaction is compared with ^0 ( L i , t) 2 Ne

(Co74,76), where E^. = *0 MeV and 38 MeV respectively and200 ,=15°. Excitation energies and spin values in Ne are

1 abobtained from Refs. Co76, Sa77, Aj78. The differential

cross section of 4.1 mb for ^F (1 A.9 2/14.5 0) is comparable

to the value of 3-3 mb for 2^Ne (15-34).

COUNTS

(MeV) 10

COUNTS

OiOoooo

12.30 MeV in the 9N(7Li,t)^9F reaction. Comparison is more difficult20for the low-spin members of the 0 band of Ne since, for example,

3”®l/2” strength appears fragmented among 7/2* states at 9F'(6.070,

6.330,7.56) (see also Bu77a). The 5” and 7 states of ^^Ne do have good 19candidates in F near 10 MeV and 15 MeV respectively (Fig. 6.5). Al-

19though additional spin assignments in F are needed to confirm the re

lationship, weak coupling can at present describe Ap-lh, a 1pha-c1uster

structure. 3p~lh, triton-cluster structure may therefore involve the

same phenomenon (Section 7-3).

6.3 Model PredictionsTheoretical investigation of triton clustering in 3p_0h states is

19complemented by extensive experimental information on F. In particu-

lar, an application of the folded-potential model to the (sd) configura-1 9tion is aided by current knowledge of the ground-state band of F

(Fig. 6.6). In contrast to the case of ^N (Fig. 5.5), a 7/2* state is

assigned to this band (B i 72) and a second 13/2+ state is identified at

high excitation energy, together with a 11/2* state (Sy77)• Low-lying

3p_0h configurations, moreover, are free from the mixing with single

particle excitations which affects 3p-Ah structure at low excitation in15 6 3N (Section 5-3). Since the ( Li, He) reaction (Fig. 6.6) and the

SU(3) shell model (S173) indicate enhanced triton clustering in the 7/2*+ 19and 9/2 states of F, they provide a good normalization for the strength

V$o) tbe triton spin-orbit interaction and the strength f of the

folded potent i a 1.Theoretical triton-cluster states show a remarkable correspondence

19 + + .to experimental levels of F. A calculated 5/2 -3/2 doublet is in

Figure 6 .

A 2N+L=6, triton-cluster band from the folded-potential

model is compared with triton-transfer data from the

^0 ( L i ,3 He) F reaction (see also Fig. 6.2, Table 6.1).

The list of experimental levels excludes the strongly

populated states at 19F* ( 6 .925,7/2“;8 .953,1111 ;9.872,11/ 2 ).

Using f=1.5l4 fm and V =0.016, we calculate the following

excitation energies (MeV):

n / 2 + 1 1 ..46

13/2* 7.■ 30

+CM 5.,46

9/2+ 2. 00

3/2+ 1 ..63

+CM\LA 0,.20

l/2+ -0,.57

precise agreement with known excitation energies (Fig. 6 .6). In addition19to a reasonable result for the ground state of F, an average position

of 1 3/2+ spectroscopic strength is predicted between the pair of 1 3/2+

states strongly populated in triton-transfer data. For the ll/2+ triton-19 * +cluster state, an experimental candidate at F (11.217,11/2 ) has a

relatively small cross section, despite the high-spin selectivity of the 6 3( Li, He) reaction. This observation suggests an important distribution

of ll/2+ cluster strength among additional levels, perhaps including a

state of unknown spin at 9.90 MeV which is resolved by the (a,p) reaction

(Ko77)• The overall correlation, however, between a cluster band pre

dicted by the folded-potential model and the ground-state band known in19 3F is evidence that (sd) triton clustering is highly developed outside

the closed-shel 1 , ^ 0 core.

This conclusion is supported by spectroscopic factors calculated

from the SU(3) shell model, e.g. large concentrations of triton-clus ter

strength in the 1 3/2* and 13/2 levels as well as in 1 1/2^ (St73, Sy77)•A result similar to the 2N+L=6 band of Fig. 6 .6 , moreover, is obtained

from a "cosh-potential" model (Bu77a), i.e. a triton-cluster model based

on a symmetrized Woods-Saxon well. Once the radius and diffuseness19parameters are fit to the ground-state band of F, this potential is

found to be similar in shape to a folded potential (Fig. 4.4) and is

applied also to excited triton-cluster configurations (Bu77a) . A nor

malization problem, however, analogous to that in the 2N+L=5 band of]c ] 9 2N (Section 5-3), arises in the 2N+L=7 band of F. Although a (sd) fp

19 *configuration (see Eq. 4.3) is indicated at F (6.925,7/2 ) by triton-

t ra ns fe r react ions (Fig. 6 . 2 , T s7 4 ) , the 5/2 member of the doublet is

more uncertain. If the parameter V<jqj were fit to a 5/2 state at

9.819 MeV, which is weakly observed in a high-resolution (<*,p) spectrum

(Ko77), the resulting predictions would include a 1 5 / 2 triton-cluster

state near ^ 0 ( Li , He) F (1 A. 10) . Mixing is expected, however, in the

case of J7T=15/2 or li/2 , because of the theoretical proximity of tri

ton- and a 1pha-c1 us ter states (Bu77a). A detailed description of the2 — 1 Acompetition between such (sd) fp and p (sd) structure is attempted by

the SU(3) shell model. Large spectroscopic factors for both the triton

and alpha clusters are calculated in a 1 1 / 2 2 state at 8.9 MeV (Mi77)» in19 *agreement with the strong population of F (8.953,11/2 ) by both the

6 3 7 -( Li, He) and ( Li,t) reactions (Fig. 6 .A). A 15/2 state, predicted

with similar mixing at 12.5 MeV, may correspond to a member of the tenta-19 *tive doublets observed at F (12.63/12.77) in triton transfer and at

1 q * 2F (12.62/12.A6) in a 1pha-partic1e transfer (Table 6.1). (sd) fp triton19 2 clustering is thus expected to be influential in F, just as p(sd)

clustering exists in the shell model of (see Section 5.3).In summary, three-nucleon transfer reactions identify candidates

for 3p-0h states at ' V' (0 . 197,5/2+;2 .7 8 0 ,9/2+; A.647,13/2+;5 +65,7/2+;

10.1ill,13/2+) and 'V' (6 .925,7/2' ; 9.872, 1 1/2* ; 12.77; 1+ 10; 15-00) . The

( Li,t) reaction, via contrast, generally confirms this configuration.3The folded-potential model, through correspondence, suggests that (sd)

triton-c1 us ter structure is important in the former states. The latter

levels appear to represent fp-shell excitations.

CHAPTER 7 A=18

The previous two chapters concern T=l/2 states of odd-A, mirror

nuclei. In this chapter, we investigate the spectrum of T=1 states in18the even-A, N=Z+2 nucleus of 0 and compare it with the spectrum of

18interspersed T=1 and T=0 states in the N=Z nucleus of F (Table 1.1).15 7TSince an unexcited N core has J =1/2 , triton-cluster states of spin

f®l/ 2 are expected in 8 0 , where j is the total angular momentum of

the triton. Their relationship to states of spin j in 9 F= 8 0+t

(Fig. 6 .6) shows the influence of weak coupling.

7.1 1 5N(6Li ,t)l8F and 15N (6 L i ,3 He)l80

Identification of probable 3p-lh states in the A=18 nuclei begins

with the role of low-lying, negative-parity states in three-nucleon trans-2 18 fer spectra. Above the (sd) ground-state band of F, such known levels

are selectively populated by the (8 Li,t) reaction at E^.=40 MeV (Fig. 7*1),

in accordance with results at E, .=30 MeV for E< 7 MeV (L i 72). The pre-L i x-1 3cedence of a p (sd) configuration over (sd)(fp) is supported by diverse

experimental evidence (see Ro73c); e.g. the first negative-parity state 18of F has spin zero (see L i 72) , which cannot arise from a d^^fy/y

coupling. Further candidates for 3p“lh structure, led by a T=0 state at1 0 l o

9.52 MeV, are plentiful at high excitation in F. In 0, where T=1

states of spin 1 , 3 and 5 appear with increasing strength (Fig. 7.2),

J7T=7 or 6 may apply to the leading peaks at 11.10 MeV and 14.61 MeV.

A correspondence to 3p~0h states of 9F with J7T=l/2+ , 5/2+ , 9/2+ and 13/2*

is indicated by their similar progression of relative cross sections in 6 1the ( Li , He) react ion (see Fig. 10 . 1a ,b ) . An analog r e l a t i o ns h i p to

Figure 7-

Figure 7.

1 1 5N(6Li,t)l8F

2 1 5N(6L i,3He)l80

3At E^.=40 MeV and ®jab= 5°, outgoing tritons and He nuclei12 16from standard targets of C and 0 provide an energy cali

bration for these spectra. Known final states of and6 3F observed in the ( Li, He) reaction generate consistent19I V C U III LIIC \ U I

1 8 1 8 excitation energies for 0. High energy levels of F are

determined by interpolation from the 2C(8Li,t) ^ 0 reaction

instead of extrapolation from 80(8Li,t)^3Ne (see footnotes18to Table 7-1). The broad contaminant peak under O '(3-555)

1 6 3 i|is from the H( Li, He) He reaction. The large peak at18 * 6F (4.85), where AL-5 in the ( Li,t) reaction, does not

appear to arise from a 1 level at 4.860 MeV.

EXCITATION

COUNTS

‘H 9)

proximity to T=0 levels of equal prominence (Fig. 7-4). Although iso

spin mixing is relevant in general, the location of large T=1 components18 18 in F can be suggested on the basis of excitation energies in 0,

which are listed adjacently in Table 7.1. More definite T=1, Tz=06 6 3assignments are obtained from the ( Li,t) and ( Li, He) reactions into

the A—16 nuclei (Section 9.1).

7.2 Other Transfer Reactions1 2 9 1 310The ( C, Be) and ( C, B) reactions at incident energies near

18 *'100 MeV (P i 77) preferentially populate the states at F (7.24,9*52)18 * 7and 0 (11.10,11.67), confirming their high-spin character. The ( LI,a)

reaction identifies additional states of lower spin at high excitation,

e.g. ^0 (16.73,17.92,20.4) (Figs. 7.3, 7.4). In comparison to ( Li,t)6 3and ( Li, He) data for the A=18 nuclei, therefore, these other three-

nucleon transfer reactions have differences equivalent to those found

in the A=15 and A=19 nuclei, where we discuss their dynamical origins

(see Sections 5.2, 6.2).

An overall difference in structural selectivity is expected from

the (7 Li,t) and (a,d) reactions, favoring 4p-2h and 2p-0h configurations7T +respectively in the A=18 nuclei. A T=1, J =6 state dominant at

^C(^Li,t)^0 (11.69) (Mo70) , however, also appears to be well populated 15 6 3 l8in the N( Li, He) 0 reaction (Fig. 7.2). Since this similarity may

- 2 4 - 1 2be interpreted as mixing between p (sd) and p (sd) fp structure, an19 *analogy exists to the 11/2 state at F (8.953), where such competition

between triton and alpha-particle clustering is indicated by both experi-7T ^

5818T=1 s t a t e s o f F, which l i e at Ex £l .04 MeV, is complicated by t he i r

ment (Fig. 6 .4 ) and theory (Sect ion 6 . 3 ) . A T=0, J =(6 ) s t a t e is

15 ,7 . J 8 Figure 7-3 N ( Li,a) 0

Excitation energies for these data are determined from the

^C(7 Li,a)^N reaction (Fig. 5-3).

Figure 7.4 Comparison

6 6 3 7Three-nucleon transfer via the ( Li,t), ( Li, He) and ( Li,a)

reactions, all measured at E^.=40 MeV and Q]ab=15°, selects

final states of the A=18 nuclei. A large background from

Coulomb break-up is subtracted from this (7 Li,a) spectrum.

EXCITATION ENERGY

COUNTS

Oi (T>O O $o o o

COUNTS COUNTS COUNTS

IbtlCo 7 7

MaMMiM

1+3+T-1,0+0"5+2-T-1,2* 1" 3" 2* 2 " 4*

T-1,44(T-1.0+)T*l,2+T-O+l.l"4_

T-1,3"5*T-1,2"4+

Ex(MeV)g.a.0.9371.0411.0801.1212.1013.0613.1343.7913.8384.2264.3984.8524.7534.9645.6056.0966.2416.5676.6446.777etc.

9.49(6+) 9.58

10.5411.38

1SN<6Li,t)18F

TABLE 7.1 A - 18V u , V ' o

E (!) x(MeV)g.a.0.95

1.132.113.123.814.234.41

4.855.606.106.206.546.787.24

8.048.228.939.249.52

10.1910.5611.011.39

11.9512.35

12.7812.8613.21

13.9414.84

15.518.3417,7618.1

'U 40 MeV 'lab

*'«i. £!(jjb/ar)3284

683645753594

7226203

80198278

E <3) dc/dO <«) x c. m.(MeV) (pb/ar)

4.455.095.52

6.186.35

7.10 887.84 1868.10 165

15N(7Ll,a)180

E (5) do/dfl («) x c. m.(MeV) ( lb/ail

2.0 25

3.63.9 4.475.10 5.52

6.26.366.97.17.818.12

205191

EX(MaV)J* Raf.

ga- 0+ AJ78

1.982 2*

3.555 4+3.634 0*3.921 2*4.456 1-5.098 3"5.530 2"

6.196 1- 01736.351 2" ", L176b6.404 3- AJ786.882 o- 01737.117 4+ AJ787.85 "8.12 5"

8.9569.0 9.0 (9.03)9.35 9.4 9.369.399.689.70 9.71 (9.72)10.27 42 10.3 10.29 4+10.60 55 10.61 123 10.5810.92 50 10.92 84 10.9111.10 232 11.10 151 11.1311.39 (2+)11.4 11.40 157 11.41 (4+)11.62 5-11.67 265 11.70 402 11.69 6*

12.50 4+12.53 166 12.53 6+etc.13.79 145 13.7814.1414.61 295 14.62 726

15.2515.95 167 16.03 48116.7316.98 17.017.9219.0 20.020.3 20.4

TABLE 7.1 (continued) 61

/i\ 1 9 *calibrated from Ne (0 .2 3 8 , 2 .7 9 4 , 5 .4 3 ), E < 10 MeV

1 c ]|c

from O (5 .2 4 1 , 8 .2 8 4 , 1 0 .4 5 , 12 .8 3 5 , 1 5 .0 5 ), Ex > 10 MeV

AE =- ± 20 keV, E < 11 MeV x± 30 keV, E > 11 MeV x

^ ± (1% - 5%), statistical~ ± 15% absolute, E < 10 MeVx- +40% absolute, E > 10 MeVx

/Q\ I Q *v 'calibrated from F (0 .1 9 7 , 2 .7 8 0 , 4 .6 4 8 , 6 .9 2 5 , 8 .9 5 3 , 10.411)

AE =- ± 15 keV, 4 MeV < E < 1 5 MeVx± 30 keV, E > 15 MeV x

^ ± (1% - 5%), statistical ~ ±15% , absolute

(5)calibrated from 15N *(5 .270, 7 .5 6 7 , 8 .5 7 1 , 10.693)AE =- ± 20 keV, E < 16 MeV

X± 40 keV, E > 16 MeV x

^ ± (2% - 10%), statistical~ ± 15% absolute, E < 10 MeVx~ ± 25% absolute, E > 1 0 MeVx

identified at 9-58 MeV in the ^N(7Li,t)^8F reaction as a candidate for + l8the 4p-2h, 1 band of F (Co77). Although other members of this band

are clearly negligible in * 3N ( Li , t) 8F data, the peak at 9.52 MeV

(Fig. 7-1) could contain a (6+ ) contribution. A more probable explana

tion of this peak, however, lies in a correspondence to the level ob

served at E =9.494 MeV ±15 keV in the 80(a,d)^8F reaction (Ma68). xWhile the large cross section in two-nucleon transfer suggests a

— 1 “3(d^^^ 7/2 6” comPonent (Ri66), mixing with p (sd)- structure would beimplied by the dominance of the same state in three-nucleon transfer.

Several other states of 8F, e.g. at 10.541 MeV and 11.384 MeV (Ma68,

Ri66), reflect a limited overlap between (8Li,t) and (a,d) spectra. The

(sd)(fp) configuration, therefore, appears to play a significant role in18these high-lying, T=0 levels of F.

7.3 Model Predictions18Interpretation of triton-transfer data for 0 by means of a calcu-

-1 3lation of p (sd) , triton-c1uster structure involves a coupling of threel8 15angular momenta. The folded-potential model of 0= N+t (Fig. 7-5) pre

dicts orbital angu1ar-momentum states, which are first split by the tri

ton spin-orbit interaction. Levels of ^F=^80+t (Fig. 6.6) fix the

strength parameter V^^ in the (sd) cluster configuration. In order to

describe the effect of a core with J7T=1/2 , we introduce a second

spin-dependent interaction ^502 ’ defined anc* discussed in Section 4.1.When its strength V 2 's adjusted approximately to the separation of18 * — -CT (5 .0 9 8 , 3 ;5 .5 3 0 , 2 ), the resulting narrow doublets prove consistentwith a "hyperfine" interaction (Fig. 4.2) rather than a strong spin-spin

coupling. A third parameter, the strength f of the folded potential V^(r),

18Triton-c1 us ter states in the 2N+L=6 band of 0 are calcu

lated from the folded potential of Fig. A.A and compared 15 6 3 18with the N( Li, He) 0 spectrum of Fig. 1 . 2 . Peaks iden-

18 +tified with positive-parity states, namely 0 (7 .1 1 7 ,A ;

1 1.6 9,6+ ;1 2.5 3 ,6+) (Aj7 8), are absent from the list of ex

perimental levels. With f=1.532 fm, V^q^=0.016 and

Vso2=0 •0032, we obtain the following theoretical excitation

energ ies (MeV):

Figure 7-5 ^ 0 = ^ N + t

5 1 7 ..76

6“ 16..99

6" 13..57

7“ 12.,62

3“ 1 1 .■ 36

A" 10.. 8 9

A“ 8.,7A

5" 8.,12

r 7..30

2 5.. 9A

3" 5..60

O" 5..02

r A..91

9N (8 L i ,9He) 80 reaction. We then find that V<.(r) for 9N+t is almost

identical to the potential for 8 0+t, which generates a triton-cluster19band in close correspondence to the ground-state band of F. Conse

quently, this normalization of the folded-potential model for 3p-lh 18structure in 0 is equivalent to a weak-coupling procedure, which joins

3 19a Pj/ 2 bo*e to (s states of F. A theoretical approach relating these two nuclei is also taken by the weak-coupling shell model, with

similar results for E < 8 MeV (El70), and by the cosh-potential cluster

model, with different interactions and normalization (Bu78). Experi

mental evidence of weak coupling is well established in the case of a4 20p1/ 2 hole plus (sd) states of Ne (Fig. 6.5, Mi 70).

-1 3Weak-coupling effects are subtle but observable in p (sd) con-— — 18figurations. In addition to the known 3 "2 doublet of 0 (Fig. 7-5),

the 4 member of a 9/2+®l/ 2 doublet predicted by the folded-potential18model has a good experimental candidate at 8.47 MeV. This level of 0

is assigned unnatural parity (0 1 7 1) and is prominent in triton-transfer

spectra (see also Fig. 7.3). The predicted position of the 1 ~2 triton-

cluster states suggests a distribution of spectroscopic strength among

0 (6.196,1 ;6 .351,2 ;7.620,1 ;7.75) (Aj78). Although population of6 3these known levels is dynamically inhibited in the ( Li, He) reaction

18 “ — 7(Fig. 7.2), low-spin states such as 0 (6.882,0 ) do appear in the ( Li,a)

spectrum (Fig. 7.3). Between Ex=8.47 and 10.60 MeV, all levels observed

in the two reactions have natural parity (0171); at 10.60 MeV, indefiniteparity and adequate cross section provide the first candidate for a

4 =7/2+®l/2 prediction. The 7 triton-cluster state represents an

64is f i t t e d to the known 5 s t a t e (Le67) s trongly populated by the

average position of spectroscopic strength which may be divided experi-

mentally, e.g. between the leading peaks at 0 (11.10;14.61 or 1 5-9 5)

(Fig. 7.5), just as 13/2+ cluster strength is found to be split between

F'* (4.647; 10.411) (Fig. 6 .6). Through an underlying correspondence to18triton-transfer data for 0 , therefore, the folded-potential model

shows useful predictive power and suggests influential clustering

phenomena.

A more evident correlation between theory and experiment is ob

tained with the SU(3) shell model. Predicted spectroscopic factors

detail the distribution and degree of triton clustering in negative-1 ftparity states of 0 (Mi 77) which, in Section 4.2, serve as examples

of SU(3) techniques. At E =7-84 MeV, a state of unknown spin andxnatural parity (0 1 7 1), which is strongly populated in triton-transfer

reactions and unaccounted for by the folded-potential model (Fig. 7*5),

can be associated with the first 5” state calculated by the shell

model (Fig. 7.6). The 52 prediction then corresponds to a known level18 * - _ at 0 (8 .1 0 , 5 )• In addition to this unexpected splitting of 5 spec

troscopic strength, a suspected division of the 7 triton-cluster state

(Fig. 7*5) is confirmed by the shell model, which thereby supports an

identification of the 7 j and 7 states with the prominent experimental

levels at ( Li ,3 He) 0 (11.10;l4.6l or 15-95)- The magnitude of the above spectroscopic factors, together with the strength of 1 and 3j

(Fig. 7-6), demonstrates the importance of triton clustering in "stretched"

angu1ar-momentum couplings, where orbital angular momentum, cluster spin

and core spin are aligned to maximum J (see Fig. 7-5). Enhanced cross6 3

s e c t i o n s for such s t a t e s are found in the ( Li , He) react ion which, for

Preliminary triton spectroscopic factors for negative-parity 18states of 0 (Mi 77) are predicted from Eq. A.5 and plotted

if S*0.05 (center). They are calculated for a p (sd) con

figuration and a 3=j®1/2 coupling which corresponds to the

triton-c1 us ter states predicted by the folded-potential

model (Fig. 7.5)* Excitation energies for the latter states

are plotted on the left-hand side of the present diagram.18Above the relevant known levels of 0 (right), which also

1 8 « —include 0* (7 .6 2 0 , 1 ;7 .7 5) (Aj7 8), the experimental levels15 6 3 18of unknown spin are obtained from the N( Li, He) 0 reac

tion. Peaks at ^0 (7.10,11.67,12.53) in Fig. 1 . 2 are asso

ciated with positive-parity states and are omitted from the18 *present diagram, although 0 (1 1 .6 7) may be related in part

18 *to the 6 prediction. Normalization at 0 (A.A5 6 , 1 )

determines the absolute excitation energies of shell-model

levels.

18Figure 7-6 SU(3) s he l l model of 0

)1 8 -

3‘ 4 *

C L U S T E R

v /tT180

r 3 io

I 1.10

S H E LL ( M i7 7 )

8.47 8.10 5 ' 7.84

6.882 0 6.404 3'

5.530 2' 5.098 3'

* ---- 4.456 T

EXPER IM EN Ti / i - l --------------- L--------------L-js^

0.30 0.15 0 ^S PE C TR O SC O PIC FACTOR

example, favors ^80 (5 *0 9 8 , 3 ) over ^8 0 (5 *5 3 0 , 2 ).

The SU(3) shell model predicts, rather than postulates, weak-coup-

1ing effects in triton-c1 us ter structure. In agreement with a 9/2+®l/2

doublet from the folded-potential model, a A^ level is calculated near18the 8.A7 MeV state of 0 (Fig. 7*6). Further theoretical evidence of

weak coupling exists in a A./3 ia doublet and a 6Y/5 lr pair, although ab 10 / I phigher level density reduces the concentration of spectroscopic strength

in these unaligned angu1ar-momentum couplings. Mixing between doublets

is found to be minor in the shell-model states of Fig. 7*6, except for

comparable 5/2 <S>1 /2 and 3/2 ®l/2 components in both 2 and 2y Despite

a basis including (sd) (fp) configurations, the admixture into 3p-lh struc

ture is predicted to be small, aside from a 1 1% component in the 6 state.

For this exception, constructive addition occurs for the spectroscopic

amplitudes of a p(sd)fp and a (sd) triton cluster. The expected cross

section is observed at ^®0“(lA.6l) (Fig. 7 *2), but (13*79) is closerto the predicted excitation energy. Overall, the SU(3) shell model of

18negative-parity states in 0 confirms the theoretical framework of tri

ton clustering from the folded-potential model and corresponds to the6 3experimental results of triton transfer from the ( Li, He) reaction. In

detail, calculated triton spectroscopic factors reveal division of clus

ter strength and favor alignment of angular momenta, but they remain

influenced by weak coupling.C. q _ 1 qIn summary, the ( Li, He) reaction identifies major p (sd) struc

ture at l80 * ( 5 . 0 9 8 , 3 ’ ; 7 . 8 A ; 8 . 1 0 , 5 ~ ; 1 1 . 1 0 ; l A . 6 l ; 1 5 * 9 5 ) . The folded-potential model and the SU(3) shell model indicate substantial triton cluster

ing in these levels. Three-nucleon transfer data are more complex for

8F, where T=0 states and (sd)(fp) configurations play an important role.

18 -*• +As an extension of the doublet structure in 0, j®1 triplets

are expected in 7 0 , where a triton cluster has total angular momentum "j

and an unexcited ^N core has spin one and positive parity (Table 1.1).7T +Mixing becomes probable among these triplets since, for example, J =3/2

+ 4*can originate from j=l/2 , 5/2 or 3/2 (Fig. 6 .6). Strong-coupling18effects, evidenced in 4p-2h configurations of F (Ro73b), may also re

duce the influence of weak coupling in 3p“2h structure. Such theoretical

complexity is beyond the scope of a folded-potential model of triton clustering. If spectroscopic strength is enhanced for aligned angular

momenta, however, relative simplicity should emerge in experimental

spectra.

8.1 ]Z|N(6Li ,t)17F and 1 N (6L i , 3He) 1 70

Three-nucleon transfer into the A=17 nuclei is dominated by three

states at 8.5, 10.7 and 14.9 MeV (Figs. 8.1, 8.2). Their excitation ener

gies suggest a relation to 1V (5.270/9.1 55,5/2+; 1 0.693,9/2+; 1 5.*»1 , (1 3/2+))

(Fig. 5-5, Ha76b). Their relative population at E|j=46 MeV, consistent

with results for E < 14 MeV at E | = 30 MeV (Ba72, see also B i 7 3a), follows^ 4 " 19 19the behavior of the first 5/2 , 9/2 and 13/2 states of Ne or F

14(Figs. 6.1, 6.2). Because the pair of p-shell holes in a N target has

spin 1+, coupling to these (sd)9 states of the A=19 nuclei is expected to

generate nine levels in 7F or 70. A preference for one member of each

triplet would then account for the observation of three leading peaks.

Their absolute, differential cross sections reflect such an enhancement;

e.g. F (1 0.7 1 ) carries 7 0% of the total strength present in

^ e (2.80,9/2*) (Tables 8.1, 6.1). If 7F (8.43) arisesfrcm a 5/2+®l+68

CHAPTER 8 A=17

Figure 8 .

1 |1*N(6Li ,t)l7F

2 ,i*N(6Li ,3He)'70

Excitation energies of the A=17 nuclei are determined from

known levels of the A=19 and A=15 nuclei, also observed at

El|=A6 MeV and 0lab=15o. The 160(6Li,t)19Ne and

^C(^Li,t) ^ 0 reactions provide calibrations which agree

within 15 keV for the energy levels of ^F. Since the low-

excitation region of ^ 0 is beyond the Q-value of 16 6 3 190 ( L i /He) F (g. s .) , an internal calibration for E < 6 MeV

supplements the independent energies at higher excitation

(see Table 8.1). The peaks at ^F (8 .A3) and ^0 (8 .A8) are both nearest in energy to 7/2+ states> known at 8.A16 MeV

±10 keV and 8 .A7A MeV ±3 keV respectively (Aj77).

EXCITATION ENERGY (M eV)

EXCITATION ENERGY

COUNTS

coupling, moreover, its spin assignment of JTI=7/2+ illustrates the align-18ment of angular momenta. As in 0 (Section 7-3), low level density

would then favor a large spectroscopic factor for this state and a small

admixture of j=9/2+ or 7/2+ structure. Further evidence concerning trans

ferred angular momenta and 3p_2h configurations is found in a comparison 6 6 3of the ( Li,t) and ( Li, He) reactions with various multi-nucleon transfer

data for the A=17 nuclei.

8.2 Other Transfer Reactions

In the (a,p) reaction at E =34 MeV (Va75) and in the ,7 Be)areaction at E =100 MeV (Ha76b), the most strongly populated state is B1 7 0 (14.9). Since a 1 3/2+ state has this property in ^F and tentatively

in (Sections 5-2, 6 .2), a spin of 13/2 ®1 =15/2 in 70 is proposed

by these authors. The sizeable peak at 10.7 MeV could similarly arise

from a 9/2+&> 1+=11/2+ state (Ha76b). In correspondence to the minor cross

section of 80(^B,7 Be)^F (0.20,5/2+) (Ha76a,c), the known 7/2+ state at

8.474 MeV is weakly populated by the ( B,7 Be)^70 reaction. This indi

cation of relatively low angu1ar-momentum transfer (see Section 5.2) con

firms the existence of a L=2, 5/2+<®l+ stretched coupling. From the

dynamic differences between three-nucleon transfer reactions, therefore,

and from the similarities between A=17 data and results for A=19 and

A=15, we conclude that the three dominant peaks of Fig. 8.1 represent

probable L=2, 4 and 6 members of a 3p~2h band. In support of this inter-17 14pretation, the folded-potential model for a 2N+L=6 band of 0= N+t

predicts a moment of inertia which is consistent to 20% with the experi

mental L(L+1) spacing.

Negative-parity states of 70 are found in two- and four-nucleon

transfer data. The (a,d) reaction (Lu69) identifies 2p-lh configura

tions at 7(T (7 -7 5 ,1 1/2" ;9 -1 5,9/2”), which have little strength in the ( Li,^He) reaction and appear analogous to minor peaks at ^N(^Li,t)^7F

(7-97,9-40) (Figs. 8.1, 8.2). This contrast to the major role of a2p-3h, 11/2 state in the ^C(^Li ,3 He)^N reaction (Fig. 5-2) illustrates

2that p(sd) transfer is dependent upon the number of holes available in

the p shell. A probable negat1ve-parity state of 70 which has different

structure, however, proves important in three-nucleon transfer data. The

dominant state in a 3 C (7 L i , t) 7 0 spectrum (Fig. 8.3) also has signifi

cant cross section at ( L i ,3 He) 7 0' (13 • 53) , a unique behavior suggesting p (sd) fp admixture into a p ^(sd) configuration (see also

Section 7.2). In the case of the three candidates for positive parity

at 7(T (8 .48,1 0.7 0 ,14.8 9), a sharp reduction of relative strength in alpha-particle transfer (Fig. 8.4) and a near absence of yield in two-

nucleon transfer (Lu69) constitute further evidence of their largely- 2 3p (sd) configurations.

The identification of this 3p~2h structure in ^0 leads to an

interpretation of the ( Li,t) reaction. A population of the above

states at ^0 (10.70,14.89) (Fig. 8.4c) can be attributed to a mechanism3 3of p(sd) alpha-particle transfer and/or a configuration with (sd) fp

admixture. Despite a primary role in J7T=5 and 7 structure of ^0(see Section 9-2), alpha-particle clustering of either type appears

secondary in these suggested ll/2+ and 15/2+ states of 70 (Fig. 8.4b).

The other prominent peaks in 3 C(7 Li,t)^70 data (Fig. 8.4c) represent - 3 \ 4candidates for p (sd) c on f i g ur a t i o ns , beginning with a 7/2 s t a t e at

Figure 8.

Figure 8 .

3 13C (7L i,t)170

For these data measured at E, .=40 MeV and 0, =10°, energyLi labcalibrations are based on the 2C(^Li,t)^0 reaction. Owing

to a small discrepancy in amplifier gain between the E and

E signals (Fig. 2.2), separate calibrations are required

above and below an excitation energy of 10 MeV (see Table 8.1).12A contribution from C impurity in the target to the

l3C(7 Li,t)170 spectrum is eliminated by a subtraction of the

calibration spectrum, weighted according to yields for

1 6 0“(g.s.,6 .1).

4 Comparison

Three-nucleon transfer via the ( Li,t) and ( Li.^He) reactions

demonstrates the mirror relationship between 7F and 7 0 .

Below, alpha-particle transfer is observed at a similar inci

dent energy but at a more forward angle. This (7 Li,t)

spectrum for ^7 0 includes the ^ 0 contamination subtracted

in Fig. 8.3.

EXCITATION

— 12

l3C (7L i , t ) l70 E L i= 4 0 MeV

0Lotf'O°

O '6 * subtracted

cviH n

ENERGY (M e V )

TABLE 8.1 17 73UN(*U,t)17F 1S*(#L1SHe)170 lv 17LI, t) O

EU-46MSV eli*40 MaV 17o(AJ77) W 160 •lab*= 10° (AJ77)

EX E (1) X dcVdfi (2) c. xn. E (3)X do/dO <*) c.m. E (3)X do'df) (4)0* ID. EX J*(MeV) (MeV) (pb/ar) (MeV) (jib/a r) (MeV) (pb/ar) (14s V)

5/2* g.s. g.a. g.a. g.a. g.a. 5/2*1/2* 0.495 0.6 0.9 0.85 0.871 1/2*1/2- 3.104 3.1 3.1 3.03 3.055 1/2-5/2* 3.857 3.85 24 3.83 3.82 187 3.841 5/2*3/2" 4.696 4.6 4.56 4.53 4.553 3/2-(»/2) 5.212 5.18 22 5.21 5.18 5.215 (9/2"]7/2" 5.672 5.65 5.69 5.69 5.698 7/2*3/2* 6.774 6.755/2' 7.0*7 7.037.49 54 7.4 7.39 7.3837.386

7.6905/2*5/2-7/2-7.97 33 7.74 7.70 7.758.410 11/2-5/2*1/2+ 8.416

etc.8.43 78 8.48 97 8.50 8.474

8.5088.90

7/2*5/2"3/2*8.89 37 8.92 34 8.95 8.972 7/2-

9.40 34 9.18 34 9.2 9.159.187 9/2”7/2-9.87 9.89 9.87 9.865

9.87810.42 10.44 56 etc.10.71 104 10.70 105 10.71 58011.13 11.23 78

11.6011.81 36711.89 66 12.02 12.00 379

12.31 12.313.01

12.41 65013.51 13.53 96 13.55 301014.17 14.22 14.214.84 151 14.89 192 14.90 <80015.5

15.8616.216.5117.1617.617.96

17.8718.1519.24

<7001040

19.9 20.2 20.0

(1) calibrated from 19Ne*(0.238, 2.794, 5.43)consistent with AE - A 20 keV

15’o (5.241, 7.276, 10.45, 12.835, 15.05)

(2) * (1% - 4%), statistical ~ a 10%, absolute

calibrated from 170*(g.s., 3.841, 8.474), E < 6 MeV

AEfrom

A 20 keV19F (0.197, 2.780, 4.648, 6.925, 8.953, 10.411), E > 6 MeV

(4) a(l%-7%», atetUUcal ~ a 10%, absolute

(5) calibrated from 160*(g.a.. 8.917, 10.353), E < 10 MeVfrom l#O*(10.353, 14.815, 16.29),E > 10 MeV

AE “ * 20 ksV, E < 16 MeV X*a 40 ksV, E > 16 MeV

8.972 MeV (Aj77). The first 1/2", 5/2” and 3/2" states of 170 (Fig. 8.3)

are also observed in alpha-particle transfer, indicating 4p-3h components

(Be70, Go71)» but shell-model calculations depict strong mixing with

2p-lh configurations (see Le72). Using theoretical states with unmixed

4p-3h structure for L=0 and L=2 (El 70), we find that the experimental

doublets have a L(L+l) spacing for L=4, 6 and 8 (Fig. 8.5). This evi

dence reveals a tentative 2N+L=8, a 1pha-cluster band of 7 0.

A comparison can be made to the 4p-4h, 0+ band of selectively

populated by the 2C (7 Li,t)^80 reaction (Co76). Weak coupling of a

P| ^ 2 Part'c e to tbe state at ^CT(16.29) would generate a 1 1 / 2 -1 3 / 2

doublet, in accordance with peaks at 70 (12.41,13.55) (Fig. 8.4c). In

the (7 Li,t) reaction at E^.=40 M e V and ©jgb®^0, *be absolute differential

cross section of the 13-55 M e V state equals 40% that of (16.29); in

the (8 Li,d) reaction at E|_j~3*4 their angular distributions are consistent (C178). Good candidates for the 8*®l/2 coupling appear at

,70 (18.15,19.24). The same 1 M e V splitting, moreover, characterizes

the effect of a Pj/ 2 neutron on two-particle tructure, i.e. a 2p-3h

doublet at (1 1.9 5 ,(9 / 2 ) ; 13.0 0 ,1 1 / 2 ) arises from the (p^/2 ^ ^ 5/2 ^1 4 * +configuration at N (8.961,5 ) (Lu69, Ri6 6). Although confirmation

requires more spin assignments, we conclude that present evidence of-3 4 17p (sd) structure in 0 exhibits the influence of weak coupling. In

contrast to the nearly equal excitation energies of 4p-lh and 4p-0h

states (Fig. 6.5), however, an increase in the moment of inertia accom

panies the formation of 4p-3h doublets (Fig. 8.5). A Pj/ 2 Particle also appears to generate greater splitting than does a Pj/ 2 bole. Corresponding phenomena may affect 3p~3h configurations described in the next chapter.

Figure 8.5 Weak coupling in 4p-3h configurations

3 ^In this plot of vs. L(L+1), the candidates for p (sd)

configurations (solid points) are obtained from ^C( 7 l_i,t) ^ 7 0

(8.972,7/2~;9.87;12.41;13.55;18.15;19.24) (Fig. 8.4c). The

positions of unmixed 4p-3h states with J 1 / 2 , 5 / 2 and 3 / 2

are calculated from the weak-coupling shell model (El70). A-4 4comparison with p (sd) , a 1pha-c1uster structure (open

points) is provided by (7 Li,t) 8 0 '(6 .049,0+;6 .9 1 7 ,2+;

10.353, V; 16.29,6+) (Co76, Aj77) and by 12C ('2C,8Be) ,60"

(22.5,(8+))(aQ)12C (Sa77).

1 7 * +In summary, triton-transfer reactions classify 0 (8.A7A,7/2_ 2 310.70;lA.89) as probable members of a p (sd) band. Alpha-particle

transfer data for ^ 0 lead us to expect a similarity between triton-13 12transfer spectra from and C targets.

As in the A=l8 nuclei, three-nucleon transfer into T«1 states of16 16N draws an analogy to the spectrum of 0, which contains both T=1

and T=0 states (Table 1.1). Since the ground state of 3C has J71 = 1/2 ,

triton clustering in involves the same theoretical spins and pari-18 15ties as 0= N+t (Fig. 7-5). A lack of experimental spin assignments

in ^N, however, prevents normalization of the folded-potential model

for 3p-3h configurations. Evidence of a relationship to 3p“Ah states

of (Table 1.1) exists despite the effect of a Pj/ 2 va ence neutron.

9.1 1 3C (6L i ,t)l60 and 1 3C (6L i ,3 He)16N

The dominant peaks at ^0 (20.49) and (7.65) in three-nucleon

transfer spectra (Figs. 9.1, 9.2) represent T=1 analog states with Tz=0

and Tz=l respectively, for their difference in excitation energy agrees with the separation between ^ 0 (12.969,2 ,T=1) (A j 77) and ^N(g.s.,2 ).

Further T=1 assignments can be made for levels at 2 2.A6 , 23.9A and 2A.63

MeV in ^0, which correspond in relative cross section to those at 9.68/

9.81, 11.21 and 11.81 MeV in (Fig. 9-Aa,b). At lower excitation, an

analog relationship between the J7T=(;+) states (Ja77) at ^0 (16.8l) and

(3 .9 7) is indicated by an energy difference equal to that of the

dominant peaks and by a population of the 17.1A MeV state in the

^N(a,d)^0 reaction (Zi70) , which prefers T=0 states. In addition to16 'c 16 * * a correspondence between 0 (18.01,J=3) and N (5-15,(2,3) ) (Ch77,

Ma78), there are two candidates at 18.AA/18.61 MeV in ^0 for the analog

of a 5+ state at 5.73 MeV in ^N. A comparison of angular distributions 6 6 3from the ( Li,t) and ( Li, He) reactions favors the upper member of the

CHAPTER 9 A=16

Figure 9-

Figure 9*

1 1 3C(6Li,t)l60

2 13C (6L i,3He)l6N

For these A=l6 spectra at E. .=44 MeV and 0. ,=10°, the ( Li,t)Li lab6 3 12and ( Li, He) reactions on a C target provide a calibra

tion of excitation energies and an identification of contami

nant peaks (see Table 9-1). Since (5-241) is equivalent

to 1 6CT(16.1) in Q-value, an extrapolation to the low-excita-

tion region of ^ 0 is replaced by known energy levels below

9 MeV (Aj77) - The J7T=(3+) and 5+ assignments in are ob

tained from Refs. Aj77 and Lu69 respectively. The peak cen

tered at (5.15) is analyzed as a multiplet in Table 9*1.

COUNTS

EXCITATION

ENERGY

COUNTS

doublet and supports the T=1 character of 80'(16.81,20.A9,2A.63) (see18Fig. 9.6). In overall contrast to F (Section 7-1), a shift in excita

tion energy and an enhancement in cross section lead to the unambiguous

identification of several T=1 states of unknown spin above E =20 MeVx

9.2 Other Transfer Reactions

Three-nucleon transfer into T=0 states of ^0 is interpreted within

the context of different mu 11i-nuc1 eon transfer data. In the low-excita-

tion region of the (8Li,t) spectrum (Fig. 9.1), which is consistent with

results at lower incident energy (Ba69,70,71 a, 0g70), the most prominent

states are 80 (11.095,A+ ;1A.A0;1A.815,6+;16.2 k). Since the (a,d) reac

tion (Ba70, Z i 70) identifies them as primarily 2p-2h configurations,9 Ap(sd) transfer appears to be important for the ( Li,t) reaction on a

'3C target. Relatively little cross section (Fig. 9.1) is found in the

positive-parity, T=0 states of Ap-Ah character at 80 (10.353,A+ ;16.29,6+).

The broad, negative-parity states at 80 (1A.59,5 ;20.9,7 ) (e.g. Sal? ) ,

which are strongly populated by the ( Li,t) reaction (Co?6), may have

significant cross section in three-nucleon transfer (Table 9-1). What-" 3 3 "A 3ever their mixture of p (sd) and p (sd) fp configurations, alpha-

particle clustering is favored in such T=0 states of an even-even AN20nucleus, an effect most evident in Ne (Appendix B). Three-nucleon

clustering is expected to develop more highly in T=1 states of ^0, which

indeed dominate the ( Li,t) spectrum. We focus, therefore, on the anal-16ogous T=1 states of N.

13 6 3 16The rather simple spectrum from the C( Li, He) N reaction

(Fig. 9.2) is further clarified by an identification of positive-parity

is confirmed to have a primarily 2p-2h configuration, and the 5.73 MeV-2 2state is assigned /£ 0+ ^ 5 / 2 5+ struct:ure (Au69) . This L=5 level of

is strongly populated by the (^Li,8He) reaction but not by ( Li,a)

(Fig. 9-3), where angular momenta are well matched (Section 5.2). A

corresponding reduction in cross section at ^C(^Li,a)^N (1 1.2 1)

(Fig. 9.A) may have similar origin. Although peaks at Ex=5-1A, 6.59 and

7.65 MeV in the ^N(^B,®B)^N reaction suggest a (p^^) ]+^5/2 A+ triplet (Ha78) , the strength of (5. 15» 7.65) in the (7 Li,a) reaction

indicates an observation of primarily different states in triton transfer.

The presence of known doublets at (5.1 30/5. 1 50,7.637/7.675) (Aj77)16 “ t? 6 3and the absence of N' (6.59) from the ( Li, He) spectrum support this

view. We are left, therefore, with four good candidates for p 3 (sd) 3

configurations, selected by both triton-transfer reactions (Fig. 9 .A)

at ,6N*(5 .15,7.65,9.81,11.81).An interpretation of these states of N emerges from experimental

1 0 irevidence of the correspondence to better known levels of 0 and PN.

Triton clustering outside a spin 1/2” core is expected to lead to a18 18similarity between N and 0 (Table 1.1, Fig. 7.5). In addition to

known 1 states at A. 3 8 7 MeV in and at A.A56 MeV in ^0, a 3 state

at 5.15 MeV in (Ma78) appears related to (5.098,3 ) (Figs. 9.A,7.A). With respect to excitation energy and relative population in

(8Li,8He) and (7Li,a) data, this tentative correspondence can be extended

to 16N*(7.65) and 180*(7.8A,(5");8.10,5') and to 16N*(11.81) and1 Q _0** (1 1 . 1 0, (7~ )) (Fig. 10.1), where spin values in parentheses are sug-

16gested by the SU(3) shell model (Fig. 7.6). A similarity between N and

80l e v e l s . In the ^ C ( a , d ) ^ 8 N react ion , the 3.98 MeV s t a t e with spin (3+ )

Figure 9-3 3C(7Li,a)^8N

The energy calibration and contaminant peaks are determined

from 2C(7Li,a)^3N data, also measured at E, .=40 MeV andL i

0lab=1O°‘

Figure 9-4 Comparison

Three-nucleon transfer into the A=l6 nuclei proceeds via the

(8Li,t), (8Li,3He) and ( 7Li,a) reactions at similar incident

energies. The laboratory angle of 10° represents a change

from 0 =15° for A=15, 19, 18 and 17.1 ab

EXCITATION ENERGY

C O U N T S

TABLE 9.1 A - I S13 6 18 13- 6 3 16 13_,7., 16mC( u,n o C( Li, He) N C( Li, a) N

ieo Eu - 44 MeV Eu - 40 MeV l#N

•l* - 10° ®lab * l°°M J* EX X X ^ . m 4> Ex(5> EX J* Ref.

(MeV) (MeV) (yb/sr) (MeV) (j4>/sr) (MeV) (j*>/er) (MeV)AJ77 0+

0+ 6.049 } 883- 6.1302* 6.917 521- 7.1172- 8.872 8.90 402+ 9.847 9.854+ 10.353 10.35 603+ 11.0804+ 11.095 11.09 1042* 11.521 11.51 843" 11.600+ 12.053 12.041- 12.442 12.472" 12.530

T=1,0" 12.797 0.120 0- AJ77T=l,2- 12.969 g.a. 2-T=1,1” 13.094 0.398 1-

3“ 13.129 13.11T*l,3- 13.254 0.297 3*

2- 13.97914.3014.40

13.9614.32

6+ 14.815etc.

14.7915.16

<240Z170 15.80

16.2415.7916.22

3.383.53

3.3553.519

1+(2+)

3*77 16.82 16.81 102 3.97 157 3.96 275 3.960 («)+Z170 17.17 17.14 100AJ77 T*l,l“ 17.29

17.8 5.054.40 4.387

5.0501-2"

Cb77 I 390 5.130J-3 18.02 18.01

18.44174108

5.155.25

5.13 1180 5.1505.232

(2.3)-(2,S)+

18.61 134 5.736.17

201 5.7306.168

5+ Lu69 (4)" Ma78

19.320.49 564 7.65 972 7.65 2317 7.637

7.675AJ77

AJ77 7- 20.9 20.9

8.859.689.81

63 358

9.779.769.8123.94 74 11.21 222 11.22 etc.24.4

24.63 201 11.8112.47

11.8112.45

26. 213.6914.43

102 13.7014.51

(1) calibrated from 150*(5. 241, 7.276, 10.45, 12.835, 15.05) AE =■ ± 30 keV

(2) ± (2% - 6%), statistical ~ ± 10%, absolute

(3) calibrated from 15N*(5. 270, 7.567, 10.693, 13.02, 15.41) AE - ± 20 keV

(4) ± (1% - 5%), statistical ~ ± 10%, absolute

(5) calibrated from 15N*(5.270, 8.571, 12.55, 15.40)AE “ ± 40 keV

<6) ± (1% - 5%), statistical ~ ± 20%, absolute

N w o u l d a r i s e f r o m w e a k c o u p l i n g o f a P j p a r t i c l e t o 3 p —Ah s t a t e s

( T a b l e 1 . 1) . B o t h 1 n " ( 1 1 . 8 1 ) a n d ' 5 n“ ( 1 5 . A l , ( 13 / 2+ ) ) ( F i g s . 9 . A , 5 . A)

16 «a r e w i t h i n 1 MeV o f t h e t r i t o n t h r e s h o l d ( B u 7 6 ) ; b o t h N ( 7 . 6 5 ) a nd

( 1 0 . 6 9 3 , 9 / 2+ ) h a v e a b s o l u t e d i f f e r e n t i a l c r o s s s e c t i o n s o f 1 m b / s r

i n t h e ( ^ L i , ^ H e ) r e a c t i o n a t £^. =44 MeV a nd ( F ' 9 * 1 0 . 1 ) . As

i n a 1p h a - c 1 us t e r s t r u c t u r e ( F i g . 8 . 5 ) , a s m a l l e r l e v e l s p a c i n g f o r t h e

13c a s e o f a C c o r e i s i m p l i e d by s u c h a c o m p a r i s o n o f t r i t o n - c 1u s t e r

s t a t e s . I n c o n t r a s t t o t h e ^ C ( 7 L i , t ) ^ 0 r e a c t i o n , h o w e v e r , d o u b l e t s

w i t h 1 MeV s p l i t t i n g a r e n o t f o u n d i n t h e ^ C L i He ) r e a c t i o n ,

w h e r e a 2 /3 p a i r a t 5 . 0 5 / 5 - 1 5 MeV p r o v i d e s t h e o n l y a v a i l a b l e e v i d e n c e

o f d o u b l e t s t r u c t u r e . * An a l t e r n a t i v e t o t h e a b o v e i n t e r p r e t a t i o n

w o u l d a s s o c i a t e ( 5 - 1 3 , 7 - 6 5 ) w i t h t h e 5 a n d 7 s t a t e s r e s p e c t i v e l y .

T h e p r e s e n t c h o i c e o f t h e 7 - 6 5 MeV a nd 1 1 . 8 l MeV s t a t e s , h o w e v e r , i s mo r e

18 15c o n s i s t e n t w i t h t r i t o n - t r a n s f e r d a t a f o r 0 a n d N. F u r t h e r d i s c u s s i o n

o f s y s t e m a t i c b e h a v i o r i n p ( s d ) c o n f i g u r a t i o n s o f t h e A= 15 t o A= 19

n u c l e i i s p r e s e n t e d i n S e c t i o n 1 0 . 1 .

6 3I n s u m m a r y , t h e ( L i , He ) r e a c t i o n , t o g e t h e r w i t h o t h e r t r a n s f e r

16 _ 3 3r e a c t i o n s i n t o N , i d e n t i f i e s c a n d i d a t e s f o r p ( s d ) c o n f i g u r a t i o n s a t

( 5 . 1 5 , 7 . 6 5 , 9 - 8 1 , 1 1 . 8 ] ) . T h e ( ^ L i , t ) r e a c t i o n a s s i g n s T = 1 , T z =0

a n a l o g s t r u c t u r e t o ^ 0 ( 1 8 . 0 1 , 2 0 . 4 9 , 2 2 . 4 6 , 2 4 . 6 3 ) .

9 - 3 A n g u l a r D i s t r i b u t i o n s

F o r a g i v e n f i n a l s t a t e , a n a n g u l a r d i s t r i b u t i o n i s r e l e v a n t t o t h e

r e a c t i o n m e c h a n i s m , a n g u 1 a r - m o m e n t u m t r a n s f e r , s p e c t r o s c o p i c f a c t o r and

a n a l o g a s s i g n m e n t . A p r e d o m i n a n t l y d i r e c t m e c h a n i s m i s c o n s i s t e n t l y i n

d i c a t e d by t h e p r e v i o u s m e a s u r e m e n t s o f a n g u l a r d i s t r i b u t i o n s i n t h e

6 6 3( L i , t ) a n d ( L i , He ) r e a c t i o n s ( s e e S e c t i o n 3 . 2 ) . A n g u 1a r - m o m e n t u m

t r a n s f e r i n t o t h e A=15 n u c l e i ( B i 7 5 ) i s f o u n d t o be a m b i g u o u s i n a n a n a l y

s i s o f s u c h d a t a . T h e s t r u c t u r e l e s s b e h a v i o r o f t h e s e a n g u l a r d i s t r i b u

t i o n s , m o r e o v e r , i m p l i e s t h a t t h e f o r w a r d - a n g 1e s p e c t r a a r e s u f f i c i e n t

f o r q u a l i t a t i v e i n f o r m a t i o n on r e l a t i v e s p e c t r o s c o p i c s t r e n g t h s . A l t h o u g h

a n a l o g a s s i g n m e n t s c a n a l s o be d e d u c e d f r o m t h e m i r r o r s p e c t r a o f T = ± 1/2

n u c l e i ( F i g s . 5 - 4 , 6 . 4 , 8 . 4 ) , t h e p r e s e n c e o f b o t h T =0 a n d T =1 s t a t e s i n

16 180 a n d F ( T a b l e 1 . 1) d e m o n s t r a t e s a n e e d f o r f u r t h e r e x p e r i m e n t a l e v i -

16 18d e n c e o f t h e c o r r e s p o n d e n c e t o N a n d 0 r e s p e c t i v e l y . T h e h i g h l e v e l

18d e n s i t y o f F ( F i g . 7 - 4a ) w o u l d h i n d e r an e x t r a c t i o n o f a n g u l a r d i s t r i

b u t i o n s , b u t t h e T =1 c a n d i d a t e s o f r e m a i n r e l a t i v e l y d i s t i n g u i s h a b l e

a t l a r g e a n g l e s o f o b s e r v a t i o n ( F i g . 9 - 5 ) . I n t h i s s e c t i o n , t h e r e f o r e ,

we i n v e s t i g a t e a n g u l a r d i s t r i b u t i o n s o n l y i n t h e ^3 C ( 8 L i , t ) ^ 8 0 a nd

^3 C ( 8 L i , 3 H e ) ^ 8 N r e a c t i o n s .

16 16 A T = 1 , ? z=0 s t a t e o f 0 , w h i c h i s a n a l o g o u s t o a T z = l s t a t e o f N,

s h o u l d h a v e an a n g u l a r d i s t r i b u t i o n o f s i m i l a r s h a p e ( s e e Ga 7 2 , B i 7 5 ) a n d

o f r e d u c e d m a g n i t u d e . A c h a n g e by a f a c t o r o f 2. 1 i n c r o s s s e c t i o n i s

i m p l i e d by t h e e x p r e s s i o n ( e . g . C e 6 4 , G a 7 3 )

^ r- a k J < T AT T nT I t T > | 2 , ( 9 . 1)dft b A c zA zc B zB 1w h e r e t h e r e a c t i o n A ( a , b ) B t r a n s f e r s c = a - b . T h e s e t w o p r o p e r t i e s a r e

e x h i b i t e d i n F i g . 9 . 6 by ( 7 . 6 5 ) a n d ( 2 0 . 4 9 ) a n d by N ( 1 1 . 8 1 )

a n d ^ 0 ( 2 4 . 6 3 ) , c o n f i r m i n g t h e i r a n a l o g r e l a t i o n s h i p . E x p e r i m e n t a l

v a l u e s o f 1 . 7 a n d 2. 1 r e s p e c t i v e l y a r e o b t a i n e d f o r t h e r a t i o o f t h e

16 '*c 16 t o T =0 s t a t e s a t 0 ( 8 . 8 7 2 , 1 0 . 3 5 3 ) , t h e 16. 81 MeV s t a t e o f 0 c o r r e

s p o n d s c l o s e l y i n a n g u l a r d i s t r i b u t i o n t o t h e 3* 96 MeV s t a t e o f as

e x p e c t e d f r o m S e c t i o n 9 . 1 . T h e c o m p a r i s o n o f ^ ( 5 . 7 3 , 5+ ) t o b o t h

Figure 9-5 G]ab =2+5

T h i s ' 3 C t6 L i ,3 H e ) 1 € *N s p e c t r u m i s c a l i b r a t e d i n e n e r g y f r o m

t h e ^ C (^ L i ,3 H e ) ^ N r e a c t i o n a t 0 ] a b = ^ 5 o . An a n a l o g o u s p r o

c e d u r e a p p l i e s t o ^3 C ( ^ L i , t ) ^ 0 d a t a f o r Ex > 16 Me V , b u t l o w

e x c i t a t i o n e n e r g i e s a r e d e t e r m i n e d f r o m t h e ^ C ( ^ L i , t ) ^ 0

r e a c t i o n a t 0, = 1 0 ° .l a b

F i g u r e 9 . 6 A n g u l a r d i s t r i b u t i o n s

13 , 6 . 1 6D i f f e r e n t i a l c r o s s s e c t i o n s a r e p l o t t e d f o r t h e C( L i , t ) 0

13 6 3 16r e a c t i o n ( o p e n p o i n t s ) a n d t h e C( L i , He ) N r e a c t i o n

( s o l i d p o i n t s ) a t E ^ . = A A MeV a n d 0j a b = l 0 ° , 15 ° , • • • » 6 0 ° . S i n c e

16 * 15 *0 ( 2 3 . 9 A) i s o b s c u r e d by c o n t a m i n a t i o n f r o m 0 ( 1 2 . 8 A) a t

15 ° £ 0 , , ^ A 0 ° , t h e a n g u l a r d i s t r i b u t i o n o f ( 1 1 . 2 1 ) a p p e a r s1 ab

w i t h o u t a n a l o g o u s d a t a f o r ^ 0 ( F i g . 9 - A ) . P o i n t s a r e o m i t

t e d f r o m t h e a n g u l a r d i s t r i b u t i o n o f ^ 0 ( 1 8 . AA) ( F i g . 9 * 1)

a t a n g l e s w h e r e t h i s s t a t e i s n o t r e s o l v e d f r o m ^ 0 ( 18 . 6 1 )

( F i g . 9 - 5 ) .

(MeV) 15

COUNTS

C O U N T S

roOo(M8

d<r/dncm (16 81,3 96) (pb/sr) do-/dnc m (8.872) {pb/sr)

d c r / d n c m (II 21) ( / j.b/sr) d < r /d n cm (18.44 ,5 .73 ) ( ju .b /sr)

d c r / d n cm (20 49 , 765 ) ( pb /s r )

coO3 +

IO 1 m '- n

<T»o* I •*(T>OJ G

•• oA

••

•• m ojo

cr»OJo

a <T»° * * OJ r*

o • 21 • • z* Zo Xi*

O • <*z O

® • cno * — , •>o ♦ ♦ooN

d c r /d n cm(24 63 , l l 81) (pb/sr)

me mbe r s o f t h e 1 8 . A A / 18. 61 MeV d o u b l e t i n ^ 0 i l l u s t r a t e s a p o t e n t i a l l y

d e c i s i v e t e s t o f a n a l o g a s s i g n m e n t s . T h e a n g u l a r d i s t r i b u t i o n o f

^ 0 'f ( l 8 . 6 l ) , t h o u g h a l i t t l e h i g h i n m a g n i t u d e , b e t t e r a p p r o x i m a t e s t h e

c u r v e o f d a t a . O v e r a l l , f o r t h e ( ^ L i , t ) a n d ( ^ L i , ^ H e ) r e a c t i o n s ,

t h e m e a s u r e m e n t o f a n g u l a r d i s t r i b u t i o n s p r o v e s t o be a w o r t h w h i l e

s o u r c e o f e x p e r i m e n t a l i n f o r m a t i o n on a n a l o g s t a t e s .

A p r e d i c t i o n o f t h e c o m p o u n d - n u c 1e u s c o n t r i b u t i o n t o t h e s e a n g u l a r

d i s t r i b u t i o n s p r o v i d e s f u r t h e r s u p p o r t f o r t h e c o n c l u s i o n ( S e c t i o n 3 - 2 )

t h a t t h e t w o r e a c t i o n s p r o c e e d p r i m a r i l y v i a a d i r e c t m e c h a n i s m . T h e

H a u s e r - F e s h b a c h mo d e l o f a s t a t i s t i c a l p r o c e s s g i v e s t h e e n e r g y - a v e r a g e d ,

d i f f e r e n t i a l c r o s s s e c t i o n by ( F e 6 0 , V0 6 A, S t 7 2 )

S - ■ £ k ^ . r r ^ h r m r l < r f -d 1 (9 .2 )

Z ( £ j £ j ; s L ) Z ( £ ‘j £' j ; s'L) ( - l ) S S P L ( c o s 0 c >m>) ,

w h e r e a r e p r e s e n t s t h e e n t r a n c e c h a n n e l , a 1 t h e o b s e r v e d e x i t c h a n n e l and

a 11 a n y r e l e v a n t o u t g o i n g c h a n n e l w i t h q u a n t u m n u m b e r s c " . L e t t i n g i a nd

I be t h e s p i n v a l u e s o f t h e p r o j e c t i l e a n d t a r g e t , we h a v e s = i + I a n d

j = £ , + s . T h e c e n t r a l q u a n t i t i e s a r e t h e t r a n s m i s s i o n c o e f f i c i e n t s T , t h e

Z - c o e f f i c i e n t s ( F e 6 0 ) a n d t h e L e g e n d r e p o l y n o m i a l s P . U s i n g t h e c o d e •

S T A T I S ( S t 7 2 ) , we c a l c u l a t e t h e f o r m a t i o n o f a co mp ou n d n u c l e u s ^ F f r o m

13 6t h e i n c o m i n g c h a n n e l J C+ L i a n d t r e a t i t s d e c a y i n t o s i x o u t g o i n g c h a n

n e l s : ^ c + ^ L i , ^ 0 + t , ^8 F + n , ^8 0+ p , ^ 0+d a n d A b o v e t h e known

d i s c r e t e s p e c t r u m o f e a c h h e a v y f r a g m e n t , t h e l e v e l d e n s i t y i s e x p r e s s e d

as ( L a 6 3 , S t 7 2 )

1/2 2p (U ,J ) = ----------- , (y. + ]) ___________ exp[l(aU )_____ ~ J J M h L ] . (9 3 )

12 a ' / ' * (U+ t ) 5 A ( 2 o 2 ) 3 / 2 P l 2 o 2

An e f f e c t i v e e x c i t a t i o n e n e r g y I N E ^ - b f i d e t e r m i n e s t h e n u c l e a r t e m p e r a t u r e

2t v i a U = a t - t , w h e r e b6 i s t h e p a i r i n g e n e r g y a n d a i s t h e l e v e l d e n s i t y

2p a r a m e t e r . A s p i n c u t - o f f i s o b t a i n e d f r o m o = I r t / 1i , w h e r e 1 i s t h e

r i g i d - b o d y moment o f i n e r t i a .

T h e r e s u l t i n g c o m p o u n d - n u c 1e u s c a l c u l a t i o n ( F i g . 9 . 7 ) c a n l a r g e l y

a c c o u n t f o r t h e e x p e r i m e n t a l c r o s s s e c t i o n o f ^ 0 ( 1 0 . 3 5 3 , A+ ) i n t h e

( ^ L i , t ) r e a c t i o n ( B a 71a ) . T h e d o m i n a n t A p - A h c o m p o n e n t o f t h i s s t a t e

( e . g . C o 7 6 ) i s e x p e c t e d t o be i n a c c e s s i b l e t o d i r e c t , o n e - s t e p , t h r e e -

n u c l e o n t r a n s f e r . F o r t h e l p - l h s t a t e a t ^ 0 ( 8 . 8 7 2 ) ( E l 7 0 ) , t h e p r o b a b l e

2 p - 2 h c o n f i g u r a t i o n s a t ^ 0 " ( 1 6 . 8 1 , 1 8 . 6 1 ) ( L u 6 9 , S e c t i o n 9 - 1) , a n d t h e

p r o p o s e d 3 p ~ 3 h s t a t e s a t ^ 0 ( 2 0 . A9 , 2 A . 6 3 ) ( S e c t i o n 9 * 2 ) , t h e m a g n i t u d e

o f e a c h t h e o r e t i c a l c u r v e i s w e l l b e l o w t h e d a t a p o i n t s . E v e n t h e s l o p e s

p r e d i c t e d f o r J 7T=5 a n d 7 a r e m i n o r r e l a t i v e t o t h e s t r o n g f o r w a r d

p e a k i n g o f m e a s u r e d a n g u l a r d i s t r i b u t i o n s . A t f o r w a r d a n g l e s a n d

Me V , t h e r e f o r e , a n e g l i g i b l e c o m p o u n d - n u c 1e u s c o n t r i b u t i o n i s i n d i c a t e d by

t h e H a u s e r - F e s h b a c h mo d e l f o r t h e ^8 C ( ^ L i , t ) ^ 0 r e a c t i o n i n t o t h e s e f i v e

s t a t e s .

An a p p l i c a t i o n o f d i r e c t r e a c t i o n t h e o r y i s r e l e v a n t b o t h t o t h e

c o n f i r m a t i o n o f t h i s r e s u l t a n d t o t h e s t u d y o f a n g u 1 a r - m o m e n t u m t r a n s f e r .

T h e t r a n s i t i o n m a t r i x f o r a d i r e c t m e c h a n i s m ( A u 7 0 )

T d i r e c t = x ^ ( r ) | V “ U I J V K £ f r ) > ( 9 - A )«3 1 6 2 3 8 6 1 3 3 14 l y 2y y y

i s c a l c u l a t e d i n t h e f i n i t e - r a n g e , 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

(FRDWBA) f r o m

F i g u r e 9 - 7 H a u s e r - F e s h b a c h mo de l

A l t h o u g h t h e a s s u m p t i o n o f J 7T = 5 , 7 f o r ^ 0 ( 2 0 . 4 9 , 2 4 . 6 3 ) i s

h i g h l y t e n t a t i v e ( S e c t i o n 9 . 2 ) , t h e 18. 61 MeV s t a t e i s a

p r o b a b l e a n a l o g o f ( 5 - 7 3 , 5+ ) ( F i g . 9 . 6 ) , a n d t h e 16. 81 MeV

s t a t e i s a s s i g n e d s p i n ( 3+ ) i n d e p e n d e n t l y ( J a 7 7 ) • T h e s e

c o m p o u n d - n u c 1e u s c a l c u l a t i o n s a r e c a r r i e d o u t f o r t h e

^3C ( ^ L i , t ) ^ 0 r e a c t i o n r a t h e r t h a n 3 C ( ^ L i , 3H e ) , b e c a u s e

^ 0 ' ( 1 0 . 3 5 3 , 4+ ) p r o v i d e s a c h e c k on t h e o v e r a l l n o r m a l i z a t i o n .

T h e p r e d i c t e d c u r v e s r e a c h t h i s u p p e r l i m i t i n m a g n i t u d e wh e n

t h e l e v e l d e n s i t y p a r a m e t e r h a s t h e v a l u e a = 0 . 1 5 2 ( H a 7 4 ) .

W i t h o p t i c a l p o t e n t i a l s f r o m T a b l e A . l , R e f . P i 74 ( # T 2 ) a n d

R e f . C o7 6 , we u s e t h e c o d e ABACUS ( A u 7 6 ) t o c o m p u t e t r a n s

m i s s i o n c o e f f i c i e n t s f o r E q . 9 . 2 .

C = */dV d*B ^ "(V V <'t' i e'('2el V Uel 'l’ lc.l,,2a> xa+ )(^ ’7a) ’ (9‘ 5)

w h e r e a r e p r e s e n t s t h e e n t r a n c e c h a n n e l , 8 t h e e x i t c h a n n e l a n d y a n y

p o s s i b l e o u t g o i n g c h a n n e l . I n a d d i t i o n t o a s s u m i n g t h a t t e r m s w i t h y£ot

a r e n e g l i g i b l e , t h i s e x p r e s s i o n a p p r o x i m a t e s a r e l a t i v e w a v e f u n c t i o n

w i t h t h e d i s t o r t e d w a v e p r o d u c e d by an o p t i c a l p o t e n t i a l

a n d a r e t h e i n t e r n a l w a v e f u n c t i o n s o f t h e p r o j e c t i l e a n d t a r g e t ; V2 a 8 •i s t h e t o t a l n u c l e a r i n t e r a c t i o n . A l t h o u g h a z e r o - r a n g e i n t e r a c t i o n i s

a s s u m e d by t h e c o d e DWUCK ( A p p e n d i x A ) , t h e c o d e PTOLEMY ( G 176 ) c a l c u l a t e s

t h e f u l l , s i x - d i m e n s i o n a l i n t e g r a l . An i n i t i a l w a v e f u n c t i o n r e p r e s e n t i n g

6 3t h e I s s t a t e o f L i - H e + t i s g e n e r a t e d by a W o o d s - S a x o n p o t e n t i a l h a v i n g

r * 1- 73 a n d a = 0 . 4 5 ( T h 6 7 , B i 7 5 ) » w h e r e t h e r a d i u s i s g i v e n by R= r QA 1/3

a n d t h e d e p t h i s f i t t e d t o t h e e x p e r i m e n t a l b i n d i n g e n e r g y . O p t i c a l

p o t e n t i a l s f o r t h e e n t r a n c e c h a n n e l ^ L i + C a n d t h e e x i t c h a n n e l 3 H e + ^ N

a r e l i s t e d i n T a b l e A . l , w h e r e t h e a l t e r n a t e p a r a m e t e r s e t c a n be shown

t o h a v e l i t t l e e f f e c t upon t h e s h a p e o f FRDWBA c u r v e s . P r e d i c t e d a n g u

l a r d i s t r i b u t i o n s a r e m o r e d e p e n d e n t upon t h e r a d i u s p a r a m e t e r o f t h e

W o o d s - S a x o n p o t e n t i a l w h i c h g e n e r a t e s a f i n a l w a v e f u n c t i o n f o r t h e b ou nd

16 13s t a t e o f N= C + t . W i t h a = 0 . 6 5 , r Q= ^-7 ' s t b e m i n i m u m v a l u e y i e l d i n g a

good r e s u l t f o r ( 3 - 9 6 , ( 3+ ) ) . C o n v e r g e n c e c h e c k s w e r e made f o r o t h e r

p a r a m e t e r s i n v o l v e d i n t h e PTOLEMY c o d e .

T h e s e FRDWBA c a l c u l a t i o n s s u c c e e d i n r e p r o d u c i n g t h e s t e e p s l o p e o f

e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n s f r o m t h e ^3 C ( ^ L i , 3 H e ) ^ N r e a c t i o n

( F i g . 9 - 8 ) . I n v i e w o f t h e f a i l u r e o f t h e H a u s e r - F e s h b a c h mo d e l t o a c

c o u n t f o r s u c h s t r o n g f o r w a r d p e a k i n g ( F i g . 9 . 7 ) , t h i s r e s u l t i s c l e a r

e v i d e n c e f o r a p r i m a r i l y d i r e c t m e c h a n i s m ( s e e a l s o G a 7 2 , B i 7 5 ) . D a t a f o r

t h e J 7T= ( 3+ ) s t a t e a t ( 3 - 9 6 ) , m o r e o v e r , a r e w e l l d e s c r i b e d t o 0 = 4 5 °c . m.

F i g u r e s 9 - 8 a , b F i n i t e - r a n g e DWBA c a l c u l a t i o n s

6 3B e c a u s e o f a r e l a t i v e s - s t a t e i n L i = H e + t , L - t r a n s f e r

i n t h e ^3 C ( 8 L i , 3 H e ) ^ 8 N r e a c t i o n i s e q u a l t o t h e o r b i t a l

16 13a n g u l a r moment um o f a N= C + t b o u n d s t a t e . T h e num

b e r o f n o d e s N i n a f i n a l - s t a t e w a v e f u n c t i o n d e p e n d s

upon t h e c o n f i g u r a t i o n o f t h e t r i t o n c l u s t e r . I n

F i g . 9 . 8a , we c o n s i d e r p ( s d ) t r a n s f e r c o r r e s p o n d i n g t o

2N + L = 5 , b e c a u s e a 2 p - 2 h c o n f i g u r a t i o n c h a r a c t e r i z e s

( 3 - 9 6 , 5 . 7 3 ) ( L u 6 9 ) a nd may a l s o a p p l y t o ( 1 1 . 2 1 ) ,

w h i c h b e h a v e s s i m i l a r l y i n F i g . 9 . 4 . I n F i g . 9 . 8 b , we

a s s u m e ( s d ) t r a n s f e r , i . e . 2 N + L = 6 , f o r t h e p r o b a b l e 3 p ~ 3h

16 16 s t a t e s o f N ( S e c t i o n 9 - 2 ) . A n a l o g s t a t e s o f 0

( F i g . 9 . 7 ) h a v e s i m i l a r a n g u l a r d i s t r i b u t i o n s i n t h e

(8 L i , t ) react ion (Fig. 9 . 6 ) .

do-/df

(/xb/sr)

by a L=3 c u r v e f r o m PTOLEMY, w h i c h i s n o r m a l i z e d i n m a g n i t u d e a t 0^ ^ - 1A ° .

A c a l c u l a t i o n f r o m DWUCK, h o w e v e r , i s b e t t e r a b l e t o f i t a L=5 c u r v e t o

t h e a n g u l a r d i s t r i b u t i o n m e a s u r e d f o r t h e 5* s t a t e a t ( 5 . 7 3 ) ( s e e

A p p e n d i x A ) . A l t h o u g h L = 4 , 5 a n d 6 l e a d t o p r e d i c t i o n s o f s i m i l a r s h a p e

( F i g . 9 . 8 ) , t h e 0 = 1A° p o i n t f a v o r s L=A f o r ( 7 . 6 5 ) a n d t h ec . m.

0 =b0 ° r e g i o n f a v o r s L=6 f o r ( 1 1 . 8 1 ) . When c o u p l e d t o t h e s p i nc . m.

1/ 2* o f t h e t r a n s f e r r e d t r i t o n a n d t o t h e s p i n 1/ 2” o f t h e ^3 C t a r g e t ,

t h e s e o r b i t a l a n g u l a r mo me n t a w o u l d i m p l y J71 = ( 3 » A , 5 ) a n d ( 5 , 6 , 7 ) f o r

t h e 7 . 6 5 MeV a n d 11. 81 MeV s t a t e s r e s p e c t i v e l y , i n a g r e e m e n t w i t h t h e

i n t e r p r e t a t i o n c h o s e n i n S e c t i o n 9 . 2 . T h e 11. 21 MeV s t a t e o f ^ N , w h e r e

p a r i t y i s m o r e u n c e r t a i n , i l l u s t r a t e s t h a t t r a n s f e r r e d a n g u l a r mo me n t a

d i f f e r i n g by o n l y o n e u n i t ( B i 7 5 ) a r e n o t d i s t i n g u i s h e d by a n g u l a r d i s -

.6 3 xt r i b u t i o n s i n t h e ( L i , H e ) r e a c t i o n .

A f i n a l q u e s t i o n c o n c e r n s t h e e v a l u a t i o n o f s p e c t r o s c o p i c f a c t o r s .

13 6 3 16T h e c o e f f i c i e n t w h i c h n o r m a l i z e s a FRDWBA c u r v e t o C ( L i , He ) N d a t a

6 3( F i g . 9 * 8 ) r e p r e s e n t s a p r o d u c t o f s p e c t r o s c o p i c f a c t o r s S. ( L i = H e + t ) x

S ^ ( ^ N = ^ C + t ) . U s i n g L=3 a n d L=5 f o r (3 • 9 6 , ( 3* ) ; 5 . 7 3 , 5 ) a n d

a s s u m i n g L=b> 5 a n d 6 f o r ( 7 . 6 5 , 1 1 . 2 1 , 1 1 . 8 1 ) ( S e c t i o n 9 - 2 ) , we o b

t a i n S. S j . =0 . 0 3 7 , 0 . 0 3 5 , 0 . 0 7 3 , 0 . 0 3 7 a n d 0 . 051 r e s p e c t i v e l y . T h e o v e r a l l

m a g n i t u d e i s r e a s o n a b l e , s i n c e S. = S ^ ] / b c o u l d a c c o u n t f o r t h e t wo l a r g

e s t p r o d u c t s . T h e r e l a t i v e s t r e n g t h i s m o r e m e a n i n g f u l , h o w e v e r , s i n c e

t h e o r e t i c a l c r o s s s e c t i o n s a r e s e n s i t i v e t o t h e r a d i u s p a r a m e t e r o f t h e

^ N = ^ C + t p o t e n t i a l . S e t t i n g S=1 f o r ( 7 . 6 5 ) , we n o t e t h a t S ^ - 1/2

f o r ( 3 . 9 6 , 5 . 7 3 , 1 1 . 2 l ) f a v o r s a common c 1 a s s i f i c a t i o n o f t h e s e s t a t e s ,

n a m e l y a s 2 p - 2 h c o n f i g u r a t i o n s ( S e c t i o n 9 - 2 ) . T h e l a r g e r v a l u e ®r e |~2/3

f o r ( 1 1 . 8 1 ) s u p p o r t s an a s s o c i a t i o n w i t h ( 7 . 6 5 ) a n d 3 p ~ 3 h s t r u c -

t u r e . A p p r o x i m a t 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 c o n t a i n e d i n t h e f o r w a r d -

a n g l e s p e c t r u m o f ( F i g . 9 - 2 ) , t h e r e f o r e , i s q u a l i t a t i v e l y c o n f i r m e d

by r e l a t i v e s p e c t r o s c o p i c f a c t o r s e x t r a c t e d f r o m t h e a n g u l a r d i s t r i b u -

t i o n s .

I n s u m m a r y , a l t h o u g h a s e a r c h f o r f u r t h e r i n f o r m a t i o n on a n a l o g

s t r u c t u r e m o t i v a t e d t h e i r m e a s u r e m e n t , a n g u l a r d i s t r i b u t i o n s f r o m t h e

( 8 L i , t ) a n d ( ^ L i , ^ H e ) r e a c t i o n s on a t a r g e t a l s o p r o v i d e s u p p o r t i n g

e v i d e n c e on t h e r e a c t i o n m e c h a n i s m , t r a n s f e r r e d a n g u l a r mo me n t a a nd

s p e c t r o s c o p i c f a c t o r s .

CHAPTER 10 Conclusion

I n t h e p r e c e d i n g f i v e c h a p t e r s , i n d i v i d u a l d i s c u s s i o n o f t h e A=15

t o A=19 n u c l e i e n t a i l s f r e q u e n t c o n s i d e r a t i o n o f p a i r s d i f f e r i n g i n m a s s .

We c o m p a r e , i n S e c t i o n 1 0 . 1 , e x p e r i m e n t a l r e s u l t s o v e r t h e e n t i r e mass

r e g i o n , a s e v i d e n c e o f s y s t e m a t i c b e h a v i o r i n t h r e e - p a r t i c l e s t r u c t u r e .

I n S e c t i o n 1 . 2 , a summa r y o f t h i s r e s e a r c h f o c u s e s on new f i n d i n g s . We

i n t e g r a t e , i n S e c t i o n 1 0 . 2 , t h e p r e s e n t r e s u l t s w i t h p r e v i o u s i n f o r m a t i o n

on t h r e e - n u c l e o n t r a n s f e r r e a c t i o n s a n d c l u s t e r s t r u c t u r e .

10. 1 S y s t e m a t i cs

3A c o n s i s t e n t i d e n t i f i c a t i o n o f ( s d ) c o n f i g u r a t i o n s i s o b t a i n e d

6 3f r o m t h e ( L i , H e ) r e a c t i o n on t a r g e t s h a v i n g f r o m z e r o t o f o u r h o l e s i n

t h e p s h e l l ( F i g . 1 0 . 1) . A c o u p l i n g o f t a r g e t s p i n t o t h e t o t a l a n g u l a r

moment um j = L ® l /2 o f t h e t r a n s f e r r e d t r i t o n ( T a b l e 1 . 1) d e t e r m i n e s f i n a l -

s t a t e s p i n v a l u e s . I n t h e A=15 t o A=19 n u c l e i , c o r r e s p o n d i n g s t a t e s w i t h

J 7T = 5 / 2+ , 3 , 7 / 2+ , 3 a n d 5 / 2+ e x h i b i t a s p i n s e q u e n c e b a s e d on t h e a l i g n

m e n t o f a n g u l a r m o m e n t a . C a n d i d a t e s f o r 3 p ~ n h , j = 9 / 2+ s t r u c t u r e r e g u l a r

l y a p p e a r w i t h l a r g e r c r o s s s e c t i o n s a n d w i t h e x c i t a t i o n e n e r g i e s a b o u t

3 MeV h i g h e r t h a n t h o s e o f t h e j = 5 / 2+ s t a t e s ( F i g . 1 0 . 1) . S i n c e t h e S U ( 3 )

s h e l l mo d e l p r e d i c t s a s p l i t t i n g o f 5 s t r e n g t h ( F i g . 7 - 6 ) , t w o l e v e l s

a t ^8 0 ( 7 - 8 4 ; 8 . 10 , 5 ) a r e a s s o c i a t e d w i t h t h e 9 / 2+ s t a t e o f ^8 F . T he

l e v e l a t ^7 0 ( 1 0 . 7 0 ) i s e q u a l i n e x c i t a t i o n t o t h e 9 / 2+ s t a t e o f ^ N ,

w h i l e t h e p e a k a t ( 7 - 6 5 ) i s s i m i l a r i n c r o s s s e c t i o n t o t h e p a i r i n

18 6 30 . R e f l e c t i n g t h e a n g u 1a r - m o m e n t u m m i s m a t c h a L =6 o f t h e ( L i , He )

19r e a c t i o n , t h e m o s t s t r o n g l y p o p u l a t e d s t a t e s o f p o s i t i v e p a r i t y i n F

h a v e s p i n 13 / 2+ . A c o r r e s p o n d i n g p n ( s d ) ^ c o n ^ ' 9 u r a 1 ' on ' s p r o b a b l e

160 s e l e c t s f i n a l s t a t e s o f t h e A=15 t o A=19 n u c l e i . T h e s e

d a t a a r e m e a s u r e d a t s i m i l a r e n e r g i e s o f j = A 0 , A A , A6 , A O, A6

r e f e r e n c e p o i n t s f o r r e l a t i v e Q - v a l u e s , c o n t a m i n a n t p e a k s

a r i s e f r o m t h e ( 8 L i , ^ H e ) AHe ( g . s . ) r e a c t i o n . E x c i t a t i o n

e n e r g i e s a n d known s p i n v a l u e s a r e g i v e n f o r p r o b a b l e

*3 + + +p n ( s d ) . c o n f i g u r a t i o n s w i t h j = 5/2 , 9/2 a nd 13/2 . T h e

+c a n d i d a t e s f o r j = 9/2 s t r u c t u r e a r e l i n e d u p . T a b l e 9*1

16 *a n a l y z e s t h e b r o a d p e a k a t N ( 5 . 15 ) ; S e c t i o n 5 - 3 i n t e r

p r e t s t h e t w o 5/2 s t a t e s a t ( 5 . 2 7 0 , 9 . 155 ) . I n f o r m a t i o n

on t h e u n l a b e l l e d p e a k s i s c o n t a i n e d i n F i g s . 5 . 2 - 9 - 2 a nd

T a b l e s 5 - 1- 9 . 1 •

Figure 10.1 T r i to n - t ra ns f er spectra

The (8L i , 8He) react ion on t arget s of ^ C, ^ C, ^ N , and

1 000-

10 E x (MeV) 5

( 1 5 . 4 1 ) . S h e l l - m o d e l c a l c u l a t i o n s f o r a n d ^ 0 ( A n 7 4 , F i g . 7 . 6 )

s u p p o r t t h i s i n t e r p r e t a t i o n a n d p r e d i c t a s e c o n d 1 3 / 2+ ® l / 2 = 7 s t a t e o f

^®0 n e a r 15 MeV . T h e s e p a r a t i o n o f ^ F ' ( 10 . 4 1 1 , 13 / 2+ ) f r o m t h e f i r s t

j = 1 3 / 2+ s t a t e i s w e l l r e p r o d u c e d by ^ 0 ( 1 5 - 9 5 ) a n d ^ 0 ( 2 0 . 2 ) , a l t h o u g h

1 8t h e 14. 61 MeV l e v e l o f 0 i s a n o t h e r c a n d i d a t e . T h r o u g h a t e n t a t i v e

b u t n e a r l y o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n a l i g n e d a n g u 1 a r - m o m e n t u m

c o u p l i n g s , t h e r e f o r e , t r i t o n - t r a n s f e r s p e c t r a r e v e a l common s t r u c t u r e i n

t h e A= 15 t o A=19 n u c l e i , b a s e d on p P ( s d ) ^ c o n f i g u r a t i o n s w i t h j = 5 / 2+ ,

9 / 2+ a n d 1 3 / 2+ .

T h i s c o n c l u s i o n i s s u p p o r t e d by e v i d e n c e o f a c o n s i s t e n t t r e n d i n

t r i t o n b i n d i n g e n e r g i e s ( F i g . 1 0 . 2 ) . F o r t h e a b o v e t h r e e - p a r t i c l e

s t r u c t u r e ( F i g . 1 0 . 1 ) , t h e i n c r e a s e i n b i n d i n g w i t h mass i s c o n t i n u o u s ,

1 7 * 1 7e x c e p t a t 0 ( 1 0 . 7 0 ) . T h e 0 l e v e l s a r e e x p e c t e d t o h a v e r e l a t i v e l y

+ 14l o w e n e r g y , b e c a u s e t h e h i g h e r s p i n ° f an u n e x c i t e d N c o r e i m p l i e s

s t r o n g e r i n t e r a c t i o n w i t h a n a l i g n e d t r i t o n c l u s t e r ( s e e F i g . 4 . 2 ) .

15 19F r o m N t o F , t h e i n c r e a s e i n moment o f i n e r t i a i s c h a r a c t e r i z e d by a

g r a d u a l l e v e l c o m p r e s s i o n f o r j = 9 / 2+ a n d 13/ 2* . T h i s t r e n d i s n o t o b

s e r v a b l e i n t h e 5 / 2 ~ 9 / 2 l e v e l s p a c i n g , w h e r e t w o m i x i n g e f f e c t s a r e

r e l e v a n t . A d m i x t u r e o f 3 p~nh s t r u c t u r e i n t o s i n g l e - p a r t i c l e e x c i t a t i o n s

a t ( 5 - 2 7 0 , 5 / 2* ) a n d ( 0 . 2 9 7 , 3 ) ( e . g . L i 7 0 ) i s e x p e c t e d t o l o w e r

— n 3 1 5 * ■+•t h e p ( s d ) c e n t r o i d b e l o w t h e p o i n t s p l o t t e d a t N ( 9 - 1 5 5 , 5/2 ) and

( 5 - 1 5 , 3 ) - I n ^ 0 = ^ N + t , m o r e o v e r , t h e r e l a t i v e p o s i t i o n o f t h e

+ + + ■+ ■+5 / 2 ®1 = 7 / 2 c o u p l i n g c o u l d be i n f l u e n c e d by m i x i n g w i t h 9 / 2 ®1 s t r u c

t u r e . A l l o w i n g f o r s u c h c o m p l i c a t i o n s , we c o n c l u d e t h a t t r i t o n b i n d i n g

e n e r g i e s e x h i b i t r e a s o n a b l y s m o o t h b e h a v i o r i n t h e A= 15 t o A= 19 n u c l e i .

95for the major s t a t e s at ^ 0 ' ( 11 .10 ) , 7 0# (14 .89) , ^ N ' (11.81) and

gure 10 2 Binding energies

E =E - E . , . . i s p l o t t e d v e r s u s . f o r d e u t e r o nc . m . x t h r e s h o l d r t a r g e t

( d ) , t r i t o n ( t ) a n d a l p h a ( a ) t r a n s f e r . T h e ( a , d ) , ( ^ L i , 3 He)

a n d ( ^ L i , t ) r e a c t i o n s i d e n t i f y c a n d i d a t e s f o r p " n ( s d ) n c o n

f i g u r a t i o n s , w h e r e n=0 t o A a nd n ' =2 ( d o t t e d l i n e s ) , 3

( s o l i d ) a n d A ( d a s h e d ) . G i v e n t h e a n g u l a r moment um a n d p a r i

t y j 71 o f a t r a n s f e r r e d c l u s t e r , c o u p l i n g t o t h e s p i n 1/2

o f 1 3 C a n d o r t o t h e s p i n 1+ o f a t a r g e t o c c u r s i n

f i n a l s t a t e s . T a b l e 10. 1 l i s t s t h e e x c i t a t i o n e n e r g i e s a n d

s p i n a s s i g n m e n t s o f l e v e l s s e l e c t e d f o r t h i s f i g u r e . I n t w o -

p a r t i c l e s t r u c t u r e , o n l y a l i g n e d a n g u l a r - m o m e n t u m c o u p l i n g s

a r e shown f o r ^ 0= ^ N + d a n d ^ 0= ^ N + d s i n c e , f o r e x a m p l e ,

a m b i g u i t y a r i s e s f r o m t h e p o p u l a t i o n o f b o t h ^ 0 ( 5 - 2 1 5 , ( 9/2 )

9 . 1 5 , 9/2 ) by t h e ( a , d ) r e a c t i o n ( L u 6 9 ) . I n t h r e e - p a r t i c l e

— l 8 l 5s t r u c t u r e , o n l y t h e known 5 s t a t e i s p l o t t e d f o r 0 = N + t ,

a l t h o u g h mo d e l p r e d i c t i o n s s u g g e s t s p i n 5 a n d A f o r

18 »»0 ( 7 - 8 A , 8 . A7 ) r e s p e c t i v e l y ( F i g . 7 . 6 ) . We do n o t i n c l u d e

+ . .t h e c a n d i d a t e s f o r a s e c o n d j = 13/2 c o n f i g u r a t i o n ( F i g . 1 0 . 1 ) .

+ 17 13I n f o u r - p a r t i c l e s t r u c t u r e , t h e 2 ® 1/2 s t a t e s o f 0 = C+a

a r e o m i t t e d b e c a u s e o f s t r o n g m i x i n g ( s e e L e 7 2 ) . O n l y c o u p -

18 1A +l i n g s t o s p i n j a r e shown f o r F= N + a : j =8 s t a t e s a r e u n -

18 19 known i n F a n d F.

TARGET NUCLEUS (CORE)

-n n *TABLE 10.1 p (sd) Candidates

Threshold L=2 L=4 L=6 L=8 Ref.

12 14 *C+cJ= N (10.272) 6.444,3* 8.961,5* Ri66i s *

C+d= N (16.160) 9 .8 2 9 ,7 /2 “; . . . 11.95, (9/2 ") ;13 .00,11/2" Lu6914 16 *N+d= O (20.737) 1 1 .0 9 5 ,4 * ; . . . 1 4 .8 1 5 ,6 * ; . . . Zi70IS 17 *

N+d= O (14.049) 5 .6 9 8 ,7 /2 " ; . . . 7 .7 5 ,1 1 /2 " ; . . . Lu69

160+d=18F*(7.526) 0.937,3* 1 .121,5* Ma68

12 IS *C+t= N (14.848) 9 .155 ,5 /2* 10.693,9/2* 15.41,(13/2*)

13 16 *C+t= N (12.394) 5 .1 5 ,3 " ; . . . 7 .6 5 ; . . . 1 1 .8 1 ; . . .14 17 *N+t= O (18.625) 8 .4 7 4 ,7 /2 * ; . . . 10 .70 ;. . . 1 4 .8 9 ; . . . Fig. 10.1IS 18 *

N+h= O (15.834) 5 .0 9 8 ,3 " ; . . . 8 .1 0 ,5 " ; . . . 1 1 .1 0 ; . . .16 19 *

0+t= F (11.700) 0 .197,5 /2* 2 .780 ,9 /2* 4.647,13/2*

12 16 *C+a= O (7.162) 6.917,2* 10.353,4* 16.29,6* 22.5,(8*) Sa7713 17 *

C+<*= O (6.361) 8 .9 7 2 ,7 /2 ‘ ;9.87 12.41;13.55 1 8 .15;19.24 Fig. 8.514 18 *

N+cx= F (4.416) 2 .5 2 3 ,2 * ; . . . 5 .2 9 8 ,4 * ; . . . 9 .58, (6 * ) ; . . . • . . Co7715 19 *

N + q ;= F (4.014) 1 .346 ,5 /2”;1 .459 ,3 /2 ‘ 3 .9 9 9 ,7 /2 " ;4 .032,9/2" 8 .2 8 8 ,1 3 /2 " ;8 .9 5 3 ,l l /2 " # • • Fig. 6.316^ 2 0 *Ofa= Ne (4.731) 1.634,2* 4.247,4* 8.776,6* 11.95,8* Co76

to<1Aj76,77,78

A l a c k o f c o m p l e t e s p i n i n f o r m a t i o n a n d m u l t i p l e t i d e n t i f i c a t i o n ,

h o w e v e r , ma k e s t h e s y s t e m a t i c b e h a v i o r o f t h r e e - p a r t i c l e s t r u c t u r e a

t e n t a t i v e r e s u l t , i n v i t i n g c o m p a r i s o n w i t h t h e b e t t e r k nown p h e n o m e n a o f

2t w o - a n d f o u r - p a r t i c l e s t r u c t u r e . M o r e o v e r , d e u t e r o n b i n d i n g

s u p p l i e s a t r e n d f r e e f r o m d e p e n d e n c e u po n t h e s u b s h e l l c o n f i g u r a t i o n ,

a n d a 1p h a - p a r t i c 1e c l u s t e r i n g o f f e r s a c a s e f r e e f r o m s p i n - s p i n i n t e r

a c t i o n w i t h t h e c o r e . R e f l e c t i n g t h e s e s i m p l i f i c a t i o n s , j = 5+ s t a t e s o f

2 p - n h c h a r a c t e r ( l_u6 9 ) a n d j !=6+ s t a t e s o f A p - n h c h a r a c t e r e x e m p l i f y a

s m o o t h i n c r e a s e i n b i n d i n g e n e r g y as a f u n c t i o n o f mass ( F i g . 1 0 . 2 ) .

We f i n d a s i m i l a r i t y i n o v e r a l l s l o p e f o r j = 13 / 2+ a nd j = 8+ a nd f o r

j = 9 / 2+ a n d j = 5+ . A l t h o u g h l e v e l s p a c i n g i s r a t h e r i r r e g u l a r i n t h e p r o b

a b l e p ~ n ( s d ) ^ + c o n f i g u r a t i o n s , t h e 2N + L =8 b a n d d e m o n s t r a t e s a n e a r l y

m o n o t o n i c i n c r e a s e i n moment o f i n e r t i a . L i m i t e d e v i d e n c e e x i s t s o n t h e

1 3n a t u r e o f m u l t i p l e t s t r u c t u r e . F o r a J C t a r g e t o f s p i n 1/2 , d u a l

p e a k s w i t h 1 MeV s p l i t t i n g a r e o b s e r v e d i n d e u t e r o n a nd a 1p h a - p a r t i c 1e

t r a n s f e r ( L u 6 9 , F i g . 8 . Ac ) b u t n o t i n t r i t o n t r a n s f e r ( F i g . 9 . 2 ) . F o r

a t a r g e t o f s p i n 1+ , a t r i p l e t a p p e a r s i n 2 p - 2 h s t a t e s a t ^ 0 " ( 1A . A 0 ;

1 A. 8 2 , 6+ ; 1 6 . 2A) (Z i 7 0) but stron ger coupling c h a r a c te r iz e s a Ap-2h band1 ft i ft i c

i n F ( R o 73 b ) . I n d i c a t i o n s o f w e a k c o u p l i n g i n 0 = N + t ( F i g . 7 * 5 )

19 15a r e s u p p o r t e d by t h e p r e s e n c e o f n a r r o w d o u b l e t s i n F= N+a ( F i g . 6 . 5 ) .

O v e r a l l , g r e a t e r k n o w l e d g e o f t w o - a n d f o u r - p a r t i c l e s t r u c t u r e p r o v i d e s■j

a s t a n d a r d f o r t r e n d s i n t h r e e - p a r t i c 1e s t r u c t u r e . T h e p ( s d ) c o n f i g u r a

t i o n s o f F i g . 10.1 show f u n d a m e n t a l c o n s i s t e n c y w i t h t h i s b r o a d e r c o n t e x t .

6 3O t h e r c o n f i g u r a t i o n s a l s o p l a y a r o l e i n t h e ( L i , He ) r e a c t i o n

1 5 .( F i g . 1 0 . 3 ) . A s t r o n g p o p u l a t i o n o f t h e 7/2 a n d 11/2 s t a t e s o f N i s

a t t r i b u t e d i n t h e s h e l l mo d e l t o t h e i r l a r g e s p e c t r o s c o p i c f a c t o r s f o r a

F i g u r e 10 . 3 R e l a t i o n t o o t h e r t r a n s f e r r e a c t i o n s

6 3T r i t o n - t r a n s f e r s p e c t r a f r o m t h e ( L i , He) r e a c t i o n a r e

r e - l a b e l l e d i n t h i s f i g u r e b u t a r e s t i l l a l i g n e d as i n

F i g . 1 0 . 1 , w h e r e a d d i t i o n a l d e t a i l s a r e g i v e n . C o m p a r i s o n

w i t h t h e ( a , d ) r e a c t i o n ( L u 6 9 ) d i s t i n g u i s h e s p e a k s a r i s i n g

f r o m p n ( s d ) ^ c o n f i g u r a t i o n s o f known s p i n i n ^ N ,

17 18 + 2a n d 0 . 0 “ ( 3 * 5 5 5 , A ) d o m i n a t e s a ( a , He ) s p e c t r u m

( J a 7 6 ) . C o m p a r i s o n w i t h t h e ( 7 L i , t ) r e a c t i o n i d e n t i f i e s

_ n hm i x i n g i n t o p ( s d ) c o n f i g u r a t i o n s a t h i g h e x c i t a t i o n i n

l 9 F ( F i g . 6 . 3 ) , l 8 0 ( Mo7 0 b) a nd 170 ( F i g . 8 . 3 ) .

^ 0 ( 7 . 1 1 7 , A+ ) may c o r r e s p o n d t o ( 1 1 . 2 1 ) .

COUNTS COUNTS COUNTS COUNTS COUNTS3 5

2p ( s d ) t r i t o n c l u s t e r ( A n 7 4 ) . F o r c o r r e s p o n d i n g t w o - p a r t i c l e m u l t i - h o l e

• 1 6 ., 17 0 u I 8 rtc o n f i g u r a t i o n s i n N, 0 a n d 0 , a t r e n d e x i s t s t o w a r d s m a l l e r c r o s s

s e c t i o n a n d l o w e r e x c i t a t i o n e n e r g y , r e l a t i v e t o 3p ~ n h S t a t e s , as p r o

g r e s s i v e l y f e w e r p - s h e l l h o l e s a r e p r e s e n t i n t h e t a r g e t . S u b s t a n t i a l

2 - 2 4( s d ) b u t l a r g e r p ( s d ) c o m p o n e n t s a r e p r e d i c t e d ( E 170 ) i n t h e s e c o n d

+ 184 s t a t e o f 0 . T h e r e i s a p o s s i b l e l i n k b e t w e e n t h i s 7 . 1 1 7 MeV s t a t e

18 16 o f 0 a n d t h e 11. 21 MeV s t a t e o f N, b e c a u s e b o t h l e v e l s l i e a b o u t

1 MeV a b o v e t h e a l p h a - p a r t i c l e t h r e s h o l d a n d h a v e l i t t l e c r o s s s e c t i o n

7i n t h e ( L i , a ) r e a c t i o n ( F i g s . 7 - 3 , 9 . 3 ) . B o t h a l p h a - p a r t i c l e a n d t r i t o n

t r a n s f e r d a t a c o n t a i n l a r g e p e a k s a t ^ F ~ ( 8 . 9 5 3 * 11/2 ) ( F i g . 6 . 4 ) , w h i c h

a r e i n t e r p r e t e d i n t h e s h e l l mo d e l a s a c a s e o f m i x i n g b e t w e e n p” ( s d )*4

2 4a n d ( s d ) f p c o n f i g u r a t i o n s (Mi 7 7 ) . S i n c e c o r r e s p o n d i n g ( s d ) . + s t r u c t u r e6

i s f o u n d a t 8 0 (1 1 . 6 9 , 6+ ) ( M o 7 0 ) a n d t e n t a t i v e l y a t ^ 0 ( 1 3 . 5 5 )

6 3( F i g . 8 . 3 ) , t h e p r o m i n e n c e o f t h e s e s t a t e s i n ( L i , He ) s p e c t r a ( F i g . 1 0 . 3 )

2r e f l e c t s c o n s i s t e n t a d m i x t u r e w i t h [ ( s d ) f p ] ^ / 2" s t r u c t u r e * * n r e l a t i o n

2t o d i f f e r e n t mu 11 i - n u c 1 e o n t r a n s f e r r e a c t i o n s , t h e r e f o r e , p ( s d ) a n d

2( s d ) f p t r i t o n t r a n s f e r i n t o t h e A= 15 t o A= 19 n u c l e i a l s o h a s s y s t e m a t i c

f e a t u r e s .

10 . 2 Summary

We c o n c l u d e w i t h a n o v e r v i e w o f t h e i n v e s t i g a t i o n i n t o t h r e e - n u c l e o n

t r a n s f e r r e a c t i o n s a n d c l u s t e r s t r u c t u r e f o r l i g h t n u c l e i . I n t h e p r e

v i o u s w o r k o f o t h e r s a n d i n t h e p r e s e n t r e s e a r c h , t h e r e i s a c o n s i s t e n t

6 6 3i n d i c a t i o n t h a t t h e ( L i , t ) a n d ( L i , He) r e a c t i o n s p r o c e e d v i a a p r e

d o m i n a n t l y d i r e c t m e c h a n i s m . T h i s r e s u l t a p p l i e s a t l e a s t t o f i n a l

s t a t e s s t r o n g l y p o p u l a t e d a t E^ . £ 4 0 MeV a n d D i v e r s e s u p p o r t

i n g e v i d e n c e i s f o u n d i n e x c i t a t i o n f u n c t i o n s , a n g u l a r d i s t r i b u t i o n s a n d

f o r w a r d - a n g l e s p e c t r a . F o r i n c i d e n t e n e r g i e s v a r i e d i n f i n e s t e p s o r

o v e r a w i d e r a n g e , t h e e n e r g y d e p e n d e n c e o f c r o s s s e c t i o n s i s e s s e n t i a l l y

f e a t u r e l e s s . S t r o n g l y f o r w a r d - p e a k e d a n g u l a r d i s t r i b u t i o n s a r e a d e q u a t e

l y d e s c r i b e d by DWBA c a l c u l a t i o n s b u t n o t by t h e H a u s e r - F e s h b a c h m o d e l .

Fr o m t h e r e l a t i v e p o p u l a t i o n o f f i n a l s t a t e s a t ® ] a b = 15 ° , s e l e c t i v i t y

w i t h i n t h e c l a s s o f h i g h - s p i n s t a t e s c a n be d o c u m e n t e d e x p e r i m e n t a l l y .

T h e o r e t i c a l e v i d e n c e on t h e o r i g i n o f t h i s s t r u c t u r a l s e l e c t i v i t y s u g

g e s t s t h a t c l u s t e r t r a n s f e r p l a y s a n i n f l u e n t i a l r o l e i n t h e r e a c t i o n

m e c h a n i s m .

6 3T h e ( L i , He ) r e a c t i o n t h e r e f o r e l e a d s t o an i d e n t i f i c a t i o n o f

p r o b a b l e 3 p _ nh c o n f i g u r a t i o n s i n t h e A=15 t o A = J 9 n u c l e i . I t s r e l a t i o n -

£s h i p t o t h e ( L i , t ) r e a c t i o n a l l o w s an a s s i g n m e n t o f a n a l o g s t a t e s ,

e s p e c i a l l y i n t h e m i r r o r s p e c t r a f o r T = ± 1/2 n u c l e i . A l t h o u g h T = 1 , 1^=0

18s t a t e s o f F a r e d i f f i c u l t t o d i s t i n g u i s h f r o m t h e T =0 l e v e l s , d o m i

n a n t s t a t e s a t h i g h e x c i t a t i o n i n ^ C ( ^ L i , t ) ^ 0 d a t a show a c l e a r c o r r e -

13 6 3 16s p o n d e n c e t o t h e T =1 s p e c t r u m o f C ( L i , He) N. C o m p a r i s o n w i t h o t h e r

t h r e e - n u c l e o n t r a n s f e r r e a c t i o n s d e m o n s t r a t e s a n u n d e r l y i n g c o n s i s t e n c y

i n s t r u c t u r a l s e l e c t i v i t y a nd a u s e f u l v a r i e t y o f d y n a m i c e f f e c t s .

W h e r e a s t h e ( a , p ) a n d ( ^ B , 7 Be) r e a c t i o n s s u r p a s s t h e ( ^ L i , ^ H e ) r e a c t i o n

i n h i g h - s p i n e n h a n c e m e n t , t h e w e l l - m a t c h e d ( 7 L i , a ) r e a c t i o n p o p u l a t e s

a d d i t i o n a l l e v e l s o f l o w e r s p i n a t h i g h e x c i t a t i o n e n e r g i e s . T h e c o n

t r a s t t o t w o - a n d f o u r - n u c l e o n t r a n s f e r d a t a p r o v i d e s a c h e c k on c a n d i

d a t e s f o r 3 p - n h s t r u c t u r e . W h i l e t h e ( a , d ) r e a c t i o n i d e n t i f i e s 2 p - ( n - l ) h

s t a t e s , t h e ( 7 L i , t ) r e a c t i o n i n d i c a t e s a d m i x t u r e i n t o A p - ( n + l ) h s t a t e s .

T h e l a r g e l y p ( s d ) c o n f i g u r a t i o n s p r o p o s e d f r o m t h e a b o v e c o m p a r i s o n s

e x h i b i t s y s t e m a t i c b e h a v i o r w i t h r e s p e c t t o a n g u 1a r - m o m e n t u m c o u p l i n g

An a p p l i c a t i o n o f s p e c i a l i z e d n u c l e a r m o d e l s t o ( L i , He ) d a t a r e

f l e c t s t h e i m p o r t a n t i n f l u e n c e o f t r i t o n - c 1 us t e r s t r u c t u r e i n t h i s mass

r e g i o n . As a f i r s t a p p r o x i m a t i o n i n a m a c r o s c o p i c a p p r o a c h , t h e f o l d e d -

p o t e n t i a l mo d e l o f t r i t o n - c 1 us t e r s t a t e s ma k e s l i m i t e d b u t s i g n i f i c a n t

p r e d i c t i o n s . Good a g r e e m e n t b e t w e e n t h e o r e t i c a l e x c i t a t i o n e n e r g i e s f o r

3 16t h e ( s d ) ^ c o n f i g u r a t i o n o f 0+ t a n d known e x p e r i m e n t a l l e v e l s i n t h e

19g r o u n d - s t a t e b a n d o f F c o n s t i t u t e s e v i d e n c e o f t r i t o n c l u s t e r i n g

o u t s i d e a c l o s e d - s h e l l c o r e . An a p p r o x i m a t e c o r r e s p o n d e n c e o f p r e d i c t e d

2 N + L=6 s t r u c t u r e t o t r i t o n - t r a n s f e r s p e c t r a r e v e a l s w e a k - c o u p 1 i n g e f f e c t s

i n ^ 0 a n d J 7r = 7 / 2+ c a n d i d a t e s i n ^ N . As a h i g h e r - o r d e r c a l c u l a t i o n f r o m

a m i c r o s c o p i c a p p r o a c h , t h e S U ( 3 ) s h e l l mo d e l p r o v i d e s a m o r e r i g o r o u s

i n t e r p r e t a t i o n o f e x p e r i m e n t a l r e s u l t s . A c o r r e l a t i o n b e t w e e n p r e d i c t e d

6 3t r i t o n s p e c t r o s c o p i c f a c t o r s a n d m e a s u r e d ( L i , He ) c r o s s s e c t i o n s l e a d s

1 8t o s u g g e s t e d s p i n v a l u e s i n 0 . D e s p i t e t h e d e t a i l e d s p l i t t i n g o f

s p e c t r o s c o p i c s t r e n g t h , t h e S U ( 3 ) s h e l l mo d e l c o n f i r m s t h e g e n e r a l o u t

l i n e o f t r i t o n - c 1 us t e r s t r u c t u r e g i v e n by t h e f o l d e d - p o t e n t i a l m o d e l .

T h r o u g h e x p e r i m e n t a n d t h e o r y , t h e r e f o r e , p a r t i c 1e - h o l e c o n f i g u r a

t i o n s a n d c l u s t e r i n g p h e n o m e n a e m e r g e as r e l a t i v e l y s i m p l e f e a t u r e s

common t o t h e s t r u c t u r e o f l i g h t n u c l e i .

and t r i t o n binding energy.6 3

I n S e c t i o n 9 - 3 , we d i s c u s s t h e 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

a n d p r e s e n t a f i n i t e - r a n g e p r e d i c t i o n o f a n g u l a r d i s t r i b u t i o n s f o r t h e

^3 C ( ^ L i , ^ H e ) ^ N r e a c t i o n ( F i g . 9 . 8 ) . We i n v e s t i g a t e h e r e t h e a d e q u a c y

o f a z e r o - r a n g e i n t e r a c t i o n , w h i c h r e d u c e s t h e c a l c u l a t i o n t o a t h r e e -

d i m e n s i o n a l i n t e g r a l o v e r r^ = ( ^ 2 0 /^ 2 3 ra a s s u m P t ' on

~ r ( ^ L i ) = r (3 H e ) = r ( t ) n e g l e c t s i n t e r n a l s t r u c t u r e b u t may a p p r o x i m a t e a

r e l a t i v e s - s t a t e o f t h e p r o j e c t i l e ( A u 7 0 ) .

B e c a u s e o f a h i g h s e n s i t i v i t y t o a d j u s t a b l e p a r a m e t e r s ( G a 7 2 ) ,

z e r o - r a n g e c a l c u l a t i o n s v i a t h e c o d e DWUCK ( K u 6 9 ) r e q u i r e s u b s t a n t i a l

1 3 6 3 16n o r m a l i z a t i o n t o t h e C ( L i , He ) N d a t a o f F i g . A . l a . T h e r e s u l t i n g

16 *p r e d i c t i o n s o f F i g . A . l b f a v o r L=6 o v e r L=A f o r N ( 1 1 . 8 1 ) b u t a p p e a r

i n c o n c l u s i v e f o r ( 7 . 6 5 ) . T h e s e t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s

show a q u a l i t a t i v e s i m i l a r i t y t o f i n i t e - r a n g e r e s u l t s ( F i g . A . 2 ) . A l

t h o u g h t h e z e r o - r a n g e a s s u m p t i o n l e a d s t o m o r e s t r u c t u r e d c u r v e s , i t

c a n p a r t i a l l y a b s o r b f i n i t e - r a n g e e f f e c t s t h r o u g h an i n c r e a s e i n t h e

r a d i u s p a r a m e t e r o f t h e f i n a l - s t a t e p o t e n t i a l . ZRDWBA a l l o w s a b e t t e r

f i t t o t h e d a t a f o r ( 5 . 7 3 ) ; FRDWBA b e t t e r r e p r o d u c e s t h e e x p e r i m e n t a l

j 6 * 6 3a n g u l a r d i s t r i b u t i o n o f N ( 7 . 6 5 ) . F o r t h e ( L i , He ) r e a c t i o n , t h e r e

f o r e , a z e r o - r a n g e i n t e r a c t i o n p r o v e s t o be a r e a s o n a b l e a p p r o x i m a t i o n

i n DWBA c a l c u l a t i o n s .

Appen dix A ZRDWBA

T h e v a l u e s o f N a n d L f o r f i n a l s t a t e s i n t h e

13 6 3 16C ( L i , He ) N r e a c t i o n a r e e x p l a i n e d i n t h e c a p t i o n t o

F i g . 9 - 8 . T h e o p t i c a l p o t e n t i a l s a r e l i s t e d i n T a b l e A . l ,

w h e r e a l t e r n a t e p a r a m e t e r s f r o m R e f s . S c 73 a n d G a 73

l e a d t o p o o r f i t s f o r t h e 5+ s t a t e a t 5 - 73 MeV a n d t h e

( 3+ ) s t a t e a t 3 - 9 6 MeV r e s p e c t i v e l y . I n o r d e r t o r e d u c e

t h e a m p l i t u d e o f o s c i l l a t i o n i n a L=5 o r L=6 c u r v e f o r

t h e h i g h - s p i n s t a t e a t 11. 21 MeV ( S e c t i o n 9 - 2 ) , a l a r g e

v a l u e o f t h e r a d i u s p a r a m e t e r r ^ = 2 .2 i s c h o s e n f o r t h e

16 13W o o d s - S a x o n p o t e n t i a l g e n e r a t i n g a N= C + t b o u n d s t a t e .

F i g u r e A . 2 C o m p a r i s o n o f z e r o - r a n g e t o f i n i t e - r a n g e p r e d i c t i o n s

T h e o r e t i c a l c u r v e s f r o m F i g . A . l a r e p l o t t e d w i t h c o r

r e s p o n d i n g r e s u l t s f r o m F i g . 9 . 8 . A l t h o u g h o p t i c a l p o

t e n t i a l s a r e t h e s a m e , d i f f e r e n t r a d i u s p a r a m e t e r s o f

r ^ = 2 . 2 a n d r Q= ^' 7 a r e u s e d i n t h e r e s p e c t i v e c a l c u l a t i o n s .

I n a d d i t i o n t o L=3 a nd L=5 f o r (3 • 9 6 , ( 3+ ) ; 5 . 7 3 , 5+ ) ,

we c o n s i d e r L=A a n d L=6 f o r ( 7 . 6 5 , 1 1 . 8 1 ) ( S e c t i o n 9 - 2 ) .

Figures A.1a,b Zero-range DWBA c a lc u l a t i o n s

®c.in dc.m.

Oc.tn.

(JS/qr/) "^up/xjp

do-/dIlcm (1181) (y-b/sr)

d<r/dftc m (765) (/u.b/sr)

Reference

Alternate

TA BL E A. 1 OPTICAL POTENTIALS

v (r) = -V f (r) - iV f (r) + V (r), where f(r) = i r i l v 1 + exp

, 1/3 ■

Projectile V

r aOr r V.

(fm) (fm) (MeV)

1.30 0.70 26.8

r0i a.1

(fm) (fm)

1.70 0.90

He 1 6 9 - 0 .2 E 1.14 0.50 6.5 + 0.177E 1.82 0.56

6Li 176.4 1.21 0.773 10.4 2.17 0.817

He 170.0 1.14 0.723 20.0 1.60 0.80

17 18T h r e e - n u c l e o n t r a n s f e r d a t a f r o m t a r g e t s o f ' 0 a n d 0 l i e b e y o n d

t h e s c o p e o f a s e a r c h f o r p n ( s d )3 c o n f i g u r a t i o n s . T h r e e - n u c l e o n c l u s t e r

i n g i n t h e A=20 a n d A= 2 1 n u c l e i , m o r e o v e r , i s e x p e c t e d t o be r e d u c e d when

v a l e n c e n e u t r o n s o f t h e t a r g e t i n t e r a c t w i t h t r a n s f e r r e d n u c l e o n s i n t h e

sd s h e l l . An i d e n t i f i c a t i o n o f ( f p ) t r a n s f e r , h o w e v e r , i s r e l e v a n t t o

20t h e c l a s s i f i c a t i o n o f h i g h l y e x c i t e d s t a t e s i n N e . T h e c o u p l i n g o f a

19 20d ^ n e u t r o n t o s t a t e s o f F i s i n v o l v e d i n F; l i t t l e - k n o w n T z = 3/2

21s t r u c t u r e i s p r e s e n t e d i n F.

T h e ^7 0 ( ^ L i , t ) 7 0 Ne r e a c t i o n ( F i g . B . l ) l a r g e l y p o p u l a t e s T =0 c o n

f i g u r a t i o n s w i t h t w o p r o t o n s a nd t w o n e u t r o n s o u t s i d e a c l o s e d p s h e l l .

S u c h s t a t e s o f an e v e n - e v e n , 4 N n u c l e u s a r e f a v o r a b l e t o a l p h a - p a r t i c l e

c l u s t e r i n g . I d e n t i f i c a t i o n o f a 1p h a - c 1 us t e r s t a t e s i n t h e ^8 0 ( 7 L i , t ) ^ 0 Ne

r e a c t i o n ( C o 7 6 , s e e F i g . 6 . 5 ) i s c o m p l e m e n t e d by r e c e n t s p i n a s s i g n m e n t s

f r o m a ^8 0 ( ^ 2 C , 8 B e ) 7 ^ N e ( a ) ^ 8 0 ( g . s . ) c o r r e l a t i o n s t u d y ( S a 7 7 ) , w h i c h sum-

20m a r i z e s t h e b a n d s t r u c t u r e o f N e . I n t h r e e - n u c l e o n t r a n s f e r d a t a ,

s i z e a b l e p e a k s o c c u r f o r h i g h - s p i n me mbe r s o f t h e 0* g r o u n d - s t a t e b a n d ,

i n c l u d i n g a 8+ s t a t e a b s e n t f r o m t h e ( 7 L i , t ) r e a c t i o n ( V a 7 3 ) • A l t h o u g h

6 — 2 0 ' * — t h e ( L i , t ) r e a c t i o n a l s o s e l e c t s a 0 b a n d b e g i n n i n g a t Ne ( 5 - 784 , 1 )

( F i g . B . l ) , t h e r e l a t i v e s t r e n g t h o f t h i s b a nd i s g r e a t e r i n a l p h a -

2 0 *p a r t i c l e t r a n s f e r d a t a , e . g . a t Ne ( 1 5 - 3 4 , 7 ) ( F i g . 6 . 5 ) . T h e c o n s i s

t e n t p r e s e n c e o f s u c h a l p h a - c 1 us t e r s t r u c t u r e i n a ^7 0 ( 8 L i , t ) 7 0 Ne s p e c

t r u m r e f l e c t s s t r o n g b i n d i n g o f t h e t a r g e t n e u t r o n t o t h e t r a n s f e r r e d

3 3^ H e . S i n c e t h i s e f f e c t s h o u l d be w e a k e r i n a ( s d ) ( f p ) c o n f i g u r a t i o n ,

we i n v e s t i g a t e n e g a t i v e - p a r i t y s t a t e s a t h i g h e x c i t a t i o n . T h e 9 s t a t e

a t 2 1 . 0 9 MeV i s d o m i n a n t i n He t r a n s f e r b u t n o t i n a l p h a - p a r t i c l e

Appendix B 2^Ne, 2^F, F

F i g u r e B.

2 , 7 0 (6 L i , 3 H e ) 2 ° F

3 ' 8 0 ( 6 L i , 3 H e ) 2 ' F

20E x c i t a t i o n e n e r g i e s a n d c o n t a m i n a n t p e a k s i n t h e Ne s p e c

t r u m a r e d e t e r m i n e d f r o m t h e ^8 0 ( 8 L i , t ) ^ 8 Ne r e a c t i o n . S i n c e

19 * 20 *Ne ( 0 . 2 3 8 ) i s e q u i v a l e n t t o Ne ( 1 3 - 0 ) i n Q - v a l u e , known

e n e r g y l e v e l s ( A j 7 8 ) a r e s u b s t i t u t e d b e l o w 8 MeV i n e x c i t a -

20t i o n . T r i t o n t r a n s f e r i n t o F shows a n a l o g o u s c o n t a m i n a -

. 16. ,6. . 3 ,, J 9 , . . . . 21t i o n f r o m t h e 0 ( L i , He ) F r e a c t i o n , b u t d a t a f o r F

r e f l e c t o n l y h y d r o g e n i m p u r i t y i n t h e t a r g e t . A d d i t i o n a l

e x c i t a t i o n e n e r g i e s a r e l i s t e d i n T a b l e B . l ; s p i n a s s i g n m e n t s

20 * 20 *a r e f r o m R e f s . A j 7 8 , E n 7 3 . Ne ( 2 1 . 0 9 ) , F ( 9 - 9 0 ) a nd

^ F ( 8 . 7 9 ) h a v e d a / d ^ c m =100 y b / s r .

t l 7 0 ( 6 L i , t ) 2°Ne

EXCITATION ENERGY

C O U N T S

4 0 0 -

EXCITATION

ENERGY (M

C O U N T S

m 00Or;

ii "o>n

<Ti 'wz I(D< ro

TABLE B . l A = 20 ,21

17 6 3 20 18 6 3 210( Li, He) F 0( Li, He) F

E t . = 46 MeV Li

e, , = is0lab

E W E (1)X X

(MeV) (MeV)

0.04 0.010.69 0.300.85 1.771.85 2.062.04 4.92.21 5.782.97 6.84.54 7.375.37 8.796.88 9.367.67 10.88.30 11.58.67 11.759.90 12.71

10.82 13.0011.50 13.6412.0 14.614.10 16.91

n\ IQ *v calibrated from F (0.197, 2.780, 4.648, 6.925, 8.953, 10.411)

AE - ± 20 keV, 2 MeV < E < 14 MeVx± 40 keV, E < 2 MeV, E > 14 MeV x x

20 * -t r a n s f e r . T o g e t h e r w i t h a r e l a t i v e e n h a n c e m e n t o f Ne ( 1 6 . 6 2 , 7 ) i n

t h e f o r m e r r e a c t i o n , t h i s o b s e r v a t i o n s u p p o r t s a c l a s s i f i c a t i o n o f t h e

t w o s t a t e s w i t h i n a 0 ^ b a nd ( S a 7 7 ) a n d s u g g e s t s an e x i s t e n c e o f l a r g e

3 20 * +( s d ) ( f p ) c o m p o n e n t s . M o r e o v e r , t h e m i n o r r o l e o f Ne ( 1 2 . 5 9 , 6 ;

1 7 - 3 0 , 8+ ) i n t h e ( ^ L i , t ) r e a c t i o n i s c o n s i s t e n t w i t h a c o n s i d e r a t i o n o f

( f p ) * * s t r u c t u r e ( S a 7 7 ) .

1 7 fi 3 70T h e 0 ( L i , He ) F r e a c t i o n ( F i g . B . 2 ) s t r o n g l y p o p u l a t e s a T =1

2 0 *s t a t e a t 9 . 9 0 MeV. As a c a n d i d a t e f o r t h e T ^=0 a n a l o g s t a t e , Ne ( 1 9 - 9 )

2 0 * +a p p r o x i m a t e s t h e s e p a r a t i o n w h i c h Ne ( 1 0 . 2 7 2 , 2 , T = l ) e s t a b l i s h e s f r o m

20F ( g . s . , 2+ ) ( A j 7 8 ) , a n d i t s a t i s f i e s a c r i t e r i o n t h a t t h e c r o s s s e c t i o n

be o n e - h a l f t h e T =1 v a l u e ( S e c t i o n 9 . 3 ) . On t h e b a s i s o f e x c i t a t i o nz

20 *e n e r g i e s , p e a k s a t Ne ( 1 4 . 8 1 , 1 5 - 9 2 ) may a l s o c o r r e s p o n d i n p a r t t o

2 ^ F ” ( 4 . 5 4 , 5 • 3 7 ) - T h e f i r s t t w o s t a t e s o f 2 ^ F , known t o h a v e J 7T= 2+ a n d

3* r e s p e c t i v e l y , c o u l d a r i s e f r o m t h e c o u p l i n g o f a c / 2 n e u t r o n t o ^ e

l / 2+ g r o u n d s t a t e o f ^ F . A l t h o u g h ^ F ( 0 . 1 9 7 , 5 / 2* ) c o u l d s i m i l a r i l y

a c c o u n t f o r t h e 4+ a n d 5* s t a t e s o f 2 ^ F ( F i g . B . 2 ) , t h e m o n o t o n i c s e q u e n c e

21o f s p i n s s u g g e s t s s t r o n g c o u p l i n g . T h e f i r s t t w o s t a t e s o f F , w i t h

_ 1 QJ = 5/2 a n d 1/2 , a l s o a p p e a r r e l a t e d t o t h e a b o v e s t a t e s o f F . S i n c e

21l e v e l s o f F a r e known o n l y b e l o w 6 MeV i n e x c i t a t i o n ( E n 7 3 ) , t h e

6 3( L i , He ) r e a c t i o n i d e n t i f i e s new s t a t e s o f t h i s T ^ = 3/2 n u c l e u s , n o t a b l y

a t 8 . 7 9 MeV a n d 12. 71 MeV. O w i n g t o s m a l l a b s o l u t e c r o s s s e c t i o n s and

20 21u n k n o w n s p i n v a l u e s i n F a n d F , t h e i n t e r p r e t a t i o n o f t r i t o n - t r a n s f e r

17 18d a t a f r o m 0 a n d 0 t a r g e t s i s a m a t t e r f o r t h e f u t u r e .

REFERENCES

A j 76 A j z e n b e r g - S e l o v e , F . , N u c l . P h y s . A 2 6 8 ( 1 9 7 6 ) 1 .

A j 77 A j z e n b e r g - S e l o v e , F . , N u c l . P h y s . A 2 8 i ( 1 9 7 7 ) 1 .

A j 78 A j z e n b e r g - S e l o v e , F . , N u c l . P h y s . A 3 0 0 ( 1 9 7 8 ) 1.

An7** A n y a s - W e i s s , N . , J . C . C o r n e l l , P . S . F i s h e r , P . N . Hudson ,A . M e n c h a c a - R o c h a , D . J . M i l l e n e r , A . D . P a n a g i o t o u , D . K . S c o t t ,D. S t r o t t m a n , D .M . B r i n k , B. B u c k , P . J . E l l i s and T . E n g e i a n d , P h y s . R ep . T 2C ( 1 9 7 * 0 2 0 1 .

Au70 A u s t e r n , N . , D i r e c t N u c l e a r R e a c t i o n T h e o r i e s ( W i l e y - I n t e r - s c i e n c e , New Y o r k , 1 9 7 0 ) .

Au 76 A u e r b a c h , E . H . , c o d e A -T H R EE , B r o o k h a v e n N a t i o n a l L a b o r a t o r y .

Ba69 B a s s a n i , G . , T . H . K r u s e , N. S a u n i e r and G. S o u c h e r e , P h y s . L e t t .3 0 B ( 1 9 6 9 ) 6 2 1 .

Ba70 B a s s a n i , G . , A . C a l a m a n d , G. P a p p a l a r d o , N. S a u n i e r and B .M .T r a o r e , N o t e C E A - N - 1 3 9 0 ( 1 9 7 0 ) 2 7 .

Ba71a B a s s a n i , G . , N. S a u n i e r , B .M . T r a o r e , G. P a p p a l a r d o andA . F o t i , S u p p l . J . de P h y s . 3 2 / 1 9 7 1 ) C 6 - 1 3 3 , 1 3 5 .

Ba7 1b B a k e r , W . L . , C . E . B u s c h , J . A . Keane and T . R . D o no g h u e , P h y s .R e v . C 3 0 9 7 0 W .

Ba72 B a s s a n i , G . , A . C u n s o l o , A . F o t i , C. G e r a r d i n , M. L e p a r e u x ,G. P a p p a l a r d o , N. S a u n i e r , A . S t r a z z e r i and M. W e r y , N o t e C E A - N - 1 6 0 0 ( 1 9 7 2 ) 1 5 .

Be69 B e v i n g t o n , P . R . , D a t a R e d u c t i o n and E r r o r A n a l y s i s f o r t h eP h y s i c a l S c i e n c e s " ( M c G r a w - H i l l , New Y o r k , " 1 ^ 9 7 ^

Be70 B e t h g e , K . , D . J . P u l l e n and R. M i d d l e t o n , P h y s . R e v . C . 2 ( 1 9 7 0 ) 3 9 5 .

Be75 B e u k e n s , R . P . , T . E . D r a k e and A . E . L i t h e r l a n d , P h y s . L e t t .56B ( 1 9 7 5 ) 2 5 3 .

B i 71 B in g h a m , H . G . , H . T . F o r t u n e , J . D . G a r r e t t and R. M i d d l e t o n ,P h y s . R e v . L e t t . 2 6 ( 1 9 7 1 ) 1***»8.

B i 72 B in g h a m , H . G . , a nd H . T . F o r t u n e , P h y s . R e v . C 6 / 1 9 7 2 ) 1 9 0 0 .

B i 7 3 a B in g h a m , H . G . , H . T . F o r t u n e , J . D . G a r r e t t a nd R. M i d d l e t o n ,P h y s . R e v . C 7 ( 1 9 7 3 ) 5 7 .

B i 7 3 b B in g h a m , H . G . , a n d H . T . F o r t u n e , P h y s . R e v . C 7 / 1 9 7 3 ) 2 6 0 2 .

B i 75 B in g h a m , H . G . , M . L . H a l b e r t , D . C . H e n s l e y , E . Newman, K.W.Kemper a nd L . A . C h a r l t o n , P h y s . R e v . C 1 i ( 1 9 7 5 ) 1 9 1 3 .

B o l 5 B o h r , N . , P h i l . Mag. 3 0 ( 1 9 1 5 ) 5 8 1 .

Bu75 B u c k , B . , C . B . D o v e r a nd J . P . V a r y , P h y s . R e v . C 1 1 ( 1 9 7 5 ) 1 8 0 3 .

Bu 7 6 B u r t e b a e v , N . T . , A . D . V o n g a i , Y . A . G l u k h o v , A . D . D u i s e b a e v ,G . N . I v a n o v , V . l . K a n a s h e v i c h , S . V . L a p t e v , A . A . O g l o b l i n ,S . B . S a k u t a , A . V . S p a s s k i i , I . B . T e p l o v and V . l . C h u e v , S o v . J .N u c l . P h y s . 2 4 ( 1 9 7 6 ) 4 5 7 .

Bu77a B u c k , B . , and A . A . P i l t , N u c l . P h y s . A 2 8 0 ( 1 9 7 7 ) 1 3 3 .

Bu77b B u c k , B . , H. F r i e d r i c h a nd A . A . P i l t , N u c l . P h y s . A 2 9 £ ( 19 7 7 ) 2 0 5 .

Bu 78 B u c k , B . , a nd A . A . P i l t , N u c l . P h y s . A 2 9 5 ( 1 9 7 8 ) 1 .

Ce6A C e r n y , J . , and R . H . P e h l , P h y s . R e v . L e t t . 1 2 ( 1 9 6 A ) 6 1 9 .

Ch76 C h u a , L . T . , F . D . B e c c h e t t i , J . l a n e c k e and F . L . M i l d e r , N u c l .P h y s . A 2 7 3 ( 1 9 7 6 ) 2 A 3 .

Ch77 Chew, S . H . , J . Lowe, J . M . N e l s o n and A . R . B a r n e t t , N u c l . P h y s .A 2 8 6 ( 1 9 7 7 ) A 5 1 .

C17A C l e m e n t , D . , and W. Z a h n , P h y s . L e t t . A 8 B ( 1 9 7 A ) 183.

C 178 C l a r k , M . E . , K.W. Kemper and J . D . F o x , B u l l . Am. P h y s . S o c .2 3 . ( 1 9 7 8 ) 5 ^ 0 .

C066 C o m f o r t , J . R . , J . F . D e c k e r , E . T . L y n k , M .O . S c u l l y and A . R .Q u i n t o n , P h y s . R e v . 1 5 0 ( 1 9 6 6 ) 2 A 9 .

C067 Col l a r d , H . R . , a nd R. H o f s t a d t e r , L a n d o l t - B 3 r n s t e i n , e d . K . - H .H e l lw e g e and H. S c h o p p e r , 12 (19 67)2*1 ( S p r i n g e r - V e r l a g , B e r l i n ) .

Co7A C o b e r n , M . E . , t h e s i s , Y a l e U n i v e r s i t y ( 1 9 7 A ) .

Co76 C o b e r n , M . E . , D . J . P i s a n o and P . D . P a r k e r , P h y s . R e v . C 1A( 1 9 7 6 ) A 9 1 .

Co77 C o b e r n , M . E . , and P . D . P a r k e r , P h y s . R e v . C 1 5 ( 1 9 7 7 ) 1929>1 6 ( 1 9 7 7 ) 9 2 0 .

De71 D6t r a z , C . , C . E . M oss , C . D . Z a f i r a t o s and C . S . Z a i d i n s , N u c l .P h y s . A 1 6 7 ( 1 9 7 1 ) 3 3 7 »

D ?77 D i x o n , W . R . , and R . S . S t o r e y , N u c l . P h y s . A 2 8 A ( 1 9 7 7 ) 9 7 .

Do7A D o v e r , C . B . , and J . P . V a r y , B r o o k h a v e n N a t i o n a l L a b o r a t o r y r e p o r t B N L - 1 9 3 3 2 ( 1 9 7 A ) .

Do75 D o v e r , C . B . , P . J . M o f f a a nd J . P . V a r y , P h y s . L e t t . 5 6 B ( 1 9 7 5 ) A .

D r 7 7 D r a i n , D . , B. Chambon, M. L a m b e r t , C. P a s t o r , N. P e r s e h a y e ,J . L . V i d a l and P . M i d y , P h y s . R e v . C 1 5 0 9 7 7 ) 5 5 1 .

E 1 70 E l l i s , P . J . , a nd T . E n g e l a n d , N u c l . P h y s . A1A AQ 9 7 0 ) 1 6 1 .

En73 E n d t , P . M . , and C. v a n d e r L e u n , N u c l . P h y s . A 2 1 A ( 1 9 7 3 ) 1 .

F a 7 5 F a l k , W . R . , A . D j a l o e i s and D. In g h am , N u c l . P h y s . A 2 5 2 ( 1 9 7 5 )A 5 2 .

F e 6 0 F e s h b a c h , H . , N u c l e a r S p e c t r o s c o p y , e d . F . A j z e n b e r g - S e l o v e ,( A c a d e m i c P r e s s , New Y o r k , 1 9 & 0 ) , P a r t B, p . 6 2 5 .

F 1 77 F i f i e l d , L . K . , T . J . M . Symons, C . H . Z im merm an, M . J . H u r s t ,F . W a t t and K.W. A l l e n , P h y s . L e t t . 6 8 6 ( 1 9 7 7 ) 1 2 5 .

Ga72 G a r r e t t , J . D . , H . G . B in g h a m , H . T . F o r t u n e and R. M i d d l e t o n ,P h y s . R e v . C 5 ( 1 9 7 2 ) 6 8 2 .

Ga73 G a r r e t t , J . D . , and 0 . H a n s e n , N u c l . P h y s . A 2 1 2 ( 1 9 7 3 ) 6 0 0 .

G 176 G l o e c k n e r , D . H . , M . H . M a c f a r l a n e a nd S t e v e n C. P i e p e r , A r g o n n eN a t i o n a l L a b o r a t o r y r e p o r t A N L - 7 6 - 1 1 .

Go71 G o l ' d b e r g , V . Z . , V . V . D a v y d o v , A . A . O g l o b l i n , S . B . S a k u t a andV . l . C h u e v , S o v . J . N u c l . P h y s . 1 2 ( 1 9 7 1 ) 1 6 .

Ha68 H a r v e y , M . , A d v . i n N u c l . P h y s . , e d . M. B a r a n g e r and E. V o g t ,1 ( 1 9 6 8 ) 6 7 .

Ha7A H a n s o n , D . L . , R . G . S t o k s t a d , K . A . E r b , C. O l m e r a nd D . A . B r o m l e y ,P h y s . R e v . C 9 ( 1 9 7 A ) 9 2 9 .

H a 7 6 a Hamm, M . , C .W. T o w s l e y , R. H a n u s , K . G . N a i r a nd K. N a g a t a n i ,P h y s . R ev . L e t t . 3 6 ( 1 9 7 6 ) 8 A 6 .

H a7 6b Hamm, M . , C .W . T o w s l e y , K . G . N a i r , R. Hanus and K. N a g a t a n i ,B u l l . Am. P h y s . S o c . H ( 1 9 7 6 ) 5 5 A .

H a 7 6 c Hamm, M . , t h e s i s , T e x a s ASM U n i v e r s i t y ( 1 9 7 6 ) .

H a7 7 H a f t e l , M . I . , R . G . A l l a s , L . A . B e a c h , R . 0 . B o n d e l i d , E . L .P e t e r s e n , I . S l a u s , J . M . L a m b e r t and P . A . T r e a d o , P h y s . R e v .C J 6 0 9 7 7 ) A 2 .

H a78 Hamm, M . , a n d K. N a g a t a n i , P h y s . R e v . C 1 7 ( 1 9 7 8 ) 5 8 6 .

He6A H e c h t , K . T . , S e l e c t e d T o p i c s i n N u c l e a r S p e c t r o s c o p y , e d .B . J . V e r h a a r , p . 51 ( N o r t h - H o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 196A ) .

H e 7 l H e c h t , K . T . , N u c l . P h y s . A 1 7 0 ( 1 9 7 1 ) 31* .

H e7 5 H e c h t , K . T . , a nd D. B r a u n s c h w e i g , N u c l . P h y s . A 2 M ( l 9 7 5 ) 3 6 5 .

H i 66 H i r d , B . , N u c l . P h y s . 8 6 ( 1 9 6 6 ) 2 6 8 .

I c 7 3 I c h i m u r a , H . , A. A r i m a , E . C . H a l b e r t and T . T e r a s a w a , N u c l .P h y s . A 2 0 4 ( 1 9 7 3 ) 2 2 5 *

J a 7 6 J a h n , R . , G . J . W o z n i a k , D . P . S t a h e l and J . C e r n y , P h y s . R e v .L e t t . 3 7 ( 1 9 7 6 ) 8 1 2 .

J a 7 7 J a u s e l - H i i s k e n , S. and H. F r e i e s l e b e n , Z . f u r P h y s . A 2 8 3 ( 1 9 7 7 )3 6 3 *

Ko77 K o u z e s , R . , D. M u e l l e r , F . C a l a p r i c e and D . J . M i l l e n e r , B u l l .Am. P h y s . S o c . 2 2 ( 1 9 7 7 ) 5 5 3 *

Ku66 K u o , T . T . S . , and G . E . Bro w n , N u c l . P h y s . 8 5 ( 1 9 6 6 ) ^ 0 .

Ku69 K u n z , P . D . , U n i v e r s i t y o f C o l o r a d o r e p o r t C 0 0 - 5 3 5 “ 6 0 6 , 6 1 3 ( 1 9 6 9 ) .

Ku77 K u an , H . M . , D . G . S h i r k and S. F i a r m a n , P h y s . R e v . C 1 5 ( 1 9 7 7 ) 5 6 9 *

L a 6 3 L a n g , D . W . , N u c l . P h y s . ^ 2 ( 1 9 6 3 ) 3 5 3 *

L e 6 7 L e e , F . D . , R . W . K ro n e and F . W . P r o s s e r , J r . , N u c l . P h y s .A 9 6 ( 1 9 6 7 ) 2 0 9 .

L e 7 2 L e m a i r e , M . - C . , M . C . Mermaz and K . K . S e t h , P h y s . R e v . C 5_(1972)328 .

L i 70 L i e , S . , T . E n g e l a n d and G. D a h l l , N u c l . P h y s . A l 5 6 ( 1 9 7 0 ) b k S .

L 1 71 L i e , S . , and T . E n g e l a n d , N u c l . P h y s . A l 6 9 ( 1 9 7 1 ) 6 1 7 .

L i 72 L i n d g r e n , R . A . , H . H . G u t b r o d , H.W. F u l b r i g h t and R . G . M ark h am ,P h y s . R e v . L e t t . 2 9 ( 1 9 7 2 ) 7 9 8 .

L i 7 6 a L i e , S . , and T . E n g e l a n d , N u c l . P h y s . A 2 6 7 ( 1 9 7 6 ) 1 2 3 .

L i 76b L i , T . K . , D. D e h n h a r d , R . E . Brown and P . J . E l l i s , P h y s . R ev .C 1 3 ( 1 9 7 6 ) 5 5 .

L u 69 L u , C . C . , M . S . Z is m a n a nd B . G . H a r v e y , P h y s . R e v . 1 8 6 ( 1 9 6 9 ) 1 0 8 6 .

M a6 8 M a n g e l s o n , N . F . , B . G . H a r v e y and N . K . G l e n d e n n i n g , N u c l . P h y s .A l 1 9 ( 1 9 6 8 ) 7 9 .

Ma73 M a i r l e , G . , and G . J . W a g n e r , Z . P h y s . 2 5 3 ( 1 9 7 3 ) 3 2 1 .

Ma78 M a i r l e , G . , G . J . W a g n e r , P . D o l l , K . T . K n o p f l e and H. B r e u e r ,N u c l . P h y s . A 2 9 9 ( 1 9 7 8 ) 3 9 .

M i 7 0 M i d d l e t o n , R . , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e onN u c l e a r R e a c t i o n s I n d u c e d by H eavy I o n s , H e i d e l b e r g ( 1 969) , e d . R. Bock and W .R . H e r i n g , p . 2 6 3 ( N o r t h - H o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 1 9 7 0 ) .

M i 7 2 M i l l e n e r , D . J . , t h e s i s , U n i v e r s i t y o f 0 x f o r d ( 1 9 7 2 ) .

H i 73 M i l l e n e r , D . J . , and P . E . H o dg s o n , N u c l . P h y s . A 2 0 9 0 9 7 3 ) 5 9 .

M i7 A M i d d l e t o n , R . , and C . T . Adams, N u c l . I n s t , and M e t h . 1 1 8 ( 1 97A)3 2 9 .

H 1 75 M i l l e n e r , D . J . , and D. K u r a t h , N u c l . P h y s . A255. ( 1 9 7 5 ) 31 5 .

M i 76 M i l l e n e r , D . J . , B r o o k h a v e n N a t i o n a l L a b o r a t o r y r e p o r t BNL-2 1 8 2 9 ( 1 9 7 6 ) .

H 1 7 7 M i l l e n e r , D . J . , p r i v a t e c o m m u n i c a t i o n .

Mo70a M o r g a n , G . L . , D . R . T i l l e y , G . E . M i t c h e l l , R . A . H i l k o and N . R .R o b e r s o n , N u c l . P h y s . A l A 8 ( 1 9 7 0 ) A 8 0 .

Mo70b M o r g a n , G . L . , D . R . T i l l e y , G . E . M i t c h e l l , R . A . H i l k o a nd N . R .R o b e r s o n , P h y s . L e t t . 3 2 B ( 1 9 7 0 ) 3 5 3 *

Mo77 M o f f a , P . J . , C . B . D o v e r and J . P . V a r y , P h y s . R e v . C 1 6 ( 1 9 7 7 )1 8 5 7 .

N a7 3 N a g a t a n i , K . , D . H . Y o u n g b l o o d , R. K e n e f i c k a nd J . B r o n s o n ,P h y s . R e v . L e t t . 3 1 0 9 7 3 ) 2 5 0 .

N e 7 0 N e g e l e , J . W . , P h y s . R e v . C 1 ( 1 9 7 0 ) 1 2 6 0 .

No70 N o r t h c l i f f e , L . C . , a n d R . F . S c h i l l i n g , N u c l e a r D a t a T a b l e sA 7 O 9 7 0 ) 2 3 3 .

0 g 7 0 O g l o b l i n , A . A . , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c eon N u c l e a r R e a c t i o n s i n d u c e d by H eavy I o n s , H e i d e 1 b e r g ( 1 9 6 9 ) , e d . R. Bock and W .R . H e r i n g , p . 231 ( N o r t h - H o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 1 9 7 0 ) .

Og73 O g l o b l i n , A . A . , S o v . J . o f P a r t i c l e s and N u c l e i 3 . ( 1 9 7 3 ) A 6 7 *

0171 O l l e r h e a d , R.W., G .F.R. A l l e n , A . M . B a x t e r and J . A . K u e h n e r ,C a n . J . P h y s . A 9 ( 1 9 7 1 ) 2589.

0 1 7 3 O l n e s s , J . W . , E . K . W a r b u r t o n and J . A . B e c k e r , P h y s . R e v .C 7 ( 1 9 7 3 ) 2 2 3 9 .

Pa 72 P a n a g i o t o u , A . D . , and H . E . G o v e , N u c l . P h y s . A 1 9 6 Q 9 7 2 ) 1A 5.

P i 73 P i s a n o , D . J . , J r . , t h e s i s , Y a l e U n i v e r s i t y ( 1 9 7 3 ) .

P i 77

R o73 a

S a 7 7 a

S a 7 7 b

S c 72

S c 7 3

S e 7 7

S I 59

S t 7 2

S t 7 3

S y 7 6 a

P i g n a n e l l i , M . , S . M i c h e l e t t i , I . l o r i , P . G u a z z o n i , F . G .R e s m i n i and J . L . E s c u d i & , P h y s . R ev . C 1 0 ( 1 9 7 4 ) 4 4 5 *

P i l t , A . A . , O . J . M i l l e n e r , H. B r a d l o w , 0 . D i e t z s c h , P . S . F i s h e r , W . J . N a u d e , W . D . M . Rae and D. S i n c l a i r , N u c l . P h y s . A 27 3 ( 1 9 7 6 ) 1 8 9 .

P i l t , A . A . , H . S . B r a d l o w , 0 . D i e t z s c h , P . S . F i s h e r , G. P r o u d - f o o t , W . D . M . Rae a nd D. S i n c l a i r , p r e p r i n t , U n i v e r s i t y o f O x f o r d ( 1 9 7 7 ) .

R i v e t , E . , R . H . P e h l , J . C e r n y and B . G . H a r v e y , P h y s . R ev .1 4 1 ( 1 9 6 6 ) 1 0 2 1 .

R o l f s , C . , A . M . C h a r l e s w o r t h a nd R . E . Azuma, N u c l . P h y s . A 199( 1 9 7 3 ) 2 5 7 .

R o l f s , C . , H . P . T r a u t v e t t e r , R . E . Azuma and A . E . L i t h e r l a n d , N u c l . P h y s . A 1 9 9 ( 1 9 7 3 ) 2 8 9 .

R o l f s , C . , I . B e r k a and R . E . Azuma, N u c l . P h y s . AJ_99.( 1 9 7 3 ) 3 0 6 .

R o o s , P . G . , D . A . G o l d b e r g , N . S . C h a n t , R. Woody I I I and W. R e i c h a r t , N u c l . P h y s . A 2 5 7 ( 1 9 7 6 ) 3 1 7 .

S a n d e r s , S . J . , t h e s i s , Y a l e U n i v e r s i t y ( 1 9 7 7 ) .

S a n d e r s , S . J . , L . M . M a r t z and P . D . P a r k e r , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e on N u c l e a r S t r u c t u r e , T o k y o ( 1 9 7 7 ) 6 2 5 , 6 2 6 , J . P h y s . So c . J a p a n 4 4 ^ ( 1 9 7 8 ) S u p p l . p . 6 4 8 .

S c o t t , D . K . , P . N . H u d s o n , P . S . F i s h e r , C . U . C a r d i n a l , N. A n y a s - W e i s s , A . D . P a n a g i o t o u , P . J . E l l i s and B. B u c k , P h y s . R e v .L e t t . 2 8 ( 1 9 7 2 ) 1 6 5 9 .

S c h u m a c h e r , P . , N . U e t a , H . H . Duhm, K . - l . Kubo a nd W . J . K l a g e s , N u c l . P h y s . A 2 1 2 ( 1 9 7 3 ) 5 7 3 .

S e n s , J . C . , S . M . R e f a e i , A . Ben Mohamed and A . P a p e , P h y s . R e v .c 16(1977)2129.S i l v e r s t e i n , E . A . , N u c l . I n s t , a nd M e t h . 1 ( 1 9 5 9 ) 5 3 *

S t o k s t a d , R . , I n t e r n a l R e p o r t No. 5 2 , 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 U n i v e r s i t y ( 1 9 7 2 ) .

S t r o t t m a n , D . , and D . J . M i l l e n e r , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e on N u c l e a r P h y s i c s , M u n i c h , 1 ( 1 9 7 3 ) 1 0 7 .

Symons, T . J . M . , t h e s i s , U n i v e r s i t y o f O x f o r d ( 1 9 7 6 ) .

Symons, T . J . M . , L . K . F i f t e l d , M . J . H u r s t , A . P a k k a n e n , F . W a t t , C . H . Z immerman and K.W. A l l e n , P h y s . L e t t . 6 3 B ( 1 9 7 6 ) 4 0 9 .

S y 7 7 Symons, L . K . F i f i e l d , E . F . G arm an , M . J . H u r s t , F . W a t t ,C . H . Z immerman and K .W. A l l e n , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e on N u c l e a r S t r u c t u r e , T o k y o ( l 9 7 7 ) 1 9 2 .

T h 6 7 T hom pson , D . R . , and Y . C . T a n g , P h y s . R e v . 1 5 9 ( 1 9 6 7 ) 8 0 6 .

T s 7 3 T s e r r u y a , I . , B. R o s n e r and K. B e t h g e , N u c l . P h y s . A213( 1 9 7 3 ) 2 2 .

T s 7 4 T s e r r u y a , I . , B. R o s n e r and K. B e t h g e , N u c l . P h y s . A 235( 1 9 7 4 ) 7 5 .

V a 7 3 a Van S t a d e n , J . , t h e s i s , H e i d e l b e r g U n i v e r s i t y ( 1 9 7 3 ) .

V a 7 3 b Van S t a d e n , J . , a nd K. B e t h g e , P r o c e e d i n g s o f t h e I n t e r n a t i o n a lC o n f e r e n c e on N u c l e a r P h y s i c s , M u n i c h , J_( 1 9 7 3 ) 4 7 3 .

V a 7 3 c V a r y , J . P . , a nd C . B . D o v e r , P h y s . R e v . L e t t . 3 1 ( 1 9 7 3 ) 1 5 1 0 .

V a 7 4 V a r y , J . P . , and C . B . D o v e r , B r o o k h a v e n N a t i o n a l L a b o r a t o r yr e p o r t B N L - 1 9 3 6 0 ( 1 9 7 4 ) .

V a 7 5 v a n d e r B o r g , K . , C . R . B in g h a m , R . J . de M e i j e r a nd A . v a n d e rWoude, KVI A n n u a l R e p o r t , G r o n i n g e n ( 1 9 7 5 ) 2 7 .

V a 7 6 v a n d e r B o r g , K . , R . J . de M e i j e r and A . van d e r Woude, N u c l .P h y s . A 2 7 3 ( 1 9 7 6 ) 1 7 2 .

V o64 V o g t , E . , D. M c P h e r s o n , J . K u e h n e r a nd E . A l m q v i s t , P h y s . R e v .1 3 6 B ( 1 9 6 4 ) 99»

We72 W e r y , M . , t h e s i s , S t r a s b o u r g ( 1 9 7 2 ) .

We73 W e r y , M . , N u c l . P h y s . A210_( 1 9 7 3 ) 3 2 9 .

Wo 7 8 Woods , C . W . , N. S t e i n and J . W . S u n i e r , P h y s . R ev . C 1 7 ( 1 9 7 8 ) 6 6 .

Y o 7 0 Y o u n g , A . M . , S . L . B l a t t a nd R . G . S e y l e r , P h y s . R e v . L e t t . 25( 1 9 7 0 ) 1 7 6 4 .

Z e 7 7 Z e l l e r , A . F . , K.W. K em per , T . R . O phe l and A . J o h n s t o n , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e on N u c l e a r S t r u c t u r e , T o k y o ( 1 9 7 7 ) 1 7 5 .

Z i 7 0 Z i s m a n , M . S . , E . A . M c C l a t c h i e a n d B . G . H a r v e y , P h y s . R e v . C2 ( 1 9 7 0 ) 1 2 7 1 .

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