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THE COLLECTIVE STRUCTURE OF TUNGSTEN NUCLEI
A D i s s e r t a t i o n
P r e s e n t e d to the F a c u l t y o f the G r a d u a t e S c hool
o f
Y a l e Un i v e r s i ty
in C a n d i d a c y for the D e g r e e of
D o c t o r o f P h i l o s o p h y
by
M a r k H e r b e r t M o r t e n s e n
D e c e m b e r , 1978
A B S T R A C T
T H E C O L L E C T I V E S T R U C T U R E OF T U N G S T E N NUCLEI
M A R K H E R B E R T M O R T E N S E N
Y A L E U N I V E R S I T Y 1978
178 180 l8lT h e n e u t r o n p a i r i n g c h a r a c t e r of s t a t e s in ’ W
, 182, 184, . . . ,a nd W h a s b e e n i n v e s t i g a t e d u s i n g the (p,t) r e a c t i o n at
an i n c i d e n t p r o t o n e n e r g y of 21 MeV. E x c i t a t i o n e n e r g i e s and
a n g u l a r d i s t r i b u t i o n s w e r e o b t a i n e d for all s t a t e s p o p u l a t e d
up to a p p r o x i m a t e l y 2 . 5 M e V of e x c i t a t i o n . E x c i t e d J ( tt) = 0(+)
178s t a t e s , m a n y p r e v i o u s l y u n r e p o r t e d , w e r e p o p u l a t e d in W at
997 keV 0%) , 1356 keV (3%), a n d 1643 keV (1%); in l8°W at
1516 keV (7%), 1095 keV (2%) a n d 2 3 5 6 keV (49;); in l82W at 1135
keV (14/), 2 5 2 0 keV (2%), 2 5 5 2 keV (1%), a n d 2 7 2 5 keV (3%); in
184W at 9 95 keV (8%); w h e r e the n u m b e r s in p a r e n t h e s e s a r e the
y i e l d s g i v e n as a p e r c e n t a g e of the g r o u n d s t a t e y i e l d s . S t r o n g
1 8 l8lL = 0 t r a n s i t i o n s w e r e a l s o s e e n in the W ( p , t ) W r e a c t i o n
to s t a t e s at 4 54 keV a n d 1864 keV of e x c i t a t i o n .
T h e f i r s t e x c i t e d 0 + s t a t e s in t h e s e i s o t o p e s c a n be d i v i d e d
into two g r o u p s a c c o r d i n g to t h e i r m o m e n t s of i n e rtia. T h e i r
p r o p e r t i e s in 4 84,W a nd 482W a n d t h o s e of all o f the t u n g s t e n
g r o u n d s t a t e s c an be well d e s c r i b e d in t e r m s o f a p a i r i n g v i b r a
tion s c h e m e a r i s i n g f r o m an e n e r g y g a p in the n e u t r o n s i n g l e
p a r t i c l e level s c h e m e o c c u r i n g at N = 108. For the lower m a s s
178 l80is o t o p e s , the f i rst e x c i t e d 0 s t a t e s in W a n d W as well
182as the 2 7 2 5 keV s t a t e in W a r e f o u n d to h a v e n e a r l y the
s a m e e x p e r i m e n t a l (p,t) r e a c t i o n Q - v a l u e . T h e i r s t r u c t u r e may,
t h e r e f o r e , be i n t e r p r e t e d as the g r o u n d s t a t e of the t a r g e t
n u c l e u s c o u p l e d to two h o l e s in a g r o u p of s i n g l e - p a r t i c l e levels
s o m e w h a t b e l o w , a n d d e c o u p l e d from, the Fermi s u r f a c e . T h e d e
c o u p l i n g c a n be e x p l a i n e d by a s s u m i n g a r e d u c e d p a i r i n g m a t r i x
e l e m e n t b e t w e e n o b l a t e a nd p r o l a t e s i n g l e - p a r t i c l e o r b i t a l s . A
s p e c i f i c p r e d i c t i o n of this s t r u c t u r e m o d e l , that the L = 0
| g |t r a n s i t i o n to the 1864 keV s t a t e in W s h o u l d not be b l o c k e d
by the u n p a i r e d n e u t r o n , is f o u n d to be f u l f i l l e d .
T h e s y s t e m a t i c s of t r a n s i t i o n s to the J ( n ) = 2(+) m e m b e r of
the K = 2 g a m m a v i b r a t i o n a l b a n d s in t h e s e n u clei a r e d i s c u s s e d
w i t h p a r t i c u l a r a t t e n t i o n to the d e t e r m i n a t i o n of the m a i n two-
n e u t r o n c o n f i g u r a t i o n s i n v o l v e d in the p o p u l a t i o n o f t h e s e states.
A l t h o u g h m o d e l c a l c u l a t i o n s i n d i c a t e f o u r p o s s i b l e c o n f i g u r a t i o n s
that c o u l d p l a y a s u b s t a n t i a l role, the c o n c l u s i o n f r o m the
s t r e n g t h s y s t e m a t i c s of the e v e n - m a s s t a r g e t s a nd the r e s u l t s
of the ^8 3 W ( p ,t ) ^ W e x p e r i m e n t , is that o n l y the (1/ 2 [510] . 5 / 2 [512])
t w o - n e u t r o n h o l e c o n f i g u r a t i o n is im p o r t a n t .
i i
In M e m o r y of my F a t h e r
H E R B E R T S V E N D M O R T E N S E N
1930 - 1977
ACKNOWLEDGEMENTS
D u r i n g m y s t a y at the 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 h a v e h ad t h e g ood f o r t u n e to be a s s o c i a t e d w i t h m a n y
fi n e p r o f e s s i o n a l s to w h o m I o w e a g r e a t deal.
F i rst, I e x t e n d my g r a t i t u d e to Prof. C h a r l e s B o c k e l m a n and
Prof. R u s s e l l B e t t s w h o h a v e s h a r e d the b u r d e n of my e d u c a t i o n in
n u c l e a r p h y s i c s . C h a r l e s , w i t h hi s rich s t o r e of e x p e r i e n c e , w a s
a l w a y s a b l e a nd w i l l i n g to p r o v i d e p e r s p e c t i v e on the p r o b l e m s
at h a n d w h i l e R u s s e l l s e r v e d as an i n e x h a u s t i b l e s o u r c e of ideas
a n d i n s p i r a t i o n . My r e l a t i o n s h i p w i t h t h e m will be long r e m e m b e r e d .
T o Mr. C h a r l e s G i n g e l l and t he s t a f f of t he e l e c t r o n i c s s h o p
I o w e a d e b t o f g r a t i t u d e for t h e i r u n t o l d h o u r s s p e n t m a i n t a i n i n g
th e h i g h l y c o m p l e x m u l t i a n g l e s p e c t r o g r a p h .
Of c o u r s e , n o n e o f t he r e s u l t s r e p o r t e d in t his t h e s i s c o u l d
h a v e b e e n o b t a i n e d w i t h o u t the skill a n d d e d i c a t i o n of M e s s r s .
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, K e n z o Sato,
R o b e r t H a r r i n g t o n , a n d M i c h a e l S i v e r t z w h o h a v e s u c c e e d e d in
b r i n g i n g the a c c e l e r a t o r up to an i m p r e s s i v e level of p e r f o r m a n c e .
Mr. Ed C o m e a u is t h a n k e d for h is f a b r i c a t i o n of m a n y of the
a n c i l l a r y d e v i c e s a s s o c i a t e d w i t h the s p e c t r o g r a p h as well as his
ca r e f u l and p a t i e n t i n s t r u c t i o n on t he m e t a l w o r k i n g c r a f t . For
t e a c h i n g me m u c h o f w h a t I k n o w a b o u t t he i n n e r w o r k i n g s o f the
a c c e l e r a t o r , I t h a n k Mr. J o h n B e n j a m i n .
S i n c e m o d e r n e x p e r i m e n t a l n u c l e a r p h y s i c s is v e r y m u c h a t e a m
ACKNOWLEDGEMENTS (cont 'd)
e f f o r t , I w a s e x t r e m e l y f o r t u n a t e in h a v i n g the h e l p a n d a d v i c e
o f M i s s S t e p h a n i e D i C e n z o , Dr. R o n a l d W h i t e , Dr. E r n e s t S e ~ ’ie
(a t h e o r i s t w h o d a r e d to e n t e r the r e a l m of e x p e r i m e n t a l p h y s i c s ) ,
Dr. J a y P e t e r s o n , a n d e s p e c i a l l y Dr. W i l l i a m C a l l e n d e r w h o a l s o
r e c e i v e s the b l a m e for e v e r y t h i n g I k n o w a b o u t the o p e r a t i o n of
the s p e c t r o g r a p h , p h o t o g r a p h i c e m u l s i o n t e c h n i q u e s , a n d d a t a
a n a 1ys i s .
In the i n t e r p r e t a t i o n o f the d a t a , I a m g r a t e f u l to Prof.
R o b e r t A s c u i t t o w h o not o n l y p r o v i d e d the c o m p l e x n u c l e a r s t r u c t u r e
a n d c o u p l e d - c h a n n e l r e a c t i o n c o d e s u t i l i z e d in the c o u r s e of this
w o r k , but a l s o p e r s o n a l l y g a v e of his t ime an d e x p e r i e n c e .
T h e p h o t o g r a p h i c e m u l s i o n s w e r e e x p e r t l y s c a n n e d by Mrs.
M a r y K ay F i t z g e r a l d , Mrs. M a r y S n y d e r , a n d Mrs. D o r o t h y P u r c e l l .
T h e i r r e m a r k a b l e s t a m i n a in the face o f h i g h t r a c k d e n s i t i e s is
m u c h a p p r e c i a t e d . I a l s o a l w a y s will v a l u e t h e i r p e r s o n a l f r i e n d
ship.
T h e a d v i c e of Prof. D. A l l a n B r o m l e y in his c a r e f u l r e a d i n g
of the m a n u s c r i p t is m u c h a p p r e c i a t e d as a re his e m i n e n t l y s u c c e s s
ful e f f o r t s in m a i n t a i n i n g the e x c e l l e n c e of the W r i g h t L a b o r a t o r y ,
in the f ace of the s h r i n k i n g n a t i o n a l c o m m i t m e n t to b a s i c r e s e a r c h .
My s t a y at Y a l e has b e e n m a d e m u c h m o r e e n j o y a b l e a n d fruitful
t h r o u g h my a s s o c i a t i o n s w i t h s e v e r a l f a c u l t y m e m b e r s a n d g r a d u a t e
s t u d e n t s . I p a r t i c u l a r l y t h a n k M i s s S t e p h a n i e D i C e n z o , Dr. S t e v e n
ACKNOWLEDGEMENTS (cont'd)
S a n d e r s , Prof. W i l l i a m l o n f o r d , Mr. R e j e a n G r o l e a u , a n d Dr. M a r t i n
S a c h s for t h e i r f r i e n d s h i p and help.
T h e e x p e r t p r e p a r a t i o n of the f i g u r e s in this t h e s i s a re
the w o r k of Mrs . S a n d y S i c i g n a n o and Mrs. Gail M e r c e r . T h e i r
t a l e n t s a re e v i d e n t .
F i n a l l y , my w i f e M i c h e l e r e c e i v e s my s p e cial t h a n k s . Not
o n l y d id she c o n t e n d w i t h m i d n i g h t s h i f t s , late a p p o i n t m e n t s , a n d
a p r e o c c u p i e d h u s b a n d , but a l s o p r o v i d e d m o n e t a r y a n d moral s u p p o r t .
T o to p it all off, s h e a l s o t y ped t h i s t h e s i s . I can n e v e r f u l l y
r e pay her.
vi
TABLE OF CONTENTS
A b s t ract
A c k n o w l e d g e m e n t s
C h a p t e r 1: I n t r o d u c t i o n ................................................ 1
C h a p t e r 2: T h e S t r u c t u r e of R a r e E a r t h N u c l e i : An
O v e r v i e w ................................................ 8
2.1 S i n g l e - P a r t i c l e S t r u c t u r e ............................ 9
2.2 C o l l e c t i v e S t r u c t u r e
2.2.1 P a i r i n g ...................................................... 16
2 . 2 . 2 R o t a t i o n a l B a n d s ..................................... 21
2 . 2 . 3 O t h e r R e s i d u a l I n t e r a c t i o n s ...................... 22
2 . 2 . k C o r i o l i s C o u p l i n g ..................................... 2k
C h a p t e r 3: T w o - N e u t r o n T r a n s f e r R e a c t i o n s : A P r o b e
of C o l l e c t i v e S t r u c t u r e ......................... 26
3-1 R e a c t i o n D y n a m i c s
3.1.1 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 . . . . 27
3 . 1 . 2 T h e O p t i c a l M o d e l ........................................ 32
3 . 1 . 3 M u l t i s t e p P r o c e s s e s .................................. 33
3-2 N u c l e a r S t r u c t u r e Input
3.2.1 R a dial F o r m F a c t o r s ..................................... 39
3 . 2 . 2 S p e c t r o s c o p i c A m p l i t u d e s ......................... ^0
C h a p t e r k: D a t a A c q u i s i t i o n a n d R e d u c t i o n ....................... ^3
A.1 E x p e r i m e n t a l P r o c e d u r e ................................. kk
k . 2 D a t a R e d u c t i o n ............................................. ^8
A . 3 A b s o l u t e C r o s s S e c t i o n M e a s u r e m e n t s .............. 57
C h a p t e r S' P r e s e n t a t i o n of D a t a ........................................58
5- 1 Q-Val u e s ...................................................... 59
5-2 R e l a t i v e a n d A b s o l u t e C r o s s S e c t i o n S c a l e s . . 61
5-3 S p e c t r o s c o p i c R e s u l t s .................................... 665.3.1* G e n e r a l C o m m e n t s ........................................ 665 . 3 . 2 T h e N u c l e u s ( l 8 4 ) W ......................................70
5 . 3 . 3 T h e N u c l e u s ( 18 2 ) W ......................................70
5 . 3 . 4 T h e N u c l e u s (1 8 1)W ......................................71
5 . 3 . 5 T h e N u c l e u s ( l 8 0 ) W ......................................73
5 - 3 . 6 T h e N u c l e u s (1 7 8 )W ......................................74
C h a p t e r 6 : S y s t e m a t i c s of C o l l e c t i v e States:
G r o u n d - S t a t e B a n d s ............................. 84
6.1 A n g u l a r D i s t r i b u t i o n S h a p e s for M e m b e r s 86of T h e G r o u n d - S t a t e B a n d s ......................
6. 2 A n a l o g to G r o u n d - S t a t e T r a n s i t i o n s :
Odd M a s s T a r g e t ....................................92
6 . 2.1 A n g u l a r D i s t r i b u t i o n S h a p e s ........................ 92
6 . 2 . 2 B l o c k i n g ................................................... 93
6 . 3 C r o s s S e c t i o n S y s t e m a t i c s of G r o u n d S t a t e s . . 100
6 .3 .1 P a i r i n g R o t a t i o n L i m i t ........................... 100
6 . 3 . 2 Full F o r m F a c t o r D W B A C a l c u l a t i o n s ............. 100
C h a p t e r J : S y s t e m a t i c s o f C o l l e c t i v e S t a tes:
E x c i t e d K = 0 B a n d s ............................. 107
7.1 G e n e r a l C o n s i d e r a t i o n s ................................ 108
7.2 O"!” S t a t e s in the H i g h - M a s s I s o t o p e s ............. 117
7.2.1 T h e P a i r i n g V i b r a t i o n M o d e l :
Ge n e r a l D e s c r i p t i o n ............................ 123
7 . 2 . 2 T h e P a i r i n g V i b r a t i o n M o d e l :
A p p l i c a t i o n to the T u n g s t e n I s o t o p e s . . . 127
7-3 0| S t a t e s in the L o w - M a s s I s o t o p e s .................135
7.4 J = 2 S t a t e s in K = 0 B a n d s ........................... 1^6
C h a p t e r 8 : S y s t e m a t i c s of C o l l e c t i v e States:
O c t u p o l e and G a m m a V i b r a t i o n a l S t a t e s . . . 148
8.1 T h e O c t u p o l e V i b r a t i o n a l B a n d s ..................... 149
v i i
TABLE OF CONTENTS (cont'd)
8.2 The Gamma Vibrat ional Bands 149
vi i i
TABLE OF CONTENTS (cont'd)
C h a p t e r 3: S u m m a r y a nd C o n c l u s i o n s ............................ . 158
9-1 L i m i t a t i o n s of t he S t u d y ......................... . 159
9.2 Pa i ri ng Vi brat i o n s .................................. . 161
9-3 T r a n s i t i o n s to J = 2 S t a t e s ...................... 163
9 • G e n e r a l C o m m e n t s .................................... 166
R e f e r e n c e s . 167
A p p e n d ix A: Sign C o n v e n t i o n f or Radial F o r m F a c t o r s . . . . 177
A p p e n d i x B: S p l i t t i n g of (p,t) S t r e n g t h in O d d - M a s s
N u c l e i ................................................ 173
A p p e n d i x C : C o m p i l a t i o n of A n g u l a r D i s t r i b u t i o n s . . . . 180
A p p e n d i x D: C a l c u l a t i o n o f the (p,t) S t r e n g t h of
the S t a t e s ..................................... 181
A p p e n d i x E: C a l c u l a t i o n of t h e B r o k e n - P a i r C o n f i g u r a t i o n
(p,t) S t r e n g t h s .................................. . 183
A p p e n d i x F : T h e F a v o r e d L = 2 T r a n s i t i o n s in the
18 3 W ( p ,t ) 1 8 1W R e a c t i o n ......................... 184
C h a p t e r 1
I N T R O D U C T I O N
1
N u c l e i in the rare e a r t h r e g i o n h a v e m a s s n u m b e r s b e t w e e n
150 a n d 192 so that on a v e r a g e t h e r e a r e a b o u t 175 n u c l e o n s . T h e
n u m b e r o f p o s s i b l e c o n f i g u r a t i o n s a v a i l a b l e to s u c h a s y s t e m is
e n o r m o u s ( ~ 1 06 b e l o w 5 M e V e x c i t a t i o n ) so that the s t u d y of the
s t r u c t u r e of t h e s e nucl e i e v e n at low e x c i t a t i o n e n e r g i e s m i g h t
s e e m i m p o s s i b l y d i f f i c u l t . It ha s b e e n f o u n d , h o w e v e r , that
c e r t a i n s t a t e s p o s s e s s w h a t is t e r m e d " c o l l e c t i v i t y " ; the w a v e -
f u n c t i o n of s u c h s t a t e s a r e c o m p o s e d of sums o f m a n y s i n g l e
p a r t i c l e c o n f i g u r a t i o n s , w i t h s p e c i a l p h a s e r e l a t i o n s h i p s
e x i s t i n g b e t w e e n the terms.
T w o t y p e s of c o l l e c t i v e s t a t e s i m p o r t a n t in n u clei in the
rar e e a r t h r e g i o n c an be d e f i n e d o p e r a t i o n a l l y :
(1) T h o s e s t a t e s w h i c h a r e s t r o n g l y e x c i t e d in i n e l a s t i c
n uc 1 e u s - n u c 1e us s c a t t e r i n g . T h e s e s t a t e s m u s t p o s s e s s
c o l l e c t i v i t y r e l a t e d to the p r o m o t i o n o f s i n g l e
n u c l e o n s f r o m the g r o u n d - s t a t e c o n f i g u r a t i o n to h i g h e r
s i n g l e - p a r t i c l e l e v e l s , s i n c e in a m i c r o s c o p i c p i c
t ure the i n e l a s t i c s c a t t e r i n g o p e r a t o r is a o n e -
b o d y o p e r a t o r . L a r g e e l e c t r o m a g n e t i c c o u p l i n g s to
the g r o u n d s t a t e s m u s t a l s o be in e v i d e n c e for t h ese
s t a t e s for the s a m e reason.
(2) T h o s e s t a t e s 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 in d i r e c t
r e a c t i o n s i n v o l v i n g the t r a n s f e r of a p a i r o f like
n u c l e o n s . T h e s e s t a t e s m u s t p o s s e s s c o l l e c t i v i t y
r e l a t e d to the a d d i t i o n or r e m oval o f p a i r s of
1.1 INTRODUCTION
3
p a r t i c l e s , w h i c h is t h u s c a l l e d " p a i r i n g
col 1ec t i v i t y ."
In the c a s e of e v e n - e v e n rar e e a r t h n u c l e i , the g r o u n d s t ate
is d o m i n a t e d by the p a i r i n g c o l l e c t i v i t y . R e a s o n a b l e s e m i - q u a n t i
t a t i v e e s t i m a t e s (Br 72) i n d i c a t e that for t w o - n u c l e o n t r a n s f e r
r e a c t i o n s , s t a t e s o t h e r tha n t h o s e in the g r o u n d - s t a t e r o t a t i o n a l
ba n d s h o u l d be p o p u l a t e d w i t h o n l y a v e r y small f r a c t i o n of the g r o u n d -
s t a t e s t r e n g t h (a f e w p e r c e n t ) . R e c e n t l y , h o w e v e r , (p,t) a n d s o m e
(t,p) r e a c t i o n s on s e v e r a l rar e e a r t h a nd a c t i n i d e n u c l e i h a v e f o und
J * 0 + s t a t e s w h i c h a r e p o p u l a t e d w i t h 10% to 15% of the g r o u n d
s t a t e s t r e n g t h (Bj 6 6 , De 70, FI 71, De 72, El 72, Ma 72a, Ma 72b,
FI 73, O o 73)- T o e x p l a i n s u c h large p a i r i n g c o l l e c t i v i t y in e x
c i t e d s t a t e s w o u l d n o r m a l l y r e q u i r e e x t r e m e l y d e t a i l e d c a l c u l a t i o n s ,
s i n c e s p e c i a l c i r c u m s t a n c e s m u s t be at w o r k to so f r a g m e n t the p a i r i n g
s t r e n g t h . As w ill be s h o w n , h o w e v e r , the s y s t e m a t i c s t u d y of a
s e r i e s o f i s o t o p e s w i t h (p,t) a n d (t,p) r e a c t i o n s c an reveal m u c h
a b o u t the d e p e n d e n c e of c o l l e c t i v e s t a t e s u p o n the u n d e r l y i n g s i n g l e -
p a r t i c l e s t r u c t u r e so long as the m i c r o s c o p i c s t r u c t u r e d o e s not
c h a n g e a p p r e c i a b l y w i t h n e u t r o n n u m b e r . S u c h s y s t e m a t i c s t u d i e s h a v e
be e n p e r f o r m e d on Sm (Hi 65, Bj 6 6 , De 70, Me 70), Yb (Oo 73),
Os (Sh 76) a n d Gd (FI 73) i s o topes. T h e i n t e r p r e t a t i o n , h o w e v e r ,
is c l o u d e d in all c a s e s but Y b by the c h a n g i n g s h a p e of the nuclei
w i t h n e u t r o n n u m b e r , b e s p e a k i n g a c h a n g e in the u n d e r l y i n g s i n g l e
p a r t i c l e s t r u c t u r e . T h e s h a p e s of t u n g s t e n n u c l e i , in c o n t r a s t ,
a r e all v e r y s i m i l a r . I n e l a s t i c a l p h a p a r t i c l e s c a t t e r i n g e x p e r i -
182 18A 186m e n t s on W, W, a n d W s h o w o n l y a v e r y small s h a p e c h a n g e
(Le 75) w h i l e e s t i m a t e s f r o m r o t a t i o n a l b a n d s t r u c t u r e s of the od d
i s o t o p e s s h o w no large u n e x p e c t e d c h a n g e in d e f o r m a t i o n b e t w e e n
77 W a n d (Be 7*0. T h u s , t u n g s t e n n u clei f o r m an e x c e l l e n t
s y s t e m for s u c h a s y s t e m a t i c study.
T h e r e h a v e b e e n m a n y e x p e r i m e n t a l s t u d i e s of the i s o t o p e s of
t u n g s t e n o v e r the p ast ten y e a r s . T h e n e u t r o n s i n g l e - p a r t i c l e
s t r u c t u r e h as b e e n p r o b e d w i t h s p e c t r o s c o p i c s t u d i e s of the o d d -
m a s s i s o t o p e s u s i n g (d,p), (d,t) (Ca 71, Ca 72) a n d (B H e , a ) (K1 73)
. . . T , 186., r e a c t i o n s as w ell as n u m e r o u s g a m m a - r a y m e a s u r e m e n t s . T h e W1 Q r(p,d) W s t u d y of A s c u i t t o , King, a n d M c V a y (As 72c, Ki 73, As 7*<,
Ki 7*0 p r o v i d e d s o m e of the f i r s t c o n v i n c i n g e v i d e n c e for m u l t i p l e -
S t e p i n e l a s t i c e f f e c t s in o n e - n u c l e o n t r a n s f e r as w e l l as t e s t i n g
d e t a i l s of the s t r u c t u r e of
T h e e v e n - m a s s i s o t o p e s h a v e not b e e n n e g l e c t e d , w i t h s t u d i e s
u s i n g (d , d 1) (Gu 71), (d,p) a n d (d,t) r e a c t i o n s (Ca 72). T h e
1 8 6, 18*1, 182,,, , . , . . . ,W ( p , t ) r e a c t i o n s to s t a t e s low in e x c i t a t i o n h ave
b e e n i n v e s t i g a t e d (Ma 72) as h a s ( p , t) (Oo 73) a nd the
g a m u t o f (t,p) r e a c t i o n s (Ca 73, Ca 76b, Ga 77). A g a i n , the w o r k
of K ing e £ aj. (As 72c, Ki 72, Ki 73, Ki 7*0 on the ^BBW ( p , t ) ' 8 W
r e a c t i o n m u s t be m e n t i o n e d as an e a r l y b ut d e f i n i t i v e s t u d y of the
e f f e c t s of m u l t i p l e - s t e p i n e l a s t i c e x c i t a t i o n in (p,t) t r a n s i t i o n s .
In a d d i t i o n , the m o r e r e c e n t w o r k o f H a n s o n £ £ a_k (Er 7**, Ha 75,
Ha 76) in w h i c h t h e s e e f f e c t s in h e a v y - i o n - i n d u c e d t w o - 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 e r e i n v e s t i g a t e d , a d d e d m u c h to o u r u n d e r s t a n
d i n g of the s t r u c t u r e o f the g r o u n d - s t a t e b a n d s of t u n s t e n nuclei.
5
T h e e x p e r i m e n t a l p a r t of the p r e s e n t s t u d y s e e k s to e x t e n d the
p r e v i o u s (p,t) m e a s u r e m e n t s to s t a t e s o t h e r tha n t h o s e b e l o n g i n g
to the g r o u n d s t a t e b a n d a nd to e x t r a c t the t r a n s i t i o n s t r e n g t h
s y s t e m a t i c s for the w h o l e r a n g e o f t u n g s t e n n u c l e i a c c e s s i b l e
w i t h the (p,t) r e a c t i o n . T h i s p a r t i c u l a r r e a c t i o n w a s c h o s e n
for s e v e r a l r e a sons:
(1) T h e m u l t i a n g l e m a g n e t i c s p e c t r o g r a p h is i d e a l l y
s u i t e d for m e a s u r i n g (p,t) r e a c t i o n s w i t h the
g o o d e n e r g y r e s o l u t i o n w h i c h is c r u c i a l for s p e c t r o
s c o p i c s t u d i e s o f n u clei in the rar e e a r t h region.
(2) T h e two n e u t r o n s in the t r i t o n a r e in a r e l a t i v e
a n g u l a r m o m e n t u m z e r o state. T h e y t h e r e f o r e f o r m
a g o o d " p a i r " so that t h e i r t r a n s f e r d i r e c t l y p r o b e s
the p a i r i n g c o l l e c t i v i t y of the s t a t e s .
(3) T h e u se of light ions in p a r t i c l e t r a n s f e r s t u d i e s
a l l o w s m a n y c a 1c u 1 a t i o n a 1 s i m p l i f i c a t i o n s to be m a d e ,
o b v i a t i n g the u se of the m o r e c o m p l e x r e a c t i o n c o d e s
n e c e s s a r y to d e s c r i b e h e a v y ion i n d u c e d r e a c t i o n s .
T h e s t u d y c o n s i s t s of (p,t) r e a c t i o n m e a s u r e m e n t s on all
f i v e of the n a t u r a l l y o c c u r i n g t u n g s t e n i s o t o p e s (A = 18 6 ,
184, 183, 182, a n d 180) at 21 M e V b o m b a r d i n g e n e r g y , all p e r f o r m e d
w i t h the s a m e a p p a r a t u s an d a n a l y z e d in a c o n s i s t e n t m a n n e r .
In e a c h c a s e s t u d i e d , e x c i t e d 0 s t a t e s (0^) w e r e f o u n d to be
p o p u l a t e d w i t h a b o u t 10% of the g r o u n d s t a t e s t r e n g t h . H o w e v e r ,
d u r i n g the c o u r s e of the s t u d y , it b e c a m e a p p a r e n t that the
c h a r a c t e r i s t i c s o f s o m e s u p p o s e d l y h i g h l y c o l l e c t i v e s t a t e s
6
a c t u a l l y c h a n g e d c o n s i d e r a b l y w i t h m a s s n u m b e r , i m p l y i n g that
they m u s t be p o p u l a t e d v ia a v e r y l i m i t e d n u m b e r o f c o n f i g u r a t i o n s .
S i m p l e m o d e l s w e r e u s e d to p r o v i d e a f r a m e w o r k for u n d e r
s t a n d i n g the s y s t e m a t i c s o f t h e s e s t a t e s . T h e s t r o n g (p,t)
e x c i t e d 0* s t a t e s in the l o w - m a s s t u n g s t e n i s o t o p e s , all w i t h the
s a m e e x p e r i m e n t a l r e a c t i o n Q - v a l u e , c a n be e x p l a i n e d by d i v i d i n g
t he s i n g l e - p a r t i c l e o r b i t a l s into tw o n o n i n t e r a c t i n g g r o u p s
a c c o r d i n g to the s i g n o f t h e i r m a s s q u a d r u p o l e m o m e n t s (Gr 7 0 -
T h e e x c i t e d 0+ s t a t e s t h e n g a i n t h e i r (p,t) s t r e n g t h t h r o u g h
p a i r i n g c o l l e c t i v i t y i n v o l v i n g o r b i t a l s d e c o u p l e d f r o m t h ose
i n v o l v e d in the g r o u n d state. T h e e x i s t e n c e o f t h e s e s t a t e s
a d d s c o n s i d e r a b l e w e i g h t to the c o n c e p t of this d i v i s i o n of the
s i n g l e - p a r t i c l e o r b i t a l s into t w o g r o u p s , w h i c h h as b e e n u n d e r
c o n s i d e r a b l e d e b a t e for s e v e r a l y e a r s (Be 72, Ch 72, Gr 71, Mo 77,
Ri 72). M o r e o v e r , t h e i r c o n s t a n t r e a c t i o n Q - v a l u e p r o v i d e s an
e x c e l l e n t s i g n a t u r e w h i c h the m o d e l s w h i c h s e e k to e x p l a i n the
d i v i s i o n o n a m i c r o s c o p i c b a s i s s h o u l d be a b l e to r e p r o d u c e .
In o r d e r to e x p l a i n (p,t) a n d (t,p) r e a c t i o n d a t a on the
h i g h - m a s s i s o t o p e s , the c o n c e p t o f p a i r i n g v i b r a t i o n s w a s a p p l i e d .
In t h e s e c a s e s , the f r a g m e n t a t i o n o f the p a i r i n g s t r e n g t h w a s
a t t r i b u t e d to an e n e r g y g a p in the s i n g l e - p a r t i c l e s p e c t r u m w h i c h
p a r t i a l l y d e c o u p l e s the l e v e l s a b o v e the g a p f r o m leve l s b e l o w the
gap. (Such a d e c o u p l i n g p r e v i o u s l y ha d b e e n s e e n o n l y in the Yb
i s o t o p e s . ) T h r o u g h the u se of s u c h m o d e l s the e s s e n t i a l f e a t u r e s
o f the d a t a c a n be u n d e r s t o o d , a l t h o u g h the s c h e m a t i c n a t u r e of
the m o d e l s p r e c l u d e s d e t a i l e d a g r e e m e n t b e t w e e n m o d e l p r e d i c t i o n s
a n d e x p e r i m e n t a l data. E v e n so, the p r e s e n t s t u d y d e m o n s t r a t e s
th e f i r s t s u c c e s s f u l a p p l i c a t i o n o f the p a i r i n g v i b r a t i o n
m o d e l to d e f o r m e d n u c l e i .
In a d d i t i o n to c o n s t i t u t i n g a d e t a i l e d s t u d y of the c o l l e c
tiv e n e u t r o n p a i r i n g s t a t e s , the p r e s e n t s t u d y a l s o p r o v i d e s
s p e c t r o s c o p i c i n f o r m a t i o n on s t a t e s in t u n g s t e n nuclei to a b o u t
2 . 5 M e V e x c i t a t i o n . S u c h d a t a a r e v a l u a b l e b e c a u s e the y d i r e c t l y
p r o v i d e e x c i t a t i o n e n e r g y d a t a , c o n f i r m i n g a n d s o m e t i m e s s u p p l e
m e n t i n g level s c h e m e s i n f e r r e d f r o m g a m m a - r a y m e a s u r e m e n t s .
Chapter 2THE STRUCTURE OF RARE EARTH NUCLEI: AN OVERVIEW
8
9
2.1 S I N G L E - P A R T I C L E S T R U C T U R E
T h e s t u d y of n u c l e a r s t r u c t u r e is the s t u d y of e x t r e m e l y c o m p l e x
m a n y - b o d y s y s t e m s . N u c l e i in the rare e a r t h r e g i o n h a v e o v e r o n e
h u n d r e d n u c l e o n s , all i n t e r a c t i n g t h r o u g h som e f o r c e s , u s u a l l y t a k e n
to be t w o - b o d y f o r c e s , g i v i n g a H a m i l t o n i a n for p a r t i c l e i:
Hi ■ - £ ' i 2 + 1 (2-"i/j
S i n c e the t w o - b o d y f o r c e s V.j a r e e x t r e m e l y c o m p l i c a t e d a n d not
f u l l y u n d e r s t o o d at p r e s e n t , it m i g h t s e e m that the c a l c u l a t i o n of
the s t r u c t u r e of c o m p l e x nuclei is i m p o s s i b l e . H o w e v e r , it has
b e c o m e a p p a r e n t that a m a j o r p a r t of t he s u m total of all f o r c e s
a c t i n g on an i n d i v i d u a l n u c l e o n in a n u c l e u s c an be r e p r e s e n t e d by
a s i m p l e p o t e n t i a l a n d that the s a m e p o t e n t i a l c an be u s e d for
all of the n u c l e o n s in a n u c l e u s (Co 7 0 - T h u s
1 . (2.2) V.. = V(R) " V . . V•J 'J
»w h e r e V.^ is a ( h o p e f u l l y ) small r e s i d u a l i n t e r a c t i o n . T h i s r a t h e r
s u r p r i s i n g f act has b e e n well e x p l o i t e d in n u c l e a r p h y s i c s in the
" n u c l e a r shell m o d e l . " O n e f o r m of the p o t e n t i a l that h as b e e n
e x t r e m e l y u s eful is the s p h e r i c a l W o o d s - S a x o n type:
\I(R) = V (R ) = V o (1 + e x p [ (R - R o ) / a ] ) ~ ' (2-3)
w h i c h has a s h a p e b e t w e e n that of a h a r m o n i c o s c i l l a t o r p o t e n t i a l
a nd a s q u a r e - w e l l p o t e n t i a l . W i t h the i n c l u s i o n of a s p i n - o r b i t
p o t e n t i a l of the f o r m 1 . •s’, ( w h e r e 1 . is the o r b i t a l a n g u l a r
10
m o m e n t u m of the n u c l e o n "i " a n d s is its spi n a n g u l a r m o m e n t u m )
a n d a s p h e r i c a l C o u l o m b p o t e n t i a l , as well as the a p p l i c a t i o n of
p h e n o m e n o l o g i c a l r e s i d u a l i n t e r a c t i o n s , the m o del has b e e n found
to r e p r o d u c e m a n y of the p r o p e r t i e s of light nuclei as w ell as
of n u clei in the lead, c a l c i u m , a nd z i r c o n i u m regions.
A t t e m p t s h a v e a l s o bee n m a d e to c a l c u l a t e the s t r u c t u r e of
the g r o u n d s t a t e s of a c t u a l m u l t i n u c l e o n s y s t e m s u s i n g s e l f -
c o n s i s t e n t m e t h o d s s u c h as H a r t r e e - F o c k a nd its m a n y v a r i a t i o n s .
T h e s e a t t e m p t s h a v e m et w i t h s o m e s u c c e s s , e s p e c i a l l y for light
n u clei (A 5 AO) (Ri 6 8), a l t h o u g h m a n y s e v e r e a p p r o x i m a t i o n s
a r e n e c e s s a r y d u e to the c o m p l e x i t y of the p r o b l e m (Va 73). O ne
i n t e r e s t i n g f e a t u r e of t h e s e c a l c u l a t i o n s is that if the s e lf-
c o n s i s t e n t f i e l d is a l l o w e d to v a r y f r o m a s p h e r i c a l s h a p e , nuclei
far f r o m c l o s e d s h e l l s u s u a l l y tak e on a d e f o r m e d p r o l a t e s h a p e w i t h
R*"/R^ ~ 1 .2 w h e r e R 1 i s the l e n g t h of the l o n g e r ( s y m m e t r y ) a x i s
a n d R that of the s h o r t e r a x i s (Va 73).
In the e v e n t of a d e v i a t i o n f r o m s p h e r i c i t y , the s p h e r i c a l
shell m o del is no l o n g e r a d e q u a t e to d e s c r i b e the s i n g l e - p a r t i c l e
s t r u c t u r e . T h e N i l s s o n m o del (Ni 55) t a kes the n e x t logical
s tep, a n d c a l c u l a t e s the s i n g l e - p a r t i c l e l e v e l s in a h a r m o n i c
o s c i l l a t o r well d e f o r m e d in the B d i r e c t i o n , w h e r e the n u c l e a r
s u r f a c e is p a r a m e t r i z e d by R = R g (1 + 82Y 2 0 ^ ' 3 u c b a d e f o r m a
tion s p l i t s the M (z - p r o j e c t i o n of the o r b i t a l a n g u l a r m o m e n t u m )
d e g e n e r a c y a nd i n d u c e s i n t e r a c t i o n s a m o n g s t a t e s of the sam e
v a l u e of M. N o r m a l l y , for c o n v e n i e n c e , h o w e v e r , o n l y the
c o u p l i n g s b e t w e e n l e v e l s in o n e m a j o r shell a r e included.
T h e d e f o r m e d shell model c a n be p u s h e d to n e a r the limit
of c o m p l e x i t y by the i n c l u s i o n of i n t e r - s h e l l c o u p l i n g a n d d e f o r
m a t i o n of the W o o d s - S a x o n well g i v e n not o n l y by but a l s o
h i g h e r - o r d e r d e f o r m a t i o n s B^, B g , . . . S u c h a c a l c u l a t i o n
w a s p e r f o r m e d in the c o u r s e of this s t u d y w h e r e AN =■= 4 m i x i n g
( m a j o r shell m i x i n g up to 4 s h e l l s away) a nd B^, B^ a n d 6^
d e f o r m a t i o n s w e r e i n c luded. T h e c a l c u l a t i o n w a s p e r f o r m e d w i t h the
c o d e D E F 2 N T of A s c u i t t o a nd S o r e n s o n u s i n g p o t e n t i a l w ell p a r a
m e t e r s s u i t a b l e for t.ne lead r e g i o n (As 72b), a n d e x p e r i m e n t a l l y
d e t e r m i n e d d e f o r m a t i o n p a r a m e t e r s for the t u n g s t e n nuclei t a ken
f r o m the l i t e r a t u r e (Le 75)- T h e r e s u l t s of s u c h c a l c u l a t i o n s
a r e s h o w n in F i g u r e 2-1 w h e r e o n the left h a n d s i d e a re the levels
as c a l c u l a t e d for a s p h e r i c a l w e l l . T h e v a l u e of w a s then
1 84v a r i e d until the a p p r o p r i a t e v a l u e for W w a s r e a c h e d , w h e n
B^ w a s v a r i e d f r o m z e r o to its p r o p e r v a l u e . T h e r e s u l t i n g s i n g l e
p a r t i c l e s p e c t r u m is s h o w n to the r i ght a n d a l s o in F i g u r e 2-2.
T h e q u e s t i o n of h o w well s u c h a c a l c u l a t i o n r e p r o d u c e s the
e x p e r i m e n t a l d a t a is a d d r e s s e d in F i g u r e 2-2. T h e c a l c u l a t e d
s i n g l e - p a r t i c l e o r b i t s for n e u t r o n s a r e s h o w n on the left a n d the
o r b i t s as e x t r a c t e d by O g l e et_ a_l_. f r o m e x p e r i m e n t a l d a t a (Og 71)
on the right. T h e g e n e r a l a g r e e m e n t is s e e n to be q u i t e g ood
a l t h o u g h i n d i v i d u a l l e v e l s d i f f e r in e n e r g y by 1 0 0 - 3 0 0 k e V .
T h e leve l s in e a c h c a s e a r e l a b e l l e d by t h e i r ' a s y m p t o t i c q u a n t u m
n u m b e r s " w h i c h a r e the d e s i g n a t i o n of the d o m i n a n t c o m p o n e n t
as a f u n c t i o n o f a n d 8^ d e f o r m a t i o n .
T h e c a l c u l a t e d s p e c t r u m for B2 = 0 . 2 2 1 ,
= -0 . 0 8 7 is s h o w n to t he r.ight a l o n g
w i t h the s h a p e s o f the Fermi s u r f a c e s for
a n d f r o m BCS c a l c u l a t i o n s as
d e s c r i b e d in S e c t i o n 2.2.
F i g u r e 2-2. T h e s i n g l e - p a r t i c l e level s t r u c t u r e c a l
c u l a t e d as d e s c r i b e d in t he text is s h o w n at
left w h i l e that e x t r a c t e d f r o m e x p e r i m e n t a l
d a t a by O g l e £t^ £]_. (Og 71) is s h o w n at
right. T h e r e l a t i v e e n e r g y s c a l e of the
" e x p e r i m e n t a l " v a l u e s a r e f r o m the r e f e r e n c e ,
w h i l e the z e r o of the s c a l e ( w hich w a s not
g i v e n ) w a s set to g i v e s o m e a v e r a g e a g r e e
m e n t w i t h the c a l c u l a t e d v a l u e s . T h e l a b e l s
i n d i c a t e the a s y m p t o t i c q u a n t u m n u m b e r s ,
K [ N N zA ] , for the o r b i t a l s .
Figure 2-1 . Energy of neutron s i n g l e - p a r t i c l e o r b i t a l s
-1
I0.20 0.221 0.2210 0 0 .0 “0 .0 0 7 I__________)
V A R I E D
SIN
GLE
-PA
RTI
CLE
BI
NDIN
G EN
ERGY
(M
eV)
SINGLE-PARTICLE LEVEL STRUCTURE
5 -------
6 —
7 -------
3/2 [65l]' 5/2 [642]
N\
■9/2 [505]
-7/2 [503] - 3 / 2 [512 ] ' 1/2 [510] ' 11/2 [615 ]
---------------------------------- 9 /2 [624]
y 7 / 2 [633 > 7 / 2 [514
1/2 [521^ 5 / 2 [512'
1/2 [660]
C A L C U L A T E D " E X P E R I M E N T A L "
15
of the level in a c y l i n d r i c a l h a r m o n i c o s c i l l a t o r b a s i s for the
c a s e of v e r y l a rge B 2 d e f o r m a t i o n . T h e q u a n t u m n u m b e r s a r e g i v e n
as K t N N^A] w h e r e K is the p r o j e c t i o n of the s i n g l e - p a r t i c l e total
a n g u l a r m o m e n t u m a l o n g the s y m m e t r y a x i s , N is th e total n u m b e r of
o s c i l l a t o r q u a n t a , is the n u m b e r of o s c i l l a t o r q u a n t a a l o n g the
s y m m e t r y a x is, a n d A is the p r o j e c t i o n o f the o r b i t a l a n g u l a r m o m e n t u m .
On som e o c c a s i o n s , this is a l s o w r i t t e n K n t N N ^ A ] w h e r e tt =
+ or the p a r i t y of the level; this d e s i g n a t i o n is r e d u n d a n t ,
h o w e v e r , s i n c e tt = + if N is e v e n a nd tt * - if N is odd. Of t h e s e
a s y m p t o t i c q u a n t u m n u m b e r s , h o w e v e r , o n l y K a n d tt a r e g o o d q u a n
tum n u m b e r s , a n d K o n l y in the limit that C o r i o l i s c o u p l i n g is
i g n o r e d (see s e c t i o n 2 .2 .3 ).
16
2.2 COLLECTIVE STRUCTUP:2.2.1 P a i r i n g
T h e N i l s s o n m o d e l has b e e n f o u n d to be i n a d e q u a t e in
e x p ’a i n i n g the c o l l e c t i v e s t r u c t u r e of d e f o r m e d n u clei so that
o t h e r r e s i d u a l f o r c e s b e t w e e n the p a r t i c l e s in the N i l s s o n levels
m u s t be i n c l u d e d . T h e p a i r i n g i n t e r a c t i o n (or m o r e c o r r e c t l y , the
m o n o p o l e p a i r i n g i n t e r a c t i o n ) is the p r i m a r y i n t e r a c t i o n used,
w i t h a H a m i l t o n i a n of the f o r m
H- * i = Hm- i + h • •total N i l s s o n p a i r i n g
(2.Mw h e r e
H . . = I G A + Ap a i r i n g u pv p v
pv
a n d A c r e a t e s a z e r o - c o u p l e d n u c l e o n p a i r in the o r b i t p a n d Gp r r p v
is the s t r e n g t h of the i n t e r a c t i o n b e t w e e n o r b i t s p a n d v. In
m o s t c a l c u l a t i o n s , the i n t e r a c t i o n s t r e n g t h is t a k e n to be the
s ame for all p a i r s of o r b i t s so that G = G, a p r o c e d u r e k n o w npv
as the " c o n s t a n t p a i r i n g m a t r i x e l e m e n t a p p r o x i m a t i o n . " T h e use
o f s u c h an i n t e r a c t i o n , w h i c h is the s a m e b e t w e e n all o r b i t a l s
r e g a r d l e s s of t h e i r s i n g l e - p a r t i c l e e n e r g i e s , is c l e a r l y an
o v e r s i m p l i c a t i o n s i n c e the i n t e r a c t i o n b e t w e e n n u c l e o n p a i r s in
o r b i t a l s d i f f e r i n g g r e a t l y in e n e r g y s h o u l d be r e d u c e d . T o c o m
p e n s a t e for this o v e r l y large c o u p l i n g , it is c o n v e n t i o n a l to
t r u n c a t e the s p a c e of s i n g l e - p a r t i c l e o r b i t s a l l o w e d to i n t e r a c t
via the p a i r i n g f o r c e to 30 or so v a l e n c e o r b i t a l s (As 72b).
In o n l y a f e w c a s e s c an the p r o b l e m so f o r m u l a t e d be s o l v e d
17
e x a c t l y ; for m o s t c a s e s the a p p r o x i m a t e s o l u t i o n of B a r d e e n ,
C o o p e r , a nd S c h r i f f e r , the s o - c a l l e d BCS s o l u t i o n , is a p p l i e d :
1$ > = 1 T ( U + V A ) 1 0>1 DL b V V V V '
w h e r e U 2 + V 2 = 1 (2.5)v v
vv 2
a n d 2/ V = the n u m b e r of p a r t i c l e s
It s h o u l d be n o t e d that the a b o v e s o l u t i o n d o e s not p o s s e s s
a d e f i n i t e n u m b e r of p a r t i c l e s , e a c h t e r m d i f f e r i n g f r o m the o t h e r s
by at least t wo p a r t i c l e s . T h e " q u a s i p a r t i c l e t r a n s f o r m a t i o n "
is u s u a l l y a p p l i e d t h r o u g h a c a n o n i c a l t r a n s f o r m a t i o n of the form:
a = U a - V a—v v v v v
a _ = U a + V a +v v V v v
w h e r e a + c r e a t e s a n u c l e o n in the o r b i t a l " v" a n d v r e p r e s e n t s thevti m e r e v e r s e of v (i.e. A * = a^a^.) . It c an t hen be s h o w n that
I* B C S ^ ‘s v a c u u m Y or q u a s i par t i c 1 e d e s t r u c t i o n o p e r a t o r
a n d that the e n e r g y of the s t a t e is a m i n i m u m if
V 2 = 2 (1 + e / /e^ + A 2 )
a n d V V
V 2 . ill - (2 7)
w h e r e
A 5 G y U V' V V V
e = E - AV V
a n d E^ is the s i n g l e - p a r t i c l e e n e r g y o f o r b i t a l v, a n d w h e r e A
is set by the r e q u i r e m e n t that the a v e r a g e n u m b e r of p a r t i c l e s
in the c a l c u l a t i o n e q ual the a c t u a l n u m b e r of p a r t i c l e s . T h e
oq u a n t i t y V is then r e c o g n i z e d as the p r o b a b i l i t y of o c c u p a t i o n
1 8
o f the level v by a n u c l e o n p a ir, the " f u l l n e s s 1 ; a n d s i m i l a r l y
2l£ as the " e m p t i n e s s " . T h e Fermi s u r f a c e is n o w no l o n g e r s h a r p
(some l e v e l s full a n d o t h e r s e m p t y ) but d i f f u s e w i t h an e n e r g y
s p r e a d of a b o u t 2A, as d e p i c t e d in F i g u r e 2-3-
A f t e r the s i n g l e - p a r t i c l e s t r u c t u r e is set, the o n l y f ree
p a r a m e t e r s in the B CS t h e o r y a r e the n u m b e r of l e vels c h o s e n to be
c o n s i d e r e d v a l e n c e p a r t i c l e s (32 in all c a l c u l a t i o n s in this w o rk)
a n d the v a l u e of the p a i r i n g m a t r i x e l e m e n t G. In this w o r k the
v a l u e o f the p a i r i n g m a t r i x e l e m e n t h as b e e n c h o s e n so that the
BCS g a p p a r a m e t e r A, is e q ual to the e x p e r i m e n t a l o d d - e v e n m a s s
d i f f e r e n c e as d e t e r m i n e d f r o m the N i l s s o n - P r i o r p r e s c r i p t i o n .
S u c h a c h o i c e is w ell j u s t i f i e d in the l i t e r a t u r e (Ni 61).
T h e a p p l i c a b i l i t y of the c o n s t a n t p a i r i n g m a t r i x e l e m e n t
a p p r o x i m a t i o n has b e e n q u e s t i o n e d in r e c e n t y e a r s b o t h on i n t u i
tiv e as well as e x p e r i m e n t a l g r o u n d s . U n d e r the a s s u m p t i o n of a
s h o r t - r a n g e f o r c e b e t w e e n o r b i t a l s , s o m e p a i r s of o r b i t a l s w h i c h
h a v e l a rge m a s s o v e r l a p s s h o u l d i n t e r a c t m u c h m o r e s t r o n g l y than
o t h e r p a i r s w h o s e m a s s o v e r l a p is s m all. In p a r t i c u l a r , G r i f f i n ,
J a c k s o n , a nd V o l k o v (Gr 71), in an e f f o r t to e x p l a i n s o m e (p,t)
d a t a in the a c t i n i d e r e g i o n (Ma 70), s u g g e s t e d that the v a l e n c e
o r b i t a l s s h o u l d be b r o k e n into t wo s u b g r o u p s , o b l a t e o r b i t a l s
( d E v/ d 62>0 ) a n d p r o l a t e o r b i t a l s ( d E ^ . / d ^ c O ) (see F i g u r e 2-1).
T h e o v e r l a p of t wo p r o l a t e o r b i t a l s ( p o l a r o r b i t s a b o u t the p r o l a t e
n u c l e u s ) wil l be of the s a m e o r d e r of m a g n i t u d e as the o v e r l a p of
t wo o b l a t e o r b i t a l s ( e q u a t o r i a l o r b i t s ) but b o t h w ill be m u c h
l a rge tha n the o v e r l a p b e t w e e n an o b l a t e a n d a p r o l a t e o r b i t a l .
s t r u c t u r e of a n u c l e u s in the N i l s s o n m o d e l .
T h e leve l s b e l o w the d o t t e d line a r e fu'l
a n d t h o s e a b o v e , e m p t y . U p o n a p p l i c a t i o n
of the p a i r i n g f o r c e v ia t he BCS t h e o r y ,
the r e s u l t is as s h o w n at r i g h t w h e r e the
Fermi s u r f a c e is d i f f u s e w i t h an e n e r g y
s p r e a d o f a b o u t 2A. T h e c u r v e a c t u a l l y
2 2p l o t s the v a l u e s o f V a n d U , the " f u l l
n e s s " a n d the " e m p t i n e s s " o f e a c h level
w h i l e the d o t t e d line r e p r e s e n t s the
e n e r g y v a l u e o f t he c h e m i c a l p o t e n t i a l , A .
Figure 2-3 . To the l e f t is schematica l ly shown the
NO PAIRING
WITH PAIRING
S i n g l e - P a r t i c l e B i n d i n g E n e r g y ( t o t h e c o n t i n u u m — )
L e v e l s F u l l L e v e l s E m p t y -
*r o>
EF
FE
CT
S
OF
PA
IRIN
G
21
T h u s , to s o m e e x t e n t , the o b l a t e o r b i t a l s a nd p r o l a t e o r b i t a l s
f o r m s e p a r a t e s u b s p a c e s w h i c h i n t e r a c t w e a k l y w i t h e a c h othe r .
T h e s u g g e s t i o n of G r i f f i n £t^ ajL w a s i n c o r p o r a t e d in c a l
c u l a t i o n s of 0+ s t a t e s in a c t i n i d e nuclei by v a n Rij a n d K a h a n a
(Ri 72) a nd p l a c e d p e r h a p s on a m o r e s o u n d t h e o r e t i c a l f o o t i n g by
Bes, B r o g l i a , a n d N i l s s o n (Be 72) w i t h t h e i r c o n c e p t of q u a d r u
p o l e p a i r i n g f i e l d s . In no c a se, h o w e v e r , w a s the a g r e e m e n t
b e t w e e n c a l c u l a t i o n s a n d e x p e r i m e n t s u f f i c i e n t to c o m p l e t e l y
s u p p o r t the a s s u m p t i o n of r e d u c e d o b l a t e - p r o l a t e p a i r i n g m a t r i x
e l e m e n t s , t h e r e b y l e a v i n g the m a t t e r u n s e t t l e d (Ch 72).
2 . 2 . 2 R o t a t i o n a l B a n d s
T h e f i n i t e d e f o r m a t i o n of c e r t a i n n u c l e i h as a
p r o f o u n d e f f e c t on t h e i r e n e r g y s p e c t r a ; w h e r e a s s p h e r i c a l nuclei
c a n n o t r o t a t e in a q u a n t u m m e c h a n i c a l s e n s e , the b r e a k i n g of
s p h e r i c a l s y m m e t r y leads to well d e v e l o p e d r o t a t i o n a l bands.
A usef u l d e s c r i p t i o n , w h e r e t he m o t i o n o f the i n d i v i d u a l n u c l e o n s
is the s a m e in a b o d y - f i x e d f r a m e of r e f e r e n c e , w h e t h e r or not' the
n u c l e u s as a w h o l e r o t a t e s , is k n o w n as the s t r o n g c o u p l i n g
m o d e l , o r a d i a b a t i c m o d e l , a n d w a s d e v e l o p e d by Boh r a n d M o t t e l s o n
(Bo 52, Bo 53)- W i t h i n this m o d e l , if a n u c l e u s is b o t h a x i a l l y
s y m m e t r i c a n d r e f l e c t i o n s y m m e t r i c (as in the c a s e for a n u c l e u s
w i t h 3 d e f o r m a t i o n s w h e r e A is e v e n ) , the n che total w a v e f u n c t i o nAfor a s t a t e o f a n g u l a r m o m e n t u m J, total a n g u l a r m o m e n t u m p r o j e c t i o n
M, a n d i n t r i n s i c a n g u l a r m o m e n t u m p r o j e c t i o n K is g i v e n by:
22
| J M K >
. 167,2(1 + 6K 0 }
_ 1/22J + 1_______(d m k Ik > + ^ d m -K R 1 I " K ^
(2.8)w h e r e D 3 ^ i s a r o t a t i o n D - m a t r i x , | l O is the i n t r i n s i c w a v e
f u n c t i o n , a n d Rj d e n o t e s a r o t a t i o n of 180° a b o u t the 1-axis.
T h e e n e r g i e s of the s t a t e s a r e g i v e n by
2E * ~ [J(J + 1) - 2 K 2 ] + E (0) (2.9)
w h e r e I is the m o m e n t o f i n e r t i a of the b a n d a n d E ^ is the
e n e r g y e i g e n v a l u e of | K > in the b o d y - f i x e d s y s t e m . For e a c h in
t r i n s i c s t a t e |K>, t h e r e t h e n e x i s t s a r o t a t i o n a l b a n d of e n e r g y
s p a c i n g J(J + 1) w i t h a n g u l a r m o m e n t u m J = K, K = 1, K = 2, ....
e x c e p t for the c a s e of K = 0 b a n d s w h e r e the f o r m of the w a v e
f u n c t i o n |JMK ) e v i d e n t 1y d e m a n d s that o n l y e v e n v a l u e s o f J
a r e a l l o w e d for p o s i t i v e p a r i t y i n t r i n s i c s t a t e s .
2 . 2 . 3 O t h e r R e s i d u a l I n t e r a c t i o n s
O n e of t he p r i n c i p a l r e s u l t s of the BCS t h e o r y
is that the l o w e s t e x c i t e d s t a t e s of e v e n - e v e n nucl e i ( n e g l e c t i n g
the m e m b e r s of the g r o u n d - s t a t e r o t a t i o n a l band) s h o u l d o c c u r at
an e x c i t a t i o n e n e r g y a b o v e 2A, a b o u t 1.5 M e V for t u n g s t e n . T he
'iT + *4*e x i s t e n c e of r o t a t i o n a l b a n d s w i t h b a n d h e a d s of J = 0 , 2 ,
a nd 2 i n s i d e the s u p e r c o n d u c t i n g g a p t hen s e e m s to r e q u i r e
tha t t h e s e s t a t e s p o s s e s s c o l l e c t i v i t y d u e to s o m e c o r r e l a t i n g
f o r c e b e s i d e s the p a i r i n g force. In o n e i n t e r p r e t a t i o n , the
J * 0+ a n d 2+ b a n d h e a d s a r i s e f r o m two of the m o d e s of v i b r a t i o n
IT + mof a B2 d e f o r m e d l i q u i d d r o p (Da 68); the J =0 s t a t e b e i n g c a l l e d a
23
" b e t a v i b r a t i o n " c o r r e s p o n d i n g to v i b r a t i o n s p r e s e r v i n g the
axial s y m m e t r y , a n d J v = 2 " g a m m a v i b r a t i o n s " a r i s i n g f r o m
v i b r a t i o n s c a u s i n g d e v i a t i o n s a w a y f r o m axial s y m m e t r y . T h e
J 11 = 2 s t a t e in this p i c t u r e c o m e s f r o m a c o u p l i n g of the
o c t u p o l e v i b r a t i o n a l a n d c o l l e c t i v e r o t a t i o n a l d e g r e e s of f r e e
dom, w h i c h will not be d i s c u s s e d here. T h e s e h y d r o d y n a m i c e x p l a n a
t i ons a d e q u a t e l y d e s c r i b e o n l y the m o s t g e n e r a l p r o p e r t i e s of
t h e s e c o l l e c t i v e s t a t e s , but the t e rms b e ta, g a m m a a n d o c t u p o l e
v i b r a t i o n s a r e still o f t e n u s e d as c o n v e n i e n t labels.
Of the m a n y i n t e r a c t i o n s u s e d to d e s c r i b e the bet a and
g a m m a v i b r a t i o n s , the m o s t f u l l y e x p l o r e d is a q u a d r u p o l e - q u a d r u -
p o l e i n t e r a c t i o n of the f o r m (Be 63, Be 6 6 , Be 67)
H = “ixl l 9 (a+a + a—a_)|^ (2.10qq A 1L pv p v p v 1
pv
w h e r e the q a r e the m a t r i x e l e m e n t s o f an o p e r a t o r ~ f ( r ) Y _ A (P) p v 202
w h e r e u s u a l l y f(r) = r , a l t h o u g h o t h e r f o r m s h a v e a l s o b e e n
u s e d (As 72a, As 7 2 B ). T h i s i n t e r a c t i o n c an lead to c o l l e c t i v e
p a r t i c l e - h o l e s t a t e s w h i c h c a n be s t r o n g l y e x c i t e d in i n e l a s t i c
s c a t t e r i n g , j u s t as c o l l e c t i v e m a s s v i b r a t i o n s s h o u l d be. T h e
u se of this i n t e r a c t i o n has p r o v e d q u i t e s u c c e s s f u l w h e n c o m
b i n e d w i t h v a r i o u s c a 1c u 1 a t i o n a 1 t e c h n i q u e s s u c h as the r a n d o m -
p h a s e a p p r o x i m a t i o n s (RPA) a n d q u a s i - b o s o n a p p r o x i m a t i o n . O ne
g e n e r a l r e s u l t of t h e s e c a l c u l a t i o n s is that the s t r u c t u r e of
e x c i t e d Ju = 0+ s t a t e s i n v o l v e s not j u s t b r o k e n - p a i r c o n f i g u r a
t i o n s ( p a r t i c l e - h o l e types) but a l s o c o n f i g u r a t i o n s i n v o l v i n g the
2A
e x c i t a t i o n of z e r o - c o u p l e d p a i r s of n u c l e o n s ( p a i r i n g type).
M i k o s h i b a , £t^ aj_. (Mi 67) p u b l i s h e d an e a r l y p a p e r on the s u b j e c t
in w h i c h the y a t t e m p t e d to c a l c u l a t e the i n t e r p l a y of the p a i r i n g
c o n f i g u r a t i o n s a nd the b r o k e n - p a i r c o n f i g u r a t i o n s in the s t r u c
t ure of e x c i t e d 0+ s t a t e s in rare e a r t h n u c l e i . T h e i r c a l c u
la t i o n s i n d i c a t e d that e x c i t e d 0+ s t a t e s w e r e not, in g e n e r a l
s i m p l y b e t a v i b r a t i o n s ( f rom the p a r t i c l e - h o l e m o d e ) in the s p i r i t
of the h y d r o d y n a m i c o s c i l l a t i o n m o d e l , but m u c h m o r e c o m p l i c a t e d
s t a t e s a r i s i n g f r o m the i n t e r p l a y o f the b e t a v i b r a t i o n a l an d
p a i r i n g d e g r e e s of f r e e d o m . In fact, the p r e c i s e c h a r a c t e r
of e x c i t e d 0+ s t a t e s w e r e f o u n d to be e x t r e m e l y d e p e n d e n t u p o n
the u n d e r l y i n g s i n g l e - p a r t i c l e s t r u c t u r e ; t hus no d i s c u s s i o n of
e v e n h i g h l y c o l l e c t i v e s t a t e s c a n i g n o r e the s i n g l e - p a r t i c l e
m i c r o s c o p i c s t r u c t u r e .
2 . 2 .14 C o r iol is Coupl i ng
O n e o t h e r r e s i d u a l i n t e r a c t i o n k n o w n to be i m p o r
tant in e x p l a i n i n g the d e t a i l s of o n e - n e u t r o n t r a n s e r r e a c t i o n s
in t u n g s t e n n u c l e i , is the C o r i o l i s f o r c e (Ca 71, Ca 72, Li 73,
So 75)- T h i s t e r m in the n u c l e a r H a m i l t o n i a n , w h i c h is n e g l e c t e d
in the s t r o n g c o u p l i n g m o d e l , c o u p l e s the s i n g l e - p a r t i c l e d e g r e e s
of f r e e d o m w i t h the c o l l e c t i v e r o t a t i o n s (hen c e it is s o m e t i m e s
c a l l e d r o t a t i o n - p a r t i c 1e c o u p l i n g ) . M i c r o s c o p i c a l l y , the t erm
has the e f f e c t of i n d u c i n g i n t e r a c t i o n s b e t w e e n s t a t e s of the
s a m e J 71 but d i f f e r i n g by o n e u nit of K. In a d d i t i o n , the i n t e r
a c t i o n has d i a g o n a l m a t r i x e l e m e n t s for K = 1/2 b a n d s l e a d i n g
2 5
to d e v i a t i o n s f r o m the usual J (J + 1) r o t a t i o n a l b a n d e n e r g y
s p a c i n g s (Da 6 8). For s i m p l i c i t y , h o w e v e r , the e f f e c t s o f
C o r i o l o i s c o u p l i n g h a v e b e e n n e g l e c t e d in this s t udy.
C h a p t e r 3
T W O - N E U T R O N T R A N S F E R R E A C T I O N S :
A P R O B E OF C O L L E C T I V E S T R U C T U R E
26
27
3.1.1 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 ion (DWBA)
C u r r e n t l y use d m e t h o d s of c a l c u l a t i n g d i r e c t n u
c l e a r t r a n s f e r 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 fall into t h r e e r a t h e r
d i s t i n c t c a t e g o r i e s c o r r e s p o n d i n g to i n c r e a s i n g c a 1c u 1 a t i o n a 1
a nd c o n c e p t u a l d i f f i c u l t y ; the D W BA, the C o u p l e d - C h a n n e l s B o r n
A p p r o x i m a t i o n ( C C B A ) , a n d the m u 11 i p 1e - s t e p s e q u e n t i a l a p p r o x
imation. T h e s i m p l e s t of the t h r e e is t he D W B A in w h i c h the
t r a n s f e r p r o c e s s is b r o k e n into t h r e e d i s t i n c t steps:
(1) T h e p r o j e c t i l e a nd t a r g e t n u c l e u s a p p r o a c h e a c h
o t h e r u n d e r the i n f l u e n c e o f t h e i r C o u l o m b a nd n u c l e a r
i n t e r a c t i o n s .
(2) T h e t r a n s f e r t a k e s p l a c e t h r o u g h a f i r s t - o r d e r p r o c e s s
(e.g., in t w o - n u c l e o n t r a n s f e r , b o t h n u c l e o n s a r e
t r a n s f e r r e d at the s a m e time, not s e q u e n t i a l l y ) .
(3) T h e o u t g o i n g p a r t i c l e a nd t he r e s i d u a l n u c l e u s m o v e
a p a r t u n d e r the i n f l u e n c e of t h e i r n u c l e a r a n d Cou -
1o m b fields.
W i t h i n the D W BA, the d i f f e r e n t i a l c r o s s s e c t i o n for the
s t r i p p i n g r e a c t i o n A ( a , b ) B w h e r e a = b + x a nd B = A + x is
g i v e n by:
<!° = " a " b ^ I I T L S J M 1 2 ( 3 J >, d2 (2irfi2 ) 2 k a L S J M
w h e r e
T L S J M = f rf3R d3p T b (p + t A /(A + 2 ) ] ^ f L S J M ( p’R)
x T ([(a - 2 ) /a] p" + R) (3 -2)3
3.1 REACTION DYNAMICS
a n d P a n d P, a r e the r e d u c e d m a s s e s of the (A + a) a n d (B + b) a b
s y s t e m s , r e s p e c t i v e l y . k a n d k, a r e t he w a v e n u m b e r s of th ea b
p r o j e c t i l e an d o u t g o i n g p a r t i c l e s , r e s p e c t i v e l y , a n d Hf a r e t he
w a v e f u n c t i o n s o f the i n c o m i n g (+) a n d o u t g o i n g (-) d i s t o r t e d
w a v e s . T h e c o o r d i n a t e s e m p l o y e d a r e d e f i n e d in F i g u r e 3~l.
T h e total f o r m f a c t o r (F, c a n be w r i t t e n
fLSJM(b’^ = / ^ ^ d3r (3’x ,r2 )<J>a U )
w h e r e A, B, a, b r e p r e s e n t all q u a n t u m n u m b e r s n e c e s s a r y to
d e f i n e the v a r i o u s n u c l e a r s t a t e s , ? a r e all c o o r d i n a t e s in a d d i
tion to Tj a n d r for the n u c l e u s B, a nd X a r e all c o o r d i n a t e s
in a d d i t i o n to T , a nd r_ for n u c l e u s a. (-)(+)A f t e r the s u i t a b l e d i s t o r t e d w a v e s Y a nd 4* a r e f o und
a b
(see S e c t i o n 3-1.2) the p r o b l e m r e d u c e s to c a l c u l a t i n g the m a
trix e l e m e n t u s i n g Eq. 3.3. For (p,t) or (t,p) r e a c t i o n s , t h r e e
c a 1c u 1 a t i o n a 1 s i m p 1 icat ions a r e u s u a l l y m a de:
(1) T h e i n t e r a c t i o n V (p") is t a k e n to be a s c a l a r i n t e r
a c t i o n V (p") = V (p) .
(2) O n l y the r e l a t i v e s - s t a t e o f the two t r a n s f e r r e d
n u c l e o n s in the t r i t o n is c o n s i d e r e d s i n c e the d-
s t a t e c o n t r i b u t i o n to the m a t r i x e l e m e n t is k n o w n
to be small for m o s t (strong) t r a n s i t i o n s (Au 70).
(3) T h e z e r o - r a n g e a p p r o x i m a t i o n is m a de. T h i s h a s the
p h y s i c a l m e a n i n g that b o t h n u c l e o n s a r e t r a n s f e r r e d
g u r e 3 - l- C o o r d i n a t e s y s t e m s u s e d in the D W B A
c a l c u l a t i o n s o f t w o - n u c l e o n t r a n s f e r r e a c
ti o n A ( a , b ) B w h e r e a = b + n^ + and
B = A + n^ + n 2 - T h e l o c a t i o n of the
v a r i o u s c e n t e r s - o f - m a s s a r e s h o w n in
th e f i g ure.
b
O c.m. of A+ri| + n
□ c.m. of b+n, +n
A c.m. of ri|+n2
31
t o g e t h e r , b e i n g e m i t t e d f r o m the n u c l e u s a a n d p i c k e d
up by the n u c l e u s B at the s a m e point. M a t h e m a t i c a l l y ,
this has the e f f e c t o f c o l l a p s i n g the s i x - d i m e n s i o n a l
integral into 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 . T h e u se of
the z e r o - r a n g e a p p r o x i m a t i o n for l i g h t - i o n i n d u c e d t r a n s
fer r e a c t i o n s is w ell j u s t i f i e d in the l i t e r a t u r e , the
p r i m a r y e f f e c t b e i n g an o v e r a l l r e n o r m a l i z a t i o n o f
s t r e n g t h of the c a l c u l a t e d c r o s s s e c t i o n s (Au 69).
U n d e r t h e s e a s s u m p t i o n s , the t r a n s i t i o n m a t r i x e l e m e n t can
be w r i t ten
t l s j « l ' ( 2 L * , r ' < \ h l < * > I y ] + ) ( R ) > ( 3 ' , , )
w h e r e the t w o - n e u t r o n f o r m f a c t o r is
$ LM (R) * '"L V LM (R) U L (R) ( 3’5)
a n d
U l (R) ' I K ( p ,v ,L ,S ,J ) B (A , B ; p , v ; J ) U y vL (R) (3-6)
pv
w h e r e p an d v a r e s i n g l e - p a r t i c l e b a s i s s t a t e s ( u s u a l l y s p h e r i c a l
shell m o del s t a t e s ) , Y. u (R) is a s p h e r i c a l h a r m o n i c , a n d K is
l m l
a j-j to 1-s c o u p l i n g t r a n s f o r m a t i o n c o e f f i c i e n t . T h e radial
i n f o r m a t i o n f or a g i v e n t w o - p a r t i c l e c o n f i g u r a t i o n [p ® v] is
c o n t a i n e d in the " r a d i a l f o r m f a c t o r " U . (R) w h i l e the B co-pvL
e f f i c i e n t s c o n t a i n the s t r u c t u r a l i n f o r m a t i o n of the s t a t e to be
p o p u l a t e d as w ell as a n g u I a r - m o m e n t u m c o u p l i n g i n f o r m a t i o n .
S i n c e the B c o e f f i c i e n t s c a n be r e l a t e d to the a m p l i t u d e of the
c o n f i g u r a t i o n [ p & v ] ^ in n u c l e u s B, t h e y a r e k n o w n as " s p e c t r o s c o p i c
32
a m p l i t u d e s . " In p a r t i c u l a r , it c an be s h o w n that
B ( A , B ; p , v ; J )
, f[a+ (u)a+ (v) ]= ( * b | - - -------------------$ r l j >' (d + Kuv)) /2 j JH (3.7)
B "B
w h e r e the b r a c k e t s [ ] a n d { } d e n o t e v e c t o r coup l i ng.
E q u a t i o n s 3-1 a n d 3 -6 i l l u m i n a t e an e x t r e m e l y i m p o r t a n t
a s p e c t of t w o - p a r t i c l e t r a n s f e r r e a c t i o n s ; T 9 ^ is a s u m o v e r m a n y
te rms [u§ )v]^ a n d s i n c e 0 (0 ) ~ |T6^ | 2 , w h a t is b e i n g p r o b e d
is not the p r o p e r t i e s of the i n d i v i d u a l terms, but the c o r r e l a t i o n
a s p e c t s of the n u c l e a r s t a t e s c o n n e c t e d by the t w o - n u c l e o n
t r a n s f e r .
D e t a i l s of the m e t h o d s u s e d in this s t u d y to c a l c u l a t e
the radial f o r m f a c t o r s will be g i v e n in S e c t i o n 3-2.1 w h i l e
s p e c i f i c e q u a t i o n s for the s p e c t r o s c o p i c a m p l i t u d e s in s e v eral
m o d e l s will be g i v e n in S e c t i o n 3.2.2.
3 . 1 . 2 T h e Opt i c a 1 M o d e 1
(±)In o r d e r to c a l c u l a t e the d i s t o r t e d w a v e s T
for input to the D W B A c a l c u l a t i o n , p o t e n t i a l s m u s t be c h o s e n to
d e s c r i b e the i n t e r a c t i o n b e t w e e n the p r o j e c t i l e a n d t a r g e t a nd
b e t w e e n the e j e c t i l e a n d r e s i d u a l n u c l e u s . T h e p o t e n t i a l s e m p l o y e d
in the p r e s e n t w o r k i n c l u d e a real p o t e n t i a l of d e p t h V q a n d s h a p e
g i v e n by a s p h e r i c a l l y s y m m e t r i c W o o d s - S a x o n wel l w i t h r a d i u s
R = r a ' ^ 3 a n d d i f f u s e n e s s a. A l s o u s e d w e r e s p i n - o r b i t po- o ot e n t i a l s (V ) o f the f o r m L • S as wel l as the C o u l o m b inter- soa c t i o n s , e a c h w i t h t h e i r o w n r a d i u s a n d d i f f u s e n e s s p a r a m e t e r s .
T h e r e m oval of f l u x t h r o u g h n u c l e a r r e a c t i o n s is s i m u l a t e d
33
by the use of imaginary po ten t ia ls , giving r ise to what is cal led
the "optical model." Normally used are such potent ia ls as a
volume absorption (W) in the shape of a Woods-Saxon well and
a surface absorption term (W ) with the shape of the d er iva t ive
of a Woods-Saxon we l l .
Since the interact ions simulated by the opt ica l model should
be manifest in the e la s t ic scattering angular d is t r ib u t io n , the
opt ica l model parameters are normally chosen to f i t the e la s t ic
scattering of the a + A system and the b + B system, at the
appropriate bombarding energy. For the present study, the pro
ton parameters were taken from the global set of Becchetti and
Greenlees (Be 69) and the t r i t o n parameters as extracted from
t r i t o n e l a s t ic scatte r ing on a number of nuclei at 15 and 20
MeV bombarding energy (Be 71) (Table 3- l ) - The e f fec ts of sub
s ta n t ia l changes in the values of the opt ica l model parameters
upon the DWBA results were not investigated during the course
of this study, but small changes ( 5 - 10%) were found to have l i t t l e
e f fe c t upon the shapes of the DWBA angular d is t r ib u t io n s .
3 .1 .3 Multi step Processes
In certa in circumstances, one or more nuclear
reaction channels are so strong that they must be e x p l i c i t l y
included in the reaction c a lcu la t io n , rather than simulated via
the opt ica l model. Such a circumstance occurs in strongly
deformed nuc le i , such as tungsten, where the members of the
rota tiona l bands are strongly coupled to each other via ine la s t ic
TABLE 3-1
OPTICAL MODEL PARAMETERS
V O c a W U I I so so
___________ (MeV)_____ (fm)_____ ( fm)____( fm)_____(MeV)_____(MeV)________ (fm)_____ (fm)_____(MeV)____ (fm)
w+p(a) - 5 7 . 0 1.17 1.25 0.75 - 1 .9 3 5 - 2 1.32 0.6*4 -2*4.8 1.01 0.75
W + t ^ -161.*4 1.20 1.25 0.72 -25 .0 - 1.30 0.8*4 -10 .0 1.20 0.72
(a) Reference Be 69
(b) Reference Be 71
(c) Convention used in Reference Ku 69
Aso(fm)
35
scat ter ing . In such a case the DWBA is no longer adequate to
describe the t ransfer process since excited states may be popu
lated not only by a d i rect p a r t ic l e t ransfer reaction, but also
by in e la s t ic scat tering (or t ransfer ) to an intermediate sta te ,
followed by t ransfer (or ine la s t ic sc a t te r in g ) . Some of the
possible routes for populating members of the ground-state band
are shown in Figure 3“2. The code u t i l i z e d in the present work
to ca lcu la te such mult is tep contr ibut ions was a version of the
CCBA code LISA of Ascuitto (As 72a, As 72B) which includes in
e la s t ic couplings to a l l orders between members of a rota tional
band (the 0+ , 2+ , and 4+ members of the GSRB in th is case) and to
f i r s t - o r d e r the two-neutron t ransfer process (see Figure 3~2) .
The e x p l i c i t inclusion of in e la s t ic exci ta t ions has an
e f fe c t upon the e la s t ic scat tering angular d is t r ibu t ions so
that the opt ica l model parameters used in a CCBA ca lcu la t ion should
in general be d i f f e r e n t from those used in a DWBA ca lcu la t ion .
Following reference As 72b, the opt ica l model parameters used
in th is work are taken from proton and t r i t o n e la s t ic scat tering
on spherical nuclei where CCBA e f fec ts are small. The inclusion
of in e la s t ic couplings when such spherical opt ica l model para
meters have been used has been shown to adequately reproduce the
e la s t ic and in e la s t ic d i f f e r e n t i a l cross sections (Ki 73).
The th ird of the ca1cu1 a t io n a 1 procedures involves the
inclusion of sequential p a r t ic l e t ransfer such as a (p,d) reaction
followed by a ( d , t ) reaction and is current ly of great interest
calcu la t ions . The wavy l ines represent
in e la s t ic couplings to a l l orders while the
solid l ines represent ( p , t ) t rans i t ions to
f i r s t order on 1y .
F i g u r e 3“2. M u l t i p l e s t e p r o u t e s i n c l u d e d in the
MULTIPLE STEP ROUTES
(Pi 74, Sh 77, Tu 78). Such e f fects seem to be important p r i
mari ly in cases where the p roba b i l i ty of d i rect t ransfer to the
f ina l state is especia l ly small (Se 75, lg 72, Ba 70) and so
except for very weakly excited sta tes , the e f fec t is usually
ignored. The p o s s ib i l i t y ex is ts , however, that the inclusion
of sequential p a r t ic le transfer e f fec ts w i l l be found to be
necessary to reproduce the absolute cross sections of even strong
two-nucleon t ransfer t rans i t ions .
39
3.2.1 Radial Form Factors
The ca lcu la t ion of the radial form factors U . (R)yvLrequires that a coordinate transformation be performed on the
wavefunctions of the two transferred nucleons from a reference
frame f ixed in nucleus a to a frame f ixed in nucleus B. A f u r
ther transformation must be made to coordinates describing the
motion of the center-of-mass of the two nucleons (r ) and the i r
r e la t iv e motion (~) . Two methods of accomplishing these t rans
formations have been used in the present study. The code DEF2NT
takes the s i n g le -p a r t i c l e wavefunctions (which have been calcu
lated on a cy l in d r ic a l harmonic o s c i l l a t o r basis) as f ixed in B.
and expands them on a spherical harmonic o s c i l l a t o r basis. The
required transformation is then carr ied out a n a l y t i c a l l y using
Moshinsky brackets for each of the terms in the spherical har
monic o s c i l l a t o r expansion.
The other method employed was used in ca lculations with the
two-nucleon t ransfer option in the DWBA code DWUCK (Ku 69).
The s ing1 e - p a r t i c 1e wavefunctions were taken from Davidson (Da 6 8 )
as calculated for a deformed harmonic o s c i l l a t o r with deformation
* 0 . 3 and the transformations performed numerical ly using the
Bayman-Ka11io method (Ba 67). In these ca lculations the individual
s in g le -p a r t ic le levels were bound at one-hal f the experimental
two-neutron separation energy 50 that no energy was allowed
in the nucleons' r e l a t iv e motion.
3.2 N U C L E A R S T R U C T U R E INPUT
AO
The spectroscopic amplitudes contain the dependence
of the t ra n s i t io n amplitude upon the structure of the state populated
in the residual nucleus. In th is section, spec i f ic equations
for these in the ad iabat ic approximation w i l l be given for two
types of states , members of the GSRB, and pure Nilsson two-hole
states , as well as s im p l i f ied expression for the d i f f e r e n t i a l
cross section 0 ( 0 ) .
(1) Ground State Band (Br 73)
B ( J fM f K f , J . M . K . j u ^ j L ) = B(JMK,000;p1u2 ;L)
= ( 1 + 6 (2J + 1)"* J [1 + ( - ) J + ' l + ] 2]V 2 oj> 0
x (-) -* i + ]2 + W1 <j 1w1j 2 -w2 |J0>
x Y (u.(A" 2 ) v.(A) w! W1. ) 6 ,. (3-8)j 1 1 J ]cu 1 J 2to2 JL
where
p = ( n , l , j , u ) ' Shell Model State
i = (cu [NN2 A ]) Physical (Nilsson) state
and the W are co ef f ic ien ts in the transformation
a+ = J W! a t (3-5)U 4- VI 1
The conceptually important information to note is that the
contr ibut ion of each s i n g le -p a r t i c l e level " i " is weighted in(A-2) (A)Equation 3-8 by the factor U . V. where the factor U measures
the emptiness of o r b i ta l i in the ground-state of the residual
3 . 2 . 2 S p e c t r o s c o p i c A m p l i t u d e s
nucleus and V measures the ful lness of the o rb i ta l in the target
nucleus ground s ta te . This factor is peaked near the center of
the Fermi surface (where U. a V.) so that the o r b i ta ls near thei imiddle of the Fermi surface are the primary contr ibutors to the
population of the members of the ground-state band in ( p , t ) reactions.
A s im pl i f ied expression can be obtained for the ground-
state cross section (Be 72) by making use of the so-called
Os approximation (Br 6 8 , Br 71) and neglecting the d i f ference in
binding energy of the various o r b i ta ls :
° ( ' 0 - [ [ / a . (9) U.(A' 2) V .( A ) ] 2 (3.10)g i 1 1 1
where a . ( 0 ) is the d i f f e r e n t i a l cross section for the L = J
pickup of a neutron pair from o rb i ta l " i " .
(2) For the case of pickup leading to a pure Nilsson two-
hole state ( i ) ' (k) ' of spin J and in t r in s ic angular momentum
projection K, the expression for the spectroscopic amplitude
has been given by Garret t e_t_ £]_. (Ga 71):
Pure N i 1sson Two-Hole State
B(JMK,000;y) u2 ;L) = [ 2 / (2J + 1) (1 + 6 ^ ) 1 ^ 6 (K,u.+iJk)
x (<j -ai, j.o). | JK> W1 Wkv 1 i 2 k 1 p p2
- <j .u. j.u). | JK > W1 W1 ) 6 (3-11)1 kJ 2 i 1 pj p2
The above expression makes i t possible to ca lcu la te the d i f f e r e n
t i a l cross section for the pickup of e i th e r a neutron pair
(i = k) or of two non-paired neutrons ( i / k ) . Within the Os
approximation and neglecting Q-effects as before leads to:
0 (J+ ) = ^ T e T ( a i k v, Vk ♦ b . k u, Uk) l 2 (3
where J is any s ta te , whose structura l information is contained
a.^ and b.^ and o.^ is calculated from Equation 3-11.
These expressions w i l l be used la te r in the present work i
discussions of the various models employed to systematize and
explain the experimental data.
Chapter A
DATA ACQUISITION AND REDUCTION
*♦3
AA
The protons used to induce the reactions discussed in this
work were obtained from the Yale MP Tandem acce lerator . Hydrogen
ions from a duoplasmatron ion source were passed through a
charge exchange canal and then accelerated to a k in e t ic energy
of 21 MeV before momentum se lection. In order to reduce s l i t
sc at te r ing , the beam was defined by a s l i t of dimension . 0 5 0" x
.005" which was located immediately a f t e r the momentum-selecting
90° bending magnet, some 35 feet before the target . An a n t i
scat ter r o l l e r s l i t of dimension . 0A0" was placed about 8"
before the ta rg et . The beam, of approximately 100 nA in te ns i ty ,
was focused and steered such that ~ 7 5 % of i t passed through a
small hole ( .012" x .065") in a ZnS-coated screen placed at the
geometrical center of the magnetic torus of the mult iangle
spectrograph (Ki 73, Ma 73). A f te r focusing, the taget replaced
the screen and photographic emulsions were used to detect the
reaction t r i to n s a f t e r momentum analysis by the spectrograph. The
approximate target thicknesses and the to ta l charge accumulated
for each experiment are l is te d in Table A - l . In each case, except
where noted, data were taken at 23 angles, from 5° to 87-5° and
from 92.5° to 167-5° in 7-5° steps.18 0The targets , with the notable exception of that of W,
were fabr icated by d i rec t vacuum deposit ion of a sub-oxide of
W0j onto 20 yg/cn! carbon backings. The separated-isotope W0
was obtained from Oak Ridge National Laboratory (0RNL) while the
natural target was composed of h igh-pur i ty natural W0^. The
A . 1 E X P E R I M E N T A L P R O C E D U R E
T A B L E fr-1
Nomi na1 Target
Mass
Mult igap Run No.
Total Charge Col 1ected ( x l 0 3yc)
TARGET COMPOSITION
Approx imateThickness of Isotopicn Compos i t ion (%)
(yg/cm2 ) 180 182 00 18/4 186 Comments
180 126 30 50 "•100% * * * * '•'-■■■
182 118 1 0 . 1 130 <.05 9* . 3 2.5 2.3 0 . 8
00 1 32 20 70 <, 1 3-5 9 0 . 0 5.6 1 . 1 Only 1st 6 angles
1814 119 8 160 <.05 1 . 8 2. 0 9/4.2 1 -9
186 129 12.5 120 < . 0 2 • 5 • 3 2.2 97.1 Only 1st 12 angles
Natura1 133 10 60 . 1 26.3 1^.3 30.7 28.5 Only 1st 6 angles
From spectrum estimated to be maximum of 2% Not seen in spectrum
46
sub-oxide of WO was made by resistance heating the samples under
vacuum in a tantalum boat u n t i l the samples changed from the ir
usual bright yellow color to deep purple. During the subsequent
evaporation of the sub-oxide onto the carbon backings, a l l but the
central portions of the backings were masked to minimize heat-
induced stresses. The isotopic analyses as given by ORNL for a l l
of the targets are given in Table 4-1.180The target of W, an isotope which is only 0.13% natura l ly
occuring, was made by J. G. Tracey of ORNL by d i rec t deposit ion of180 2 m eta l l ic W on a 30 yg/cm carbon backing in an isotope separa
tor . This resulted in a remarkably iso top ica l ly pure target of 180m e ta l l ic W with only a thin oxidized layer . The upper l im i t
182of 2% W contamination of the target l is ted in Table 4-1 was
determined from the ( p , t ) results since no d i rec t isotopic analy-
s i s was poss i b l e .
The photographic emulsions used for detecting the t r i to ns
were obtained from I l f o r d , Ltd. In view of the low ionizing power
of t r i to n s , the most sensit ive emulsions ava i lab le (K 5) were used.
The plates were developed in Kodak D5 developer for t h i r t y minutes
at a temperature of 5°C and f ixed in Kodak Rapid Fixer with Har
dener for two hours at 10°C. The reduced temperature during
development and f ix in g gave good contrast between the tracks and the
background, although longer developing and f ix in g times as well
as stronger solutions than usual were required because of the de
creased chemical a c t i v i t y . A d d i t io n a l ly , for the and ^
target experiments, .005" thick acetate f o i l s were placed in front
of the emulsions during exposure to slow down the t r i to ns in
order to increase th e i r energy loss in the emulsions and thereby
make the tracks more evident.
After development, the photographic plates were scanned in
0 . 5mm str ips by microscopists using a 20x microscope and d a rk - f ie ld
i l luminat ion . The raw data were reduced with the computer program
ALLPLOT which corrects for the spectrograph’ s changing so lid angle
with plate pos i t ion, performs a laboratory to center-of-mass
conversion of the y ie ld s , and then plots and tabulates the data with
respect to reaction Q-value. The conversion from distance along
the plate to reaction Q-value requires that the magnetic f i e ld
(B) and the plate distance (D) to radius-of-curvature (p) conver
sion for the gap in question be known. The D to p conversion
has been measured and found to be independent of B to a good
approximation (Ko 69)- All of the results reported in th is work
were obtained using the la tes t D to p c a l ib r a t io n , CALIBM. Since
the f r inge f ie ld s in each gap are d i f f e r e n t and, in general,
unknown, an empirical approach - known as the e f f e c t i v e f i e l d method
was employed to determine the e f fe c t iv e magnetic f i e l d in each
gap. For each gap, the plate posit ion of a peak well separated
from the others (usually the 0 or 2* states) was determined.
Then, assuming a reaction Q-value for the peak, an e f fe c t iv e f i e ld
for the gap was determined and used to ca lcu la te the reaction
Q-value corresponding to each bin , assuming the nominal reaction
masses. In a l l cases, the e f fe c t iv e f i e l d so calculated d i f fered
from the nominal value of the magnetic f i e l d by less than 0 . 1%.
Two main contaminant peaks were seen in the experiments.
4.2 D A T A R E D U C T I O N
49
These arose from the ' 70 ( p , d ) ^ 0 (g . s .) reaction with oxygen in the13 12target and from the C(p,d) C reaction with the carbon backing.
In p r in c ip le , incident deuterons or t r i to ns of the same magne
t ic r i g i d i t y can be distinguished by the brightness of th e i r
tracks; in prac t ice , however, th is d is t in c t io n is d i f f i c u l t
and was judged not worth the extra scanning time. In no case,
however, were the t r i to n s confused with deuterons from the
W(p,d) reactions because of the great d i f ference in magnetic
r i g i d i t y of these two reaction products.
Representative spectra for the separated-isotope target
experiments are shown in Figures 4-1 to 4-5 for laboratory angle
of 20°. Peaks assigned to the various nuclei are numbered for
la te r id e n t i f i c a t io n . In these spectra, the data are compressed
into one m i l l im eter bins and the y ie lds of the reaction groups
corresponding to the population of the ground-state band are
reduced in magnitude by a factor of f i v e . In each case the resolu
t ion was of the order of 20 keV, ar is ing p r im ar i ly from f i n i t e
beam spot size (10 keV) , straggling in the target (5 keV) and
spectrograph aberrations.
In most cases the peak yie lds were determined by summing the
counts as taken from the ALLPLOT output and subtracting the appa
rent background. Overlapping peaks were deconvoluted by eye or
through the use of the Yale in te ra c t iv e version of the program
AUTOFIT (Ma 73) which f i t s peak posit ions and heights with a
reference peak shape taken from a selected peak in the measured
Figures A- 1to A-5 . T r i ton spectra at a laboratory angle
of 20° for the separated-isotope W(p,t)
experiments. The data are shown in 1 mm
bins. The peaks are numbered for id e n t i
f ic a t io n purposes and correspond to the
numbers in Tables 5~h to 5_8 and to those
in Appendix C.
-6.5 -6.0Q - V A L U E (M e V )
COUNTS/CHANNEL
C O U N T S / C H A N N E L
CO
UN
TS
/C
HA
NN
EL
300
2 0 0
100
-9.5 -9.0 -8.5Q -V A L U E ( M e V )
COUN
TS/C
HANN
EL
180 W(p,t)l78WEp=2l MeV
© = 2 0 °150
1 0 0
50
-10.0 -9.5
Q VALUE (MeV)
♦erf
spectrum. In p a r t ic u la r , most of the yie lds for the high-1 80lying states in the complicated W spectra were analyzed with
AUTOFIT. The errors quoted in the d i f f e r e n t i a l cross sections
r e f le c t s t a t i s t i c a l errors as well as approximate f i t t i n g errors
where appropr ia te , but do not include e f fec ts due to scanning
re p ro d u c ib i l i t y , estimated good to 5%, not possible errors in the
absolute cross sections.
4.3 ABSOLUTE CROSS SECTION MEASUREMENTStl, l i 4. f 184, 1 8 2 , 180.., .The absolute cross sections for the W(p,t)
reactions were measured in a separate experiment which u t i l i z e d
a solid state detector-te lescope and on - l in e computer data acqui
s i t io n to simultaneously detect and iso la te e l a s t i c a l l y scattered
protons and reaction t r i to n s . Proton e la s t ic scat te r ing angular
d is t r ibu t ions were measured for each target at a bombarding energy
of 21 MeV to establ ish an absolute cross section scale (see
Section 5.1) and then both t r i t o n and proton spectra were taken
at various lab angles. The 0 and 2+ members of the ground state
band were unresolved in these experiments, so th e i r summed y ie ld
was compared to the e la s t ic proton y ie ld to establ ish the ab
solute ( p , t ) summed cross section.
Chapter 5
PRESENTATION OF DATA
58
59
The ( p , t ) experiment on the natura1-tungsten target provided
accurate r e l a t iv e Q-values for the lowest L = 0 t rans i t ions for the18G I8A 183 182 w(p, t ) reactions, shown in Table 5*1. The values
obtained assume a Q-value of -4 .470 MeV for the 4 88W (p , t ) 4 8**W (g . s .)
t ran s i t io n (Ma 65) and are believed to be accurate to w ith in
5 keV r e la t iv e to one another. The L = 0 t ra n s i t io n in the1 Q o 1 0 1 1 ft 1
W(p,t ) W reaction is to the j - [510] level in W, which
occurs at an ex c i ta t io n energy of 458 keV (Da 71). Thus the
Q-value to the ground state in ' 84W is determined to be -5-769
MeV, in disagreement with the value of — 5 • 8 10 MeV quoted in
Reference Oo 72 but in exce l lent agreement with the value of
-5 .770 MeV appearing in the 1965 Mass Table (Ma 65).1 8 0As W is so rare in nature, the ground-state Q-value for
18 0 178the W(p,t ) W(g.s.) reaction could not be obtained from the
natura1 - tungsten target experiment (predicted y ie ld .1 10 counts).13 12The C(p,d) C (g .s . ) peak in the spectra from the separated-
isotope experiment was therefore used to determine the e f fe c t iv e
f i e ld and then the Q-value of the ground-state t ra n s i t io n to
' 78W determined to be -6 .844 ± .015 MeV.
5.1 Q - V A L U E S
60
TABLE 5-1
(p , t ) Q-VALUES FOR LOWEST L = 0 TRANSITIONS
Residual Previous Values (MeV).Mass (MeV) (b) (c) (d)
184 -4.470 -4 .470 -4.474
182 -5 .107 -5.120 -5.124
181 -6.227
180 -6.247 -6.271 ( - 6 . 2 6 1 ) -6 .273 ± .007
178 -6.844 ± .015 -6.843
(a) Assuming Q = -4 .470 for 1 *^W ( P l t) ^ W ( 0 * ) , errors ±.005 MeVunless indicated.
(b) Reference Ma 65
(c) Reference Oo 73
(d) Reference Ca 76
61
5.2 RELATIVE AND ABSOLUTE CROSS SECTION SCALES
Table 5- 2 l i s t s the r e la t iv e d i f f e r e n t i a l cross sections for
the lowest L = 0 t ransi t ions as determined from the natural target
experiment. Only s t a t i s t i c s and scanning r e p ro d u c ib i l i ty in t r o
duce uncerta inty into the r e la t iv e cross section values for the
' 86W(p,t ) and 8^W(p,t) reactions. The r e la t iv e t r i to n yie lds 183 182from the W and W in the target are less cer ta in due to th e i r
nearly equal Q-values. These d i f f e r by only 20 keV while the
experimental resolution was 25 keV, necessitat ing separation of
the two peaks out of the composite, using deta i led peak f i t t i n g
rout i nes.
The absolute cross section scales for the ^86W(p,t)
reactions were determined using the proton scat ter ing method
described in the previous chapter. Figure 5- l shows the measured1 8 0proton e la s t ic scatte r ing angular d is t r ib u t io n on W as well
as the normalized opt ica l model prediction using the proton para
meters of Table 3“ 1• The agreement is completely sa t is fa c tory .
Table 5~3 l i s t s the results for each target so studied; also
included are the values of FACTOR adopted for each multigap run,
where 0 (0 ) (pb/sr) = (yield)/FACTOR. The quoted errors for FACTOR
are a l l estimated absolute errors and vary between 10% and 15%.
62
TABLE 5-2
RELATIVE CROSS SECTIONS FOR LOWEST L = 0 TRANSITIONS FROM NATw(p, t)
Residual do Rel 6a_ RelMass Yield @ 20° dft 20° Yield g 27-?~ dn 27?°
184 1163 ± 3A 1.12 ± . OA 2267 ± A8 1.27 ± .OA182 1112 ± 3A 1.00 1915 ± AA 1.00181 326 ± 30 0.29 ± .OA 355 ± AO O.AO ± .05180 1160 ± AO 0.57 ± .08 17A8 ± AO 1.07 ± .10
Figure 5~1. E las t ic scattering angular d is t r ib u t io n1 8 0for 21 MeV protons on W plot ted as
r a t io to Rutherford. The error bars
are smaller than the data points. The
ca lcu la t ion is an opt ica l model prediction
using the proton parameters of Table 3- l
and was normalized to the data.
RUTHERFORD
65
TABLE 5-3
ABSOLUTE (p . t ) CROSS SECTIONS
do/d?)(0+ + 2+ ) (yb/sr )Residual 9 9 Multigap
Mass 27.5° 35° 50° Run No. Factor
184 - - - 129 14.9 + 1 . 8
182 322 + 32 330 + 34 119 12.9 + 1.3
181 - - - 133 23.9 + 3.6
180 181 + 14 205 + 15 118 1 3 . 8 Hr 1 . 8
178 809 ± 75 235 + 31 228 + 33 126 6.7 + 1 . 0
66
5.3-1 General Comments
The spectroscopic results for each of the individual
experiments w i l l now be discussed with p a r t ic u la r a t ten t io n to the
ground-state bands, the two rota tiona l bands presumably b u i l t
on the quadrupole mass v ibrat ions (the beta (K = 0+ ) and. 71 +. .gamma (K = 2 ) bands) and the octupole v ibra t iona l band
. Tt - .(K = 2 ) . The various states populated are enumerated in Tables
5-A through 5 _8 and the complete set of angular d is t r ib u t io n s
for these states are given in Appendix C.
Spin assignments are made to low-lying states by the compari
son of th e i r ex c i ta t io n energies with previous level scheme
determinations and by comparisons of th e i r angular d i s t r i b u
t ion shapes with those of levels of known spin. For 0+ sta tes ,
the highly c h a ra c te r is t ic o s c i l la to r y angular d is t r ib u t io n s
make th e i r spin assignment unambiguous, even for weakly populated
states . Other than the above exceptions, no attempt has been
made to assign spins, as the angular d is t r ib u t io n for (p , t )
reactions on deformed nuclei are not c h a ra c te r is t ic of the
transferred angular momentum due to complications r e f le c t in g the
presence of mult istep processes.
The general features of the spectra can be seen to change186 18 4smoothly but markedly with mass number. Whereas the W(p,t) W
spectra show many rather strongly populated states between 1 and180 1733 MeV e x c i ta t io n , the W(p,t) W reaction leads to much less
excited state strength. This feature can be understood as ar is ing
5.3 S P E C T R O S C O P I C R E S U L T S
67
from two factors , the e f fe c t of the Coulomb b a r r ie r and the e f fec t
of changing angular momentum matching condit ions. The Coulomb
b a r r ie r for t r i to n s on tungsten is about 12 MeV, which corre
sponds to an ex c i ta t io n of 2 .5 MeV in 2.7 MeV in
and s t i l l higher in the higher mass isotopes. Thus, the reduc
t ion in overa l l population of excited states for the low-mass
tungstens can be understood as a b a r r ie r e f fe c t . In addi t ion ,
angular momentum matching also plays a ro le . The best matching 1 78for W at 1 MeV ex c i ta t io n is L = 0, and increases by approxi
mately one uni t of angular momentum for each MeV more posi t ive1 fi f\ 1 fii*Q-value. A value of L = 3 is reached for the W(p,t ) W(g.s.)
t ra n s i t io n . Thus, higher spin states (J >_ 3) should be more
strongly populated in the higher mass isotopes.
To invest igate the overal l e f fects of these two factors ,2ty p ic a 1-conf igurat ion (the ( 2 f 7 / 2 ) conf iguration) two-part ic le
l fi 7 1 finpickup DWBA ca lculations were performed for the W(p,t) W
reaction for various Q-values, and angular momentum transfers of
0 and 2 uni ts . These ca lculations u t i l i z e d the code DWUCK (Ku 6 9 )
and the opt ica l model parameters of Table 3"1 with bound-state
radius parameters; rg = 1.25 fm and a = 0.60 fm. The angular
d is t r ib u t io n shapes so calculated were found to change only s l i g h t l y
with Q-value while th e i r magnitudes varied considerably. Figure
5_2 presents the calculated r e la t iv e cross sections for L * 0
and 2 t rans i t ions as a function of Q-value, both normalized to 1 at
Q = -6.271 MeV. The posit ions of the various ground-state t ransi t ions
on the Q-value scale are shown for o r ie n ta t io n as well as the
for t ransferred angular momentum 0 and
2 as a function of reaction Q-value.
The ca lculations employed the opt ica l
p a r a m e t e r s of T a b l e 3-1 a n d a t y p i c a l -
2conf iguration form factor ( 2 f 7 / 2 ) .
Also shown are the Q-values for the
various ground-state ( p , t ) t rans i t ions .
F i g u r e 5_ 2. R e l a t i v e s t r e n g t h s o f (p,t) r e a c t i o n s
RE
LA
TI
VE
S
TR
EN
GT
H
- 4 - 5 - 6 - 7 - 8 - 9Q - V A L U E ( M e V )
70
Q-value corresponding to the t r i t o n having only the Coulomb separa
t ion energy. The f igure demonstrates q u a n t i ta t iv e ly the results
previously discussed q u a l i t a t i v e l y .
5 .3 .2 The Nucleus l8/*W
The spectrum for the 8^W(p , t ) ^ r e a c t i o n taken
at a laboratory angle of 20° is shown in Figure 4-1 where the peak
numbers correspond to those in the table of ex c i ta t io n energies,
Table 5*4 and to the angular d is t r ib u t io n s in Appendix C - l . Only184the low-lying states of W were invest igated, up to an exc i ta t io n
energy of 1.2 MeV, although a complete spectrum is shown. A
search was made for J77 = 0+ states excited with greater than 10%
of the ground-state strength below an exc i ta t io n energy of 3 MeV.
Only the known state at 995 keV f i t t e d these c r i t e r i a .
5 .3 .3 The Nucleus l82U18 4 182The spectrum for the W(p,t ) W reaction taken at
a laboratory angle of 20° is shown in Figure 4-2 where the peak
numbers correspond to those in Table 5_5 and to the angular d i s t r i
butions in Appendix C-2. All peaks below 3 MeV exc i ta t io n were analy
zed and angular d is t r ibu t ions extracted. Doublets were unfolded
by eye where possible ,or with the program AUTOFIT where necessary.
The ground-state ro ta t iona l band (GSRB) is strongly in evidence7T +as well as the bandhead of the well-known f i r s t excited K = 0
band (P 4) , the J71 = 2+ bandhead (#5) and the 4+ member ( r 8 ) ofTT + - TTthe K = 2 gamma band, and the 3 member of the K = 2 octupole
band (#7) . The 2+ member of the f i r s t - e x c i t e d K71 = 0+ (o |) band ( r 6 )
is found at
Bes i d
a higher e x c i ta t io n energy than the 2+ and is consi
derably mor: weakly excited.
Ins the well established 0+ state at 1138 keV, three
other O' st« tes were located, at 2520 keV (#27), 2552 keV (#28)
and an especia l ly strongly populated one at 2725 keV (#31).
The assignment for the 2520 keV state Is somewhat uncertain as the
peak appears
5 . 3 . A
to be an unresolved doublet.
The Nuc1eus ^8 Wl83 l 8 lA sample spectrum for the W(p,t) W reaction
taken at a laboratory angle of 20° is shown in Figure A-3. The
iden t i fy ing peak numbers correspond with those in Table 5-6 and
the angular c is t r ib u t io n s in Appendix C~3- All states below
2 MeV e x c i ta t io n were analyzed
The most prominent peak in the spectrum corresponds to a
t rans i t ions (#5) . This t ra n s i t io n presumably leads
bandhead of the band b u i l t on the ? - [ 5 1 0 ] in t r in s ic1 O o
[510]) since the odd neutron of the W ground state
strong L = 0
to the Jn = i state (i i ~is assigned to that s i n g le -p a r t i c l e state (Ar 6 6 ) . Based on the l e
vel schemes o' References Da 71 and Ca 72, the - [510] state at
^58 keV shoulc not be completely resolved from the 3/2 state at
A50 keV and barely resolved from the 7/2 state at A7G keV. The
highly o s c i l l a t o r y nature of the angular d is t r i b u t i o n , however, unam
biguously id e n t i f ie s peak #5 as due to an L = 0 t ra n s i t io n and not
L = 2 or A t ra n s i t io n which would be required for populating the 3/2
and 7/2 s t a t e ; , respective ly . The maximum possible contr ibut ion
72
from the 3 / 2 and 7 / 2 states to the value of the d i f f e r e n t i a l
cross section at the second maximum of peak US may be estimated
by noting that they would f i l l in the minimum in the angular d is
t r ib u t io n at 12.5°, where the d i f f e r e n t i a l cross section is 4.2
pb/sr. The two states are then seen to add at most 5 pb/sr to
o[q) at the second maximum of peak §S where 0 ( 0 ) = 280 pb /sr , a
negl ig ib le contr ibut ion. Besides the L = 0 t ra n s i t io n to the
bandhead of the 5 - [510] band, L = 2 t rans i t ions to the next two
members of i ts ro ta t iona l band are seen, the 3 / 2 and 5 / 2 being
peaks #7 and #8 respectively .
Conspicuously absent from the spectrum is a strong peak
corresponding to an L = 0 t ra n s i t io n to the i ~ [521] s ta te ,
which would be a doublet with the L = 4 t ra n s i t io n to the
7/2 7/2 - [514] s ta te . Peak #4 corresponds to the correct exc i ta t ion
energy for th is doublet, but i ts angular d is t r ib u t io n shows very
l i t t l e o s c i l l a t i o n , requiring that i ts L = 0 contr ibut ion be small.
Assuming that the odd £ neutron is only weakly coupled to
the gamma v i b r a t i o n , the pickup strength of the 2y should s p l i t
between two states , a 3/2 and a 5/2 . These states would be
formed by the two possible couplings of the angular momentum of the
odd p a r t ic l e with that of the gamma v ib ra t io n . They should also
occur at approximately the same ex c i ta t io n energy above the
i - [ 5 1 0 ] s ta te , as the 2y is above the ground states of neighboring
even-mass nucle i . Such states are not seen. Instead, a strong
L = 2 t ransfer to the 5/2 5/2 - [512] state at 365 keV (#3) is in
73
evidence with a strength comparable to that of the gamma vibrat iona l
bandheads in neighboring even-mass tungsten nucle i . The implica
tions of these observations w i l l be discussed in section 8 . 2 .
A strong L = 0 t ran s i t io n is located at an ex c i ta t io n energy
of 1864 keV and is interpreted as ar is ing from a t ran s i t io n to a
state formed by the coupling of the odd neutron to the excited
0 state in at 1516 keV. The occurence of another state at
1892 keV with a very forward-peaked angular d is t r ib u t io n required
that the two peaks be f i t t e d out of the composite, resu l t ing in a
somewhat larger uncertainty in the values of the experimental
d i f f e r e n t i a l cross sections at forward angles.1 ftn5 .3 .6 The Nucleus W
l 82Table 5~7 l i s t s the states populated in the W(p,t)1 8 0 W reaction where the ident i fy ing peak numbers are consistent with
the designations in the sample spectrum (Figure 4-4) and the angular
d is t r ib u t io n s in Appendix C-4. Considerable emphasis was placed
upon the extract ion of spectroscopic information since the structure 1 80of W previously had not been extensively probed with p a r t ic le
t ransfer reactions and i ts level scheme not well established
(Gr 75)- The J71 = 2+ state reported at 1 1 18 keV in a ( d .d 1)
experiment (Gu 71) is consistent with our observation of a state
at 1120 keV (#9) with an angular d is t r ib u t io n s im i la r to that for
the known J^K = 2+2 state in ' 8°W at 903 keV and is therefore
assigned J^K = 2+2. The state at 1321 keV (#12) is s im i la r ly
id e n t i f ie d with one member of a doublet seen in the ( d . d ') study
74
at 1319 keV and is assigned J^K = 4+2 on the basis of a comparison
with the 488W (p , t )^ 8\ / ( 4 2) t ra n s i t io n angular d is t r ib u t io n shape.
The 1083 keV state has an angular d is t r ib u t io n s im i lar to that of
the 3 state populated in the 4 80W(p , t ) 4 8\ / reaction and has been
previously id e n t i f ie d (Gu 7 0 -
Four non-L = 0 t rans i t ions corresponding to ex c i ta t io n energies
in l8 °W of 746 keV (#4) , 831 keV (US), 1012 keV (#6) and 1034
keV ("7) were observed at the l im i t of detect ion, about 2 pb/sr
peak cross section. Due to the d i f f i c u l t i e s of observation, however,
t h e i r ex c i ta t io n energies are assigned larger uncerta inties (±10 keV)1 80and in fact th e i r assignment as states in W must be considered
t e n ta t iv e . No evidence for the 0+0 state suggested in Reference
Gr 66 to l i e at 908 keV was found. In addi t ion , the (d ,d ' ) work
on 480W reported no excited 0*0 or 2*0 states , al though the present
work shows a very strongly populated 0 0 state at 1516 keV e x c i ta t io n ,
with an associated 2+ band member at 1589 keV. Also, three other
weakly excited 0+ states were located, at 1695, 1824, and 2356 keV.1 785 .3 .7 The Nuc 1 eus W
13 12The C(p,d) C(g.s) peak from the target backing
obscures thespectrum above 2 MeV ex c i ta t io n at forward angles,
as shown in the sample spectrum in Figure 4-5- The states below
2 MeV ex c i ta t io n energy populated in the reaction are enumerated
in Table 5-8 and th e i r angular d is t r ibu t ions displayed in Appendix
C-5 . Within th is small ex c i ta t io n region, three excited 0 states
were located. The lowest at 997 keV (#4) is the most strongly
75
populated with 3% of the ground state y ie ld . The others, at 1356
keV (#8 ) and I 6A3 keV (#11) exc i ta t io n energy, are populated with
3% and ]% of the ground state y ie ld , respective ly . The assignment of
0 for the 997 keV state is consistent with the suggestion in
References Ga 70 and Ca 76 of a 0+ state at 1001 keV; the 2+ and
A members of the ro ta t iona l band also agree with the previous
assignments. As discussed in the above references, the 3 member
of the octupole band l ies only 9 keV from the 2y, result ing in a
composite peak (ft6 ) which was indeed wider than the other peaks
in the spectrum.
76
TABLE S-k LEVELS POPULATED IN 186W(p, t ) 1 8\ /
State H 0
1
2
3
A5
j V a)0 + 0
2+ 0
A+06 + 0
l 82W(2+0)
2+ 2
foV;( 3v ( d )
2 0 ; 2 2
A+2 '
Exci ta t ion Energy( 6 )
(keV)
0
109
36A7AA
901
995
1122
Prev i ous Experiments
(keV)
0
1 1 1
36A 7*8
903
10021 0 06
1121 1 130 1 13*
(a) do/dn) MAX(yib/sr)
7*0
1 AA 25
c(c)
37
57
22
3 2 1 2 1 A 1221
(a) From Reference Ki 73 unless otherwise indicated
(b) Exci ta t ion energies ±10 keV
(c) Upper l im i t
(d) Strengths expected to be vanishingly small for unnaturalpar i ty states
77TABLE 5-5
LEVELS POPULATED IN 18 \ / ( p , t ) 182W
ita te # j V a)
Exci ta t ionEnergy ^
(keV)
Previous ^ Exper i ments
(keV)ds/dn)(p b/sr
0 0+ 0 0 0 7001 2+ 0 100 100 200
2 4+0 330 329 263 6+ 0 690 680 6
4 0+ 0 1135 11 38 100
5 2+ 2 1219 1221 60
6 2+ 0 1266 1257 12
7 3~2 1372 1374 78 4+2 1444 1443 12
9 (5)" 1626 1621 310 ( 5 ) ' 1663 D 1660 511 1767(D) 1512 5" 1814 I 8 l 0 d) 2
13 1824 8_ +14 2 2 1853 1851 8
15 (5 ,6 ,7" ) 1890 1960.4 20
16 (5,6) 1961 1960.9 22
17 2094 518 2117 319 2154 1420 2175 1721 (3") 2209 2209 1522 2251 / I \ 423 2278 2274 * 1424 2311 D 1325 2331 D 726 2363 3427 0+(c) . / \ 2520(D) 1428 0 2552 10
29 “ 2625 M 30
78
State §
TABLE 5-5 (cont'd)
LEVELS POPULATED IN 181*W(p, t ) 1 ®2W
Excita tion prpu:n,nEnergy Experiments^3 da/df^MAX
J^K (keV) (keV) (yb/sr)
3° , . 2689d (c)31 o 2725
32 277533 2815
1222105
(a) Assignments and ex c i ta t io n energies from Nucl. Data Sheets14 (1975) unless otherwise indicated.
(b) Exci ta t ion energies ±5 keV below 2 MeV, ±10 keV above 2 MeV
(c) Assignment th is work.
(d) From references Gu 71, Ki 73, Je 77
D Doublet
M Mult ip le t
79
TABLE 5-6
LEVELS POPULATED IN l 83W ( p , t ) , 8 l W
Excita tion Previous ,(b ) . ( a ) d o / d 9 W/ \ Energy Experiments
State H J*K [NN_A] K ’ (keV) (keV)________ (pb/sr)
0 9 / 2 + 9 / 2 [624] 0 01 1 1 / 2 + 9 / 2 [6 2 A] - 1132 1 3 / 2 + 9 / 2 [6 2 A] - 2 5 03 5 / 2 - 5/ 2 [ 5 1 2 ] 365 366 70
0 r t / 2 - 1/2(521]• ^ „ f385) ?) 7/2 - 7/2[5141, ko9
[3 / 2 - 1/21521]; ^ ;A50 | 28q1 / 2 - 1 / 2 [ 5 1 0 ]) (A5 8 !
•7/2 - 5 / 2 [ 5 1 2 ] j ^ TA76') ?2
'5/2 - 1/2 [521 IA8819/2 - 7/2[5lA]^d) 531 p ? ' - 3/43/2 - 1 / 2 [ 5 1 0 ] / (529,'
8 5 / 2 - 1 / 2 [5 1 0 ] 5 6 0 5 6 0 639 9/2 - 5/2[512] 610 '611 11
10 71A 715 1311 78A 812 5/2 - 3 /2[512] 807 807 1113 1093 71A 1193 815 1262 616 1377 717 1A37 618 1 5 18 819 1667 1*<20 / 1712 721 1. 2- [510] x 18°W(o| ) vc) ’ 86A AA22 1892 3523 19A5 5
TABLE 5-6 (cont'd)
LEVELS POPULATED IN 1®3W(p, t ) ’ 81W
State tf J^K [NN A] 'a^2k2526
Exc i ta t ion(b)Energy
(keV)
2015203/42067
PreviousExper i ments
(keV)____(a) d 0 / d t ! ) « A X
(pb/sr)
128
17
(a) From Reference Ca 12 unless otherwise indicated
(b) Calculated assuming 5/2 - [512] state at 365 keV e x c i ta t io n ,error ±5 keV
(c) Assignment th is work
(d) From Reference Ca 71
81
1 MDL L p-/
LEVELS POPULATED IN 18 2w(P, t ) l8ow
State " j V a)
Exci ta t ion (b)Energy
(keV)
Previous ^ Exper i ments
(keV)do/dfi) Mi(yb/sr)
0 o+o(d) 0 0 500
1 2+ 0 109 104 180
2 4+0 332 338 22
3 6+ 0 688 689 44 (7*6) (e)
( 8 3 1 ) (e)( 2 )
5 ( 2 )6 ( 1 01 2 )
( 1 0 3 * ) (e)( 2 )
7 ( 1 03 6 ) (3)8 3"2 1083 1083 139 2+ 2 1120 1118 21
10 1 190 311 1229 (1234) 2
12 * + , 5" 1321 1319 8
13 1359 414 1382 415
o+o(c)2+0 (c)
1470 516 1516 3617 1589 5118 1635 8
19 o+o(c) 1695 11
20 “ 1740 D 721 1824 6
22 1906 21
23 19** 1724 2057 10
25 2095 10
26 2164 2527 2203 8
28 2212 10
29 2265 10
82
TABLE 5-7 (cont'd)
LEVELS POPULATED IN l 82W ( p , t ) l8°W
Excita tion PreviousE n e r g y ^ E x p er im en ts^ da78” ^MAX
State # J K (keV) (keV) _____ (ub/sr)30 2293 5
31 0+0 (c) 2356 18
32 2400 7
(a) From reference Gr 75 unless indicated
(b) Error ±5 keV below 2 MeV, ±10 keV above 2 MeV
(c) Assignment this work1 fill 1 ft?(d) Doublet with W ( p , t ) W (1135 keV)
(e) States near lower l im i t of observation, er ror ±10 keV.
D Doublet
83
State0123456789
10 1 1 121314151617181920
(a)
(b)
(c)
(d)
D
M
T A B L E 5- 8
LEVELS POPULATED IN 18°W(p, t ) ' 78W
j V a) 0 + 0 2+ 0 4+0 6 + 0 o+o ( c 2 + 0
(2 + 2 + 3 _2 ) 4+0 o+o^c)
o+o^c)
Exc i tat ion
E n e r g y ^
(keV)______
0
105 343 699 997
1083 111/ D 1 278 1356 1435 1450 1643
-1720 D -1790 M -1825 - i 8 6 0 M - I 9 6 0 M
2030 2 0 6 0 2091 2116
P r e v i o us
E x p e r imen t s
( k e V )______
0106343694
( 1 0 0 1 ) (d)1083
( 1111 + 1 1 2 0 )1276
(a)d°/d % A X
(yib/sr)741294
214
63 69 16
6 22
2 6 9
10 3 1 2 6
21 11 7
From reference Ca 76 unless otherwise indicated
Exci ta tion energies ±5 keV
Assignment this work
Suggested in Reference Ca 76
Doublet
Mu 1 1 i p 1et
C h a p t e r 6
S V S T E M A T I C S OF C O L L E C T I V E S T A T E S :
G R O U N D - S T A T E B A N D S
8 k
The next three chapters are devoted to the systematic
properties of the major bands, with p a r t ic u la r a t ten t ion
being paid to the extract ion of microscopic s t ructura l informa
tion through the study of such systematics. Chapter 6 deals with
t rans i t ions between members of the same superfluid band; the grou
state bands in the even-even nuclei and the "favored" band in the
odd-mass nucleus case where the one quas ipar t ic le is l e f t undis
turbed. Chapter 7 then explores the systematics of the excited
K11 = 0+ bands while Chapter 8 contains a discussion of the gamma
v ibrat iona l band.
86
6.1 ANGULAR DISTRIBUTION SHAPES FOR MEMBERS OF THE GROUND-STATE BAND
Figure 6-1 displays the angular d is t r ib u t io n s for t ransi t ions
to the ground-state rota tiona l bands (GSRB). The shapes of the angu
lar d is t r ib u t io n s for gound-state t ransi t ions are in each case very
s im i la r while the shapes of the L = 2 t rans i t ions vary considerably,
although smoothly, with mass. In contrast, as discussed in the
previous chapter, one-step DWBA calculations predict s im i lar shapes
for a l l masses. I t has been shown, however, (As 72b, Ki 72, Ki 73)
that the inclusion of in e la s t ic ex c i ta t io n e f fec ts in the transfer
process is necessary to describe the r e la t iv e magnitudes and shapes
of the angular d is t r ib u t io n s for t rans i t ions to the members of the
GSRB and K77 = 0* band in the 8^W (p , t ) 8 W reaction at 18 MeV
bombarding energy. During this study, a s im i lar ca lcu la t ion was182 18 0performed for the W(p,t) W reaction at 21 MeV according to the
procedure out l ined in section 3 -1 .3 . The results are shown in
Figure 6-2 as so lid l ines and the DWBA predictions as dotted l ines .
Only one normalization constant was used for a l l of the curves.
The shape and magnitude of the ground-state angular d is t r ib u t io n
is described equally well by both the DWBA and CCBA ca lcu la t ions ,
indicating that mult is tep in e la s t ic e f fects for strong L = 0 t ran
si t ions are minimal. The slow drop of the 2* angular d is t r ib u t io n
from 0° to 50° is not reproduced by the DWBA, but is well described
by the CCBA ca lcu la t ion . This lack of o s c i l l a to r y structure in
( p , t ) t rans i t ions to 2+ 0 states seems to be a general feature in
F i g u r e 6-1
Figure 6-2.
and k+ members of the ground-state rota tional, , , 1 8 6 , \ 8 k , 1 8 2 , 1 8 0 . . , .band for W(p,t) reactions at
E = 2 1 MeV. The error bars contain Ps t a t i s t i c a l errors only and are smaller
than the data points i f not shown.
A n g u l a r d i s t r i b u t i o n s for the 0+ , 2 ,
Angular d is t r ib u t io n s for the 0+ , 2+ ,
and k+ members of the GSRB for the1 Q O 1 Qri
W(p,t ) W reaction at E = 2 1 MeV.PThe CCBA predictions are shown by the
solid l ines where only one normalization
constant was used. The DWBA predictions,
s im i la r ly normalized, are shown as dotted
l ines. Both ca lculations employed the
opt ica l parameters of Table 3“ 1 and form
factors as described in the tex t .
( f c s p r 3©
da/dD,
(^b/sr)
dcr/
dfl
(/xb
/sr)
© cm (degrees)
90
the rare earth region (e.g. As 72b, Ki 73).
One obvious defic iency of the CCBA ca lcu la t ion is the over
prediction of the magnitude of the 2+ angular d is t r ib u t io n . We9f i n d , however, that the magnitude depends se ns i t ive ly upon the value
of the pair ing matrix element, which was chosen so that A = P.,.N
Thus, in the presence of a large energy gap in the s in g le -p a r t ic le
spectrum, there can be a rather sensit ive dependence of this mag
nitude on the energy of only a few o r b i ta ls near the middle of the
Fermi surface. Further, in the nuclear structure ca lcu la t ion ,* l8°wsince the deformation parameters or w are not known, the same
l 82values as for W were used. In l ig h t of these fac ts , the results
presented here are considered sa t is fa c to ry .
The a b i l i t y of the ca lcu la t ion to reproduce the shapes of
the 2* angular d is t r ib u t io n s in both the ^8BW ( p , t ) ' B\ / and the l 8 2 180W(p,t ) W reactions is very g r a t i f y in g , yet the reproduction
of the data without physical insight into the cause of the changing
shapes is not enough. This insight was gained through studies of10 1 I
( C, C) reactions on the A = 186, 184, and 182 isotopes of
tungsten by Hanson et a_k (Er 74, As 75, Ha 75, Ha 76) where the
shapes of the 2* angular d is t r ib u t io n s were also observed to change
with target mass. These data, in conjunction with deta i led CCBA
analyses, showed that although the strengths of the m u l t ip le -
step t rans i t ions remained rather constant through the tungsten iso
topes, the p ro b a b i l i ty of the d i re c t L = 2 t ransfer changed
considerably. The d i re c t route was found to be weak for the
91
^ ( p , t ) ^8^W(2* ) t ra n s i t io n due to an e f f i c i e n t cancel la t ion between
the contr ibut ions from the various o r b i ta ls near the Fermi surface
(the contr ibut ions of each o r b i ta l being weighted by a UV factor
as described in Section 3 . 2 .2 ) . As the target mass decreases,
the d i rec t quadrupole t ransfer strength increases, although staying
small, as d i f f e r e n t o r b i ta ls become important in the t ransfer
process due to the changing UV factors. The net e f fe c t thus, is
a changing r a t i o of ( two-step route to 2* ) / ( d i r e c t route to 2^) .
Since these two amplitudes add coherently and have d i f f e r e n t angular
dependences, the result is changing angular d is t r ib u t io n shapes
as a function of mass.
92
6.2 ANALOG TO GROUND-STATE TRANSITION: ODD MASS TARGET
6.2 .1 AnguIar D i s t r i but i on Shapes
In the ad iabatic approximation, the wavefunction of
an odd-mass nucleus is |jMK> ~ D3 ^|k> where the in t r in s ic wave
function in the BCS theory is given by | K> = a* Such states
are known as "one qu a s ip a r t ic 1e states" since th e i r structure is
described as one quas ipart ic le in o r b i t K, coupled to the BCS
vacuum (which is also the qu a s ip a r t ic1e vacuum). The spectro
scopic amplitudes B as given in Equation 3-8 then depend upon
the " i n t r i n s i c form factor" <K|A+ |K‘ > (where A is a two-quasi
p a r t ic l e creation operator) which now becomes
< + > ■ > . < .-> + 2 ) i»K ; ; » ; , i t ^ > « . uUnder the assumption that A* operates only on the core an8
not on the q u a s ip a r t ic 1e in o rb i ts K and K' , then
< + > ■ > . <*'AS+ 2 ) | A > < A>> ^ K, ( 6 . 2 )
where the f i r s t factor is recognized as the in t r in s ic form factor
for the neighboring even-mass targ \ nucleus t ra n s i t io n . Thus,
w ith in the model, there should ex is t "favored" ( p , t ) t rans i t ions
in odd-mass nuclei analagous to t rans i t ions in the neighboring
even-mass nuc le i , but with the s l ig h t added complication of the odd
q u a s ip a r t ic 1e . Angular d is t r ibu t ions for such t rans i t ions from
the W ground s ta te , the one q u a s ip a r t ic 1e (q .p . ) i - [510]\ 8 \sta te , to the one q.p. j - [510] state in W via L =■= 0, and
via L = 2 to the 3/2 and 5/2 members of the rota t iona l band are
93
182 180shown in Figure 6-3- The angular d is t r ibu t ions for W-* W
trans i t ions to the 0+ and 2+ members of the GSRB are also shown for
compa r i son.
Both the 3/2 and 5/2 members of the i - [510] band can be
reached via d i re c t L = 2 t rans i t ions since the angular momentum
of the odd quas ipar t ic le can couple with L = 2 to give 3/2 or
5/2. Thus, the quadrupole t ransfer strength should s p l i t between
these two states. In the ad iabatic l i m i t , the s p l i t t i n g follows
the rule
p(e) (3 /2") 2 (6 3)o ( 8 ) (5 /2" ) 3
at a l l angles (see Appendix B ) . Experimental ly, although the
summed cross sections for the f i r s t six angles give the correct
r a t io ( 3 / 2") to ( 5 / 2_) of (132 ± 6 ) : (184 ± 8 ) = (2.15 ± -14) : 3,
the shapes of the two angular d is t r ibu t ions are not in fact the
same, having very d i f f e r e n t behavior at forwar, angles (Figure 6 -3 ) .
Such a d i f ference can be a t t r ib u te d to d i f f e r e n t mu 11 ip ie-step
routes populating the two states . This extremely in te re s t ing , but
complicated, s i tua t ion w i l l be more f u l l y explored in Appendix F.
6 . 2 . 2 B1ocki ng
One test of the assumption behind Equation 6 .2 , that
the presence of the odd p a r t ic l e does not modify the in t r in s ic form
fac to r , is to compare the d i f f e r e n t i a l cross sections for the
L = 0 t rans i t ions ' 83W( p, t ) ' WQ - [510]) and ^82W(p , t ) ^ 89W(0^)
which, according to Equation 6 .2 , should be the same. In f a c t ,
although th e i r shapes are very s im i la r , th e i r absolute magnitudes
182 180Figure 6-3. Angular d is t r ibu t ions for the W(p,t ) W
reaction at E = 2 1 MeV to the 0+ and 2+Pmembers of the GSRB and for the 83W(p, t ) 8 ' W
reaction at Ep = 21 MeV to the 3 /2 - .
and 5/ 2 - members of the i - [ 5 1 0 ] band.
The solid l ines are drawn to connect the data. 182,, 180,, points for the W -*■ W transi t ions
while the same l ines, but renormalized
by a factor of 0 . 6 , are shown on the 183,, 1 8 1 .. , . . . . . . . rW -v W angular d is t r ib u t io n s for
compa r i son.
96
differ by about 40%, the odd-mass target transition being weaker.
Such as effect is well documented in superfluid nuclei (e.g. Hi 72,
Hi 75, Sh 76) and has been given the name "blocking." As discussed
previously, the (p,t) transition to the ground state (or ground
state analog) of a superconducting nucleus involves the coherent
removal of a neutron pair from many levels. When a single particle
occupies a level, there can be no pair pickup from that level,
thus reducing the total pickup strength. Although blocking effects
of factors of two in (p,t) reactions are not unknown (see, for
example Sh 76), the large value seen here merits explanation. In
an attempt to provide such an explanation, the standard set of
calculated single-particle levels was used with the code DEF2NT,
in conjunction with the DWBA code DWUCK, toca lc ul at e the strength
182 180of the W-> W ground-state transition. Precisely the same
calculation was then performed, except that the 7 - [5 1 0 ] level
was effectively blocked by artificially respecifying its binding
energy to place the level well above the Fermi surface. Its
UV factor was then such that it contributed negligibly to the result.
The calculation predicts a (p,t) blocking effect of 45%, in reason
able agreement with experiment. The calculated values of A did
not, however, agree well with the experimental values of P^ for
either ^8 ^W nor ^83W, reflecting not unexpected deficiencies in
the single-particle spectrum used. When the calculation was re
peated, with the value of the pairing matrix chosen so that A = PN >
a blocking effect of 36% was predicted, again in good agreement with
the experimental value of 40%.
The size of the blocking effect may seem surprisingly large
since the ground states are populated by coherent pair pickup from
about 30 orbitals. In fact, two effects conspire to give this large
blocking effect. First, in all cases the orbital blocked is the
one which in the neighboring even nuclei has the largest UV factor
and so would normally contribute a reasonable fraction of the total
strength. Secondly, in the present case the blocked orbital is
the 2 - [510] which has a large intrinsic L = 0 pair pickup strength
Figure 6-A shows the relative probability of L = 0 (and L = 2)
pair pickup from each of the orbitals included in the full c a l
culation. The values were determined by calculating transfer form
factors for each of them individually with DEF2NT and matching on
a Hankel function tail. The resulting form factors were then used
as input to the code DWUCK for the ^82W ( p ,t) ^8°W (Q = -7-8 MeV)
reaction. The strength was taken to be the value of the differentia
cross section at the second maximum. Also shown on the figure are
182 180the values of the UV factors appropriate for the W+ W transi
tion. As discussed, the 5 - [510] orbital, located at a binding
energy of 5-9 MeV, is seen to have a large UV factor and carries
a large intrinsic pickup strength besides, so that its blocking
leads to a considerable reduction in the available L = 0 strength.
Figure 6 - k . L = 0 and L = 2 transfer strengths for the
orbitals shown in Figure 2-1, calculated as
described in the text. The solid curves
give the UV weighting factors appropriate
1 ft? l ftnto the W(p,t) W reaction. Also
indicated for orientation purposes is
the level occupied by the unpaired
1 83neutron in the W ground state, the
* - [510] orbital. The signs of the
L = 2 transfer strengths indicate the
signs of their form factors at the
nuclear surface.
L = 0 TRANSFER STRENGTHS FOR VARIOUS N ILSSO N ORBITALS
t------------ ,------------ ,------------ ,------------ ,------------ j------------ 1------------ r
Ul£in
SINGLE-PARTICLE BINDING ENERGY (MeV)
L~2 TRANSFER STRENGTHS
100
6.3 CROSS SECTION SYSTEMATICS OF GROUND STATES
6.3-1 Pairing Rotation Limit
In the pairing-rotation model (Br 73) the ground
states of neighboring even-even superfluid nuclei are described
by a single intrinsic wavefunction, <*>Drc., but with various orien-BC 5
tations in par t i c 1 e-number space; 4* = ^ ^ ^ B C S w ^ere ^ 's t ie
number of particles and <j> is the gauge angle in this space.
Within this model, using the Os approximation, the (p,t) cross
sections between ground states of even isotopes are given by
o(0)°S a (| -)2 o(e) (0) (6. A)
where the last term contains only the kinematics of the reaction
under consideration. To compare with experiment, the last factor
can be taken from the DWBA calculations of Section 5-3 while
A = P^. The value of G is expected to change only slowly with mass
so that a constant value is used. Table 6-1 gives the results
obtained; the predictions are clearly not fulfilled. Not only do
the pairing-rotation values change by only 10% rather than by 50%
182 180as found experimentally, but the prediction is that the W-* W
transition should be the strongest while it is actually the weakest.
Thus, the pa ir in g- ro ta ti ona1 limit does not reproduce the data.
6.3-2 Full Form Fac tor DWBA C a 1c u 1 a t i ons
In a further attempt to reproduce the ground-state
cross sections, the BCS form factor for each ground-state transi
tion was calculated with the code DEF2NT, using the deformation
parameters and pairing matrix elements shown in Table 6-2. These
TABLE 6-1GROUND-STATE (p,t) STRENGTHS IN THE
PAIRING-ROTATIONAL LIMIT
Mass PN (a) PH (AVERAGE) (b) f (Q) (P ’t)(C) ( pr)2 f (Q)) R E L . (d) EXPT. (e)
186 .797
18* .720
182 .809
180 .805
178 .832
758 1 .05 713 7*0
765 1.05 720 720
807 1.00 750 500
818 0.92 698 7**
(a) Calculated using the Nilsson-Prior Prescription (Ni 63), masses from Reference Ma 65.
(b) Average of two neighboring values
(c) Kinematic factor from figure 5_2
1 8 4 182(d) Pairing-rotational prediction, normalized to the W (p,t) W (g.s.) transition
, o(e) Taken to be the differential cross section at the second maximum ~27±
o
102
were then used as input to the code DWUCK along with the optical model
parameters of Table 3-1. The resulting values of the differential
cross section at the second maximum are listed in Table 6-2 and
shown in Figure 6-5. The predicted curve is arbitrarily normalized
1 8 4 182to the result of the w(p,t) W transition. The results of the
calculations show a slowly increasing strength with decreasing mass,
in reasonable agreement with the (p,t) results.
182 180The decreased strength of the W(p,t) W transition, however,
is not reproduced. To investigate whether this problem can be
traced to deficiencies in the underlying single-particle spectrum,
the calculations were repeated including only the 13 orbitals shown
in Figure 2-2 as valence orbitals. The energies of these were
adjusted to agree with the "experimental" scheme shown on the right
of Figure 2-2. The relative strengths of the ground-state transi
tions so calculated were identical to those shown in Figure 6-5.
it is thus concluded that the explanation for the failure of the
calculation to correctly reproduce the trends can be traced to the
assumptions inherent in the BCS theory itself. The BCS wavefunction
does not correspond to a state with a definite number of particles
and is, in fact, a wave-packet in particle-number space. Therefore,
the solution for a given nucleus has some of the properties of
neighboring even nuclei mixed in. Thus, the BCS technique can be
expected to describe only the average properties of a series of
isotopes. Moreover, the applicability of the BCS theory to a
given situation is directly related to the degeneracy of the orbits;
TABLE 6-2
W(p,t)W(g.s.) CROSS SECTION CALCULATIONS
St ructure 1 nput
Mass B (a) -2 — B (a)£4---- G (b)£ n
186 . 209 - . 0 8 8 .1*3 .797
184 .221 -.087 .1*0 .720
183 .219 1 O KD . 1 5 7 .772
182 .219 - . 0 6 9 .1*6 OO
O
181 . 2 1 9 - . 0 6 9 .150 • 772
180 .219 -.069 .1*3 OO 0
178 .219 1 O CT' UD .138 .832
Kinematic Input and Results(d)
Residual Q - V a l u e ^ d° /d^ S E C O N D MAX EXPT.Mass (MeV) (pb/sr)___________ (ub/sr)
184 -4.470 681 740 ± 75
182 -5 . 120 720 720 ± 70
181 -6 . 2 2 8 646 320 ± 42
180 -6.271 836 500 ± 75
178 -6.843 855 744 ± 75
(a) deformations from Reference Le 75, scaled to rn 1 82
1.25F, W deformation from Reference Ap 70, other1 82deformations taken to be the same as for W
(b) Value chosen so that A = PN
(c) From Reference Ma 651 8*4 183
(d) Normalized to W (p,t) W experimental result
gure 6-5- Cross sections for the ground-state transi
tions for the tungsten isotopes in (p,t)
reactions at E = 2 1 MeV. Experimental P
values are shown with error bars reflecting
both statistical and relative cross section
uncertainties. The curve gives the results
of calculations as described in the
t e x t .
(doV
dfl)s
ecQnd
m
ax(^
b/sr
)
RESIDUAL MASS
106
only in the limit of very large degeneracies is the BCS technique
expected to properly describe the system. In the case of deformed
nuclei, the s i n g 1 e - p a r t i c 1e levels are only doubly degenerate, and
even worse, the gap in the single-particle spectrum at N = 108
further reduces the effective degeneracy in the vicinity of the
Fermi surface. Thus, in such cases, the BCS theory provides an
inadequate nuclear structure model, requiring that more sophisti
cated methods be used, such as an exact diagonalization of the
pa i r i ng f o r c e .
Chapter 7
SYSTEMATICS OF COLLECTIVE STATES:
EXCITED K71 = 0+ BANDS
107
108
7.1 GENERAL CONSIDERATIONS
Of all of the many J71 = 0 states in deformed even-even
nuclei, the ground state exhibits the greatest pairing collectivity.
(It is this fact that makes that particular state the ground state.)
This same collectivity is also responsible for the strong two-nu
cleon transfer transitions which connect the ground states. In the
BCS theory, the ground state contains essentially all of the pairing
strength so that excited 0 (0*) states should be populated with
only a few percent of the ground-state strength in two-nucleon
transfer reactions (Be 66, Br 73). Thus, the observation of sub
stantial excited L = 0 strength is worth of note. There are,
however, known circumstances in which 0+ states can demonstraten
large (p,t) or (t,p) strength;
(1) Near closed shells where the inter-shell spacing is
larger than the pairing interaction. In such a case
the single-particle levels below and above the gap are
nearly non-interacting so that the pairing correlation
can produce strong two-neutron transfer transitions
by the addition or removal of nucleon pairs from levels
above or below the gap, thereby populating different
states. The concept of pairing vibrations (Be 66) is
useful in such cases and has been used with success
on the Zr, Ni, Ca, and Pb nuclei.
(2) For deformed nuclei where a gap in the energy spacing
of the Nilsson orbits occurs, the levels above and
109
below the gap can to a certain extent decouple, thereby
splitting the L = 0 transfer strength. Such an occurence
has been seen experimentally in the Yb (Oo 70) and perhaps
in the Cm (FI 77) isotopes.
(3) When a rapid change in ground-state shape occurs, an
excited state of the residual nucleus may be more closely
related to the target nucleus ground state than is
the residual nucleus ground state leading to strong
L = 0 transitions to 0* states. The Sm isotopes are known
to demonstrate such an effect (Bj 66, Br 73).
(4) In the Actinide nuclei, there occurs a region of non-
uniform distribution of oblate and prolate single-particle
levels around the Fermi surface. Excited L = 0 transitions
of 15% to 20% of the ground state strength have been
seen experimentally (Ma 70, Ma 72a, Ca 72b, Fr 74), and
are interpreted as being due to the partial decoupling
of the prolate orbitals which compose the bulk of the
Fermi surface from the oblate orbitals, which lie
below the surface (Gr 71, Be 72, vR 72).
With the exception of (3), the above splitting of the L = 0
strength arises from the decoupling of one group of single-particle
orbits from another. Without this splitting, the ground state
would collect almost all of the available pairing collectivity
and hence almost all of the L = 0 two-neutron transfer strength
discussed previously. Inherent in the discussion so far, however,
is the assumption that the L = 0 transitions for (p,t) reactions
occur through the pickup of zero-coupled neutron pairs, both
neutrons being in the same Nilsson orbit. Certain excited 0+
states, especially beta vibrations, however, should include broken
pair configurations (P»v )j7Tk _ q + q where p and v are different
Nilsson levels. Within the space of the 13 orbitals shown in
Figure 2-2, only two broken-pair configurations can be involved
in J^K = 0+ 0 states. Moreover, it is calculated that the p r o
bability of pickup for these configurations is smaller than the
probability of pair pickup by factors of 5 to 20 on the average
(see Appendix E). In light of these considerations, the effect
of broken-pair configurations on the population strength of 0
levels in this mass region will be ignored. Therefore, in the
discussions that follow, substantial excited L = 0 strength is
interpreted in terms of the splitting of the pairing strength
due to the decoupling of groups of levels from each other.
Figure 7~1 shows the Q-value corrected L = 0 (p,t) strengths
plotted versus excitation energy in the residual nucleus. Table 7
tabulates the relevant information and Figure 1-2 displays the
angular distributions for the first excited 0+ states (0j) their
+ + + associated 2 states, and the 2 states. In each case, the 0.y 1state is populated with 10% to 1 5% of the ground-state strength.
A further notable feature of the data is the discontinuity for
+the even nuclei in the excitation energy of the 0j state in
i ftnW where it has a value of 1.5 MeV rather than the 1.0 to 1.1
I l l
TABLE 7-1
Res i d u a 1 Mass
L
Exc i tat i on Energy (keV)
= 0 STRENGTHS
do/d n)s e c o n d MAX.
(pb/sr)
V a)(p,t>
(pb/sr)Percent o
g.s. Stren<
184 0 740 705 100
1002 57 54 8
182 0 720 686 100
1 140 110 110 16
2512,,(b) ,5 <bi 2 (b)
2571 10 14 2
2706 23 33 5
181 454 320 320 100
1864 48 65 15
180 0 500 500 100
1516 44 64 13
1695 10 15 3
1824 6 10 1
178 0 744 809 100
997 67 97 12
1356 22 37 5
1643 9 17 2
(a) do/dfi)s e q o n D MAX after Q _Va1ue correction
(b) Doublet, upper limit given
Figure 7*1
Figure 1-2
Figure 7*3
Relative strengths of L = 0 transitions,
after Q-value correction, vs. excitation
energy in the residual nucleus. The favored
L = 0 transitions have been reduced in
size by a factor of 10 for graphical purposes.
Angular distributions for the first two
TT ^members of the first excited K = 0
band, and for the lowest 2+ 2 states. The
error bars reflect statistical errors only.
+ 178The values for the 2 in the W residual
Y
nucleus also contain contributions from the
3 state which was unresolved from it.
Difference in energy between various J^K = 0 0
states and their associated 2+ 0 states for
the ground-state bands and 0* bands in the
even tungsten isotopes. Note the suppressed
zero.
^(p,
t)
(arbit
rary
units
)
, 86W(pft ) ,84W
, 8 4 W ( p , t ) 182 w
I n1x —10
l 8 3 W(p,t) 181 w
h i f e
l 8 2 W(p,t) l 8 0 w
. . 1 I - . . A
, 8 0 W(p,t) l 7 8 w
, 1 I_____x i10
3 2 1 0EXCITATION E N E R G Y ( M e V )
0 25 50 75 0 25 50 75 0 25 50 75 0 25 50 75© c m (degrees)
E(2+)-
E(0+)
(keV)
116
MeV as in the other isotopes. Moreover, the energy spacing between
the 0j states in and ^78W and their associated 2+ states
are substantially different from those of the ground states or
o| states in the higher mass isotopes (Figure 7“3 and Table 7_ 1),
reflecting different deformations and therefore substantially
different structures for these states. The first excited 0+
states therefore can be divided into two distinct groups correspon-
a - + - * 178, 180.. . . 182, 184, 186..ding to states in W and in W.
7.2 0j STATES \K_ THE HIGH-MASS ISOTOPES+ 1 ft? 1 gi*
The first excited 0 states in W and W are strongly
populated in (p,t) with 16% and 8% of their respective ground state
strengths and, as previously discussed, must therefore possess
special properties. The calculations of Mikoshiba et^ a_l_. (Mi 67)
IS?predicted that the Oj state in W is primarily a neutron pair
vibrational state and not a beta vibration. Further, (d ,d 1)
experiments on showed a much weaker coupling between
the ground and 0* bands than would be expected for beta vibrations
(Gu 71). However, RPA calculations of Ascuitto and Sorenson
(As 72b) with conventional pairing and surface-peaked quadrupole-
quadrupole forces fit both the B (E 2; g.s.-*2+ 0j) values and the
ratio of the (p,t) strength of the ^88W ( p , t)^BAW(o|) transition
at Ep = 18 MeV to the ground state transition strength. No d e
tailed comments were m a d e , however, as to the main configurations
involved in the excited 0+ state.
As has been noted before (Ba 7 3 ) , there exists an energy
gap in the single-particle level scheme of tungsten above
108 neutrons. The existence of such a gap is demonstrated in
Figure 7-4 where the two-neutron separation energy ($2 N ) ^or t ie
even-mass tungsten isotopes is plotted against the mass numbers.
The general trend of the plot is to lower values of with
increasing mass. This trend is explained by the fact that a l
though the potential well deepens somewhat, as more neutrons are
added it requires less energy to remove a neutron pair since the
Figure l ~ k . The two-neutron separation energies
(S ^ f ° r various tungsten isotopes.
The two parallel lines are to emphasize
the smooth variation of S„., vs. mass2N
above and below N = 108 (A = 182) and
the striking discontinuity at N = 108.
S2
n (
MeV
)
17T W O - N E U T R O N S E P A R A T I O N
E N E R G I E S F O R T H E\ XI IM^ %N \ \\ j\
N.
1 U N o O 1 L. IM 1C i u r c c >
%s, s 1\ \ \s \ \
V VV V \k \ \ \L \V v %
176 178 180 182 184 186 188M A S S N U M B E R
120
Fermi surface is moving towards zero single-particle energy. The
sudden change in S2N between A = 182 and A = 18A bespeaks a
larger than average energy spacing between the N = 108 and 110
neutron single-particle levels. The "experimental" single-particle
level scheme of Figure 2-2 also exhibits an energy gap of about
1 MeV above the 9/2 - [62A] orbital (N = 108).
The existence of a gap in the single-particle energy spectrum
should lead to a partial decoupling of the levels above and the le
vels below the gap, splitting the L = 0 strength. The extent of the
decoupling depends upon the ratio of the size of the gap to the
energy spread of the Fermi surface, about 2 A . Since A = P , we
find R = (GAP)/(2A) = 0.67 for the present case, whereas for the
N = 10A gap in Yb, R = 0.60 and in Sn, R = 0.A5. Thus the
decoupling in the case of tungsten should be even greater than
in Yb or Sn where the effect has been previously observed (Br 73).
This decoupling is graphically demonstrated in Figure 7 -5 where the
Fermi surfaces for various even-mass tungsten isotopes are shown.
The energies of the single-particle levels are taken from the
"experimental" values of Figure 2-2 and the pairing force constant
chosen so that A = Pjg. As seen from the figure, the A = 182
nucleus has the levels below the gap almost completely full
2(£ U = 0.6) and the levels above the gap almost completely
2empty (£ V - 0.6), the pairing force just barely being able to
overcome the gap. Then, to a reasonably good approximation, the
ground states of the A > 18A tungsten nuclei as formed by adding
for each of the even-mass tungsten isotopes.
The calculations were performed within the
framework of the BCS theory where the
(constant) pairing matrix element was
chosen so that A = P^. The energies of
the single-particle orbits are those of the
"experimental" values of Figure 2-2.
1 82The fact that in W the orbitals above
the gap are almost completely empty and the
orbits below the gap almost completely
full suggests the applicability of the
pairing vibrational model.
gure 7*5- The shapes o f the Fermi surfaces are given
123
neutron pairs to orbitals above the gap, while those of the A < 182
nuclei result from the removal of pairs from orbitals below the
gap. A model developed to describe such a scheme is the "pairing
vibration model" of Bes and Broglia (Be 66) based on a suggestion
by Bohr and Mottelson (Bo 6 A ) . This model is only schematic,
however, so that although it may contain much of the essential
physics, it may well be inadequate to explain the experimental
details. In the next section the major results of the pairing
vibration model will be discussed (much of the discussion taken
from References Br 73 and Bo 75), while Section 7.2.3 will consider
the application of the model to tungsten nuclei.
7.2.1 The Pairing Vibration M o d e l : General Descr i pt ion
The schematic pairing vibration model is most useful
in situations where there exists a gap in the single-particle
energy level scheme at N = N q large enough to overcome the pairing
force. The vacuum of the model is taken to be the ground state of
the N = N q nucleus, including ground-state correlations if necessary.
Two operators are then defined; A r which (to first approximation)
removes a neutron pair from the levels below the gap, and A whicha
adds a neutron pair to the levels below the gap. It should be n o
ted that these creation and destruction operators are but a linear
combination of those previously defined, A* and A ^ , except that
now the space of single-particle levels those operators act on has
been divided into two subspaces, each with their own destruction and
creation operators. The 0 states within this model are labelled by
124
(M , N ) where N is the number of removal quanta and N are the r a r ^ a
number of addition quanta needed to form the state from the vacuum.
In the approximation in which these quanta are non-interacting
(harmonic approximation), simple energy relationships describe the
relative binding energies of the nuclei in the various states. Figure
7-6 shows these relationships, where the energy scale is the total
energy relative to that of the ground-state energy of the
nucleus, after subtraction of a term linear in the neutron number.
The coefficient of the linear term is chosen as per the convention
of Reference Bo 75 so that the (1,0) and (0,1) states both c o rr e
spond to the same phonon energy.
As shown on Figure 7“6 also, the pairing vibration model in
the harmonic approximation can describe the relative intensities
for two-nucieon transfer reactions. The relevant matrix elements
are (Bo 75) :
<N ,N +1|A+ |N , N > = /n T Tr a ' a r ’ a a ( 7 1 )
<N ' '- 1 , n.Ja In , n,> = /n~r ’ a 1 r 1 r ’ a r
so that the relative intensities are
(N , N ) -*• (N , N + 1 ) o(e) = (N + l)xr a r a a
(7.2)
(N , N ) -*■ (N + 1 , N ) o ( 6 ) = (N + l)yr a r a r
where x and y can be different, reflecting the differences in
structure, degeneracy, level density, etc. between orbitals above
and orbitals below the gap.
Selection rules for two-particle transfer also come from
Figure 7_6.vibration theory for the relative energies
of 0+ states and for the (t,p) and (p,t)
cross sections. The vacuum is taken to be
the N q nucleus while all of the states
are labelled by (N , N ) as described 7 r a
in the text. The figure is taken from
Reference Br 73-
The pred ic t i ons of the harmonic pair ing
ENER
GY(P
HONO
N UN
ITS)
127
the model. For instance, a (t,p) transition to the first excited
0 state in an N > N q nucleus is not allowed since it would require
changing both quantum numbers. Similarly, (p,t) transitions to e x
cited 0 states in the N > Ng nuclei are not allowed. It is
extremely interesting to note that within the model, one, and
only one, excited 0+ state can be excited in both (p,t) and (t,p)
reactions, the (1,1) state in the N = Ng nucleus.
Anharmonic effects to a certain extent can be included within
the framework of the model, but with some difficulties. The
expression for the energies of the states can be generalized to
include a first-order interaction between phonons by adding the
term
(7.3)
<5E (N ,N ) = j V N (N - 1) + V N N +r a rr r r ra r a
+ *V N (N - 1) aa a a
where V , V , and V are phenomenological parameters. The r r r 3 33
modifications necessary to include such effects in the expression
for the transition strengths are not known, however. Thus, in the
presence of large a n h a r m o n i c i t i e s , the pairing vibration model
loses its appeal as a detailed nuclear structure model, but,
hopefully, still retains its ability to organize the data into a
conceptually simple pattern.
7.2.2 The Pa i r i ng Vi brat ion Model : A p p 1 i cat ion to theTungsten Isotopes
The experimental results for the tungsten isotopes
within the framework of the pairing vibration model are shown in
128
Figure 7“7- In the figure, the zero of the energy scale has been
set by assuming equal energies for addition and removal p h o n o n s ,
giving a phonon energy of 575 k e V . The predicted locations of the
0+ states in the harmonic approximation are shown as dotted lines
in the figure while the numbers give the values of the (p,t) or
(t,p) differential cross sections at the second maximum, the (p,t)
values for Ey = 21 MeV and the (t,p) values E^ = 15 M e V . In
parentheses are also shown the values for the inverse reaction.
1 Oa | l87The W(t,p) data were from Reference Ga 77, the W(t,p)
1 Qj. 1 0 /from Ca 73, and the ’ W(t,p) results from Ga 11. The
180 182 +W(t,p) W(Oj) value is a preliminary value from work done by
R. R. Betts and the author (Be 78), a sample spectrum for which is
shown in Figure 7-8. Not shown in Figure 7_7 are the results of
the (t,p) transitions to the 0* state in the A = 18A, 186, and
188 isotopes, which are extremely small (given as 2.8 pb/sr and
less than 1.5 and 6 pb/sr, respectively) as is predicted by the
+ 180 178model. The 0^ states in W and W are also omitted because,
as discussed before, they fall into a different structural c a t e
gory. These states will be discussed in Section 7-3-
The energy spectrum of the tungsten isotopes shown in Figure
7-7 shows large a n h a r m o n i c i t i e s , although the excitation energy
+ 182of the 0^ state in W is well described. If a first order
interaction is allowed between the phonons where V = V andr I a aV = 0 (see Equation 1 . 3 ) , as shown by dots, the energies of the r d
ground states are well described, but the excitation energies of the
Figure 7*7
Figure 7-8.
The experimental tungsten isotope spec
trum within the framework of the pairing
vibration model. The solid lines give the
experimental energies of the various states
while the dotted lines are the predictions
of the model in the harmonic approximation
and the series of dots the prediction
when a first order phonon-phonon interaction
is allowed (V ' = V = 610 U e V ) . Ther i 39
numbers give the experimental strengths
of the transitions while the values for
the inverse reactions are shown in
p a r e n t h e s e s .
180 182Sample spectrum from the W(t,p) W
reaction at E = 15 MeV and a laboratory
angle of 26.25° from Be 78. The figure
shows the very strong population of the
first excited 0+ state (labelled in the
figure as 0*) relative to that of the
ground state.
ENER
GY
(I PH
ONON
UN
IT = 5
75
keV)
i 1------------1----------- 1----------- r
EX P ER IM E NT AL W - I S O T O P E S P E C T R U M
— Experiment .— Harmonic Approximation ) First Order Interaction '
5 7
7 4 4 ~ I O O ^ * 1 1 0 3147 0 0 )
5 0 0 A / 2 7 3 ( 2 8 3 ) \ ( 7 2 0 )
0±
1 7 6 17 8 1 8 0 18 2 1 8 4 1 8 6
I
4 0 0
1 8 8
COUN
TS
PER
'/2 mm
ST
RIP
132
Oj states in W and W are predicted to be too high by several
MeV. Thus, the agreement between experiment and the model p r e
dictions of the relative energies is only marginally acceptable.
In contrast to the failure of the model to predict the c o r
rect energies of the states, the mode 1’ predictions of which states
should be strong and which should be weak in (p,t) and (t,p)
transitions are very well fulfilled. Most notable are the results
182 +for the W(0j) state which is populated with about 15% of the
ground-state strength in the (p,t) reaction and about *0% of the
ground-state strength in the (t,p) reaction. The detailed nu me ri
cal predictions of these strengths are not borne out, however, again
reflecting the large anharmonic effects; the transitions to e x
cited states are always weaker than the ground-state transitions.
112 1 1 A(Such a reduction in strength is also seen in the Sn(t,p) Sn
11 A +reaction where the gap occurs above Sn, the excited 0 state
is excited with 11% of the ground-state strength.) Further limits
on the applicability of the pairing vibration model to the present
+ 182 180case are evident from the fact that the Oj states in W and W
are at least weakly populated in inelastic scattering whereas the
model calls for zero strength. This result comes from the fact
that within the model the 0+ states are built out of the groundn
states by the excitation of pairs of neutrons across the gap,
while the inelastic scattering operator creates p a r t i c 1e-hole
pairs. An admixture of proton b e ta -v ib ra ti on a1 strength in the states
however, could explain this result. A much more serious limitation
133
1Oo iis seen in the results of the JW(d,t) W reaction where the
Oj state is populated with 23% of the ground-state strength (KI 73)-
i p 7The model would describe the ground state of W as the vacuum
coupled to one particle in the +[510] orbital above the gap. The
removal of the particle should then populate only the vacuum, the
'82W ground state. A similar argument holds for the '8 ^ W ( d ,p ) '84W(o|)
transition which has 23% of the ground-state strength. In addition,
] Oi 1 ft 1the occurence of blocking in the W(p,t) W ( + [ 5 1 0 ] ) transition
indicates that the H 5 1 0 ] orbital does in fact take part in the
182 180ground-state transition in the W(p,t) W reaction, in d i sa gr ee
ment with the (first order) model predictions.
One interesting prediction of the model is that there should
exist an excited 0 state at 1.0 MeV excitation in with
the designation (2,1). Such a state would not be excited in
(p,t) but would be strong in (t,p) transitions. The prediction
I 7 Qis difficult to check directly since W is unstable, but a
serious effort to locate the state through either (a,xn) reactions
l82 l80or an especially high-statistics W(p,t) W experiment, or
even an a 1pha-1ransfer stripping reaction on Hf is called for.
Even though the limitations of the model as applied to the
tungsten isotopes are apparent, the ability of this very simple
model to organize the data is impressive. Unfortunately, there is
no simple way to include anharmonic effects in the model so that
a more detailed discussion in the framework of the pairing vibration
model would not be useful. The use of another model, which includes
the phonon-phonon interactions, is clearly called for. Such
a model can be developed with the use of the Interacting Boson
Approximation of Arima and Iachello (Ar 75, Ar 76) and no doubt
will be found in the future to be of great value in explaining
these data in a quantitative manner
7.3 Oj STATES IN THE LOW-MASS ISOTOPES
+ i fin 1 7ftBefore discussing the Oj states in W and W, it will
be helpful to review their properties:
(1) They both are excited strongly in (p,t), with 13%
and 12% of their respective ground state strengths.
(2) They possess similar moments of inertia which are
substantially different from those of the 0* states
in the higher-mass tungsten isotopes and of all of the
ground states.
(3) They occur at very different excitation energies, 1516
and 997 KeV respectively.
The last property would seem to preclude any reasonable
possibility of relating the structure of these two states with a
simple model. When the L = 0 transfer strength versus experimental
Q-value Is plotted as in Figure 7“9» however, a new simplicity
emerges. Both states occur at nearly the same Q-value, -7.8 M e V ,
which corresponds to a two-neutron separation energy of
16.3 MeV. In addition, substantial L = 0 strength is seen in
l 82W at nearly the same Q-value. This constancy of can be
explained through the assumption that in populating the constant-
Q states (Qc states) neutron pairs in each case are removed from
the same single-particle level or group of levels. Since the
single-particle potential well does not change significantly with
a small change in mass, each state should have the same S 2^.
These Q^ states, being two neutron holes in a particular group of
135
after Q-value correction vs. experimental
Q-values. The favored L = 0 transitions
have been reduced in size by a factor of
10 for graphical purposes.
Figure 7*10. Neutron single-particle level scheme
calculated as described in Chapter 2.
The orbitals occupied by the unpaired
nucleon for various isotopes are shown
along with the spin-parity of those
levels. The purpose of the figure is to
point out the clustering of single-particle
levels near a binding energy of 8 M e V .
Figure 7~9. Re l at i ve s trengths of L = 0 t r a n s i t i on s
J{p
,t)
(arb
itrar
y un
its)
l 8 6 W ( p , t ) l 8 4 W
1 x ®l 8 4 W ( p , t ) 182 w
L1 1 0
. . . . . . .. -
l 8 3 W ( p , t )
//
1 8 1 w
| x i oV
l 8 2 W ( p , t ) l 8 0 w
. . . I 1 ■*------------------Ty--------l 8 0 W ( p , t )
J i l l
£00
x i 5i i
- 9 - 8 - 7 - 6 - 5 - 4Q - V A L U E ( M e V )
B I N D I N G E N E R G Y ( M e V )
r o o o CDT
r o O
nrol<o ro|— roloj
139
levels, coupled to the target ground state, can be called pair
vibrations in a macroscopic picture. These pair vibrations, h o w
ever, have a qualitatively different structure from the pair
vibrations in the high-mass isotopes (Mo 77).
The calculated single-particle level scheme is displayed
again in Figure 7 ~ 10 in order to point out the grouping of o r b i
tals below the Fermi surface at a single-particle binding energy
of about 8 MeV. If the states can be described as two neutron
holes in this group of orbitals, coupled to the ground state of
the target nucleus, then to first order S„„ = 16 MeV. Under nor-ZN
mal circumstances, however, these orbitals should be strongly
coupled via the pairing interaction to those participating in the
Fermi surface. The excited states would then display the normal
few percent of the ground-state (p,t) strength. A close look
at the microscopic nature of the single-particle levels, however,
shows that while the Fermi surface is composed almost entirely
of oblate levels, four out of the six levels in the group are
prolate (the i [510] , the 3/ 2[6511 , the 5/2[512] , and the 5/2[6 A 2 ]
orbitals). Under the assumption of a reduced pairing matrix
element between oblate and prolate orbitals, as discussed in Sec
tion 2.2, these four prolate orbitals would be somewhat decoupled
from the Fermi surface so that the excited 0+ state could display
substantial (p,t) strength. A calculation of the (p,t) strength
of an excited 0+ state formed by coupling a dineutron hole in the
102 -f. -f.four orbitals to the ground state of W predicts a(0 0 )/o(0 ) - 0.10
c 9
140
in reasonable agreement with the data (see Appendix D for a de scrip
tion of the c a 1c u 1 a t i o n a 1 procedure). The two transitions, to
the ground state of the residual nucleus through a coherent re
moval of a neutron pair from the oblate orbitals (resulting in
a net lowering of the Fermi surface), and to the Qc states through
the removal of the dineutron from the decoupled prolate orbitals,
can be schematically pictured as in Figure 7-11.
This model for the Qc states makes a strong prediction for
IBS l8lthe W(p,t) W reaction. As discussed in Section 6.2, the odd
neutron causes the 2 - [5 1 0 ] level to be blocked, decreasing the
strength of the favored L = 0 transition to the ^ [510] state in
IS]W. The situation is schematically represented in Figure 7-12
where the favored L = 0 transition (the analog to the ground-state
transition in even-even nuclei) is shown at the top. The transition
involves the coherent removal of a dineutron from the oblate
orbitals, leaving the unpaired neutron and the neutron pairs in the
prolate orbitals completely undisturbed. The strength of the
transition is less than that of the neighboring even target nuclei
since the L = 0 pickup strength of the blocked orbital is not
available in the transition. The transition shown at the bottom
of the figure, where the odd particle and the oblate orbitals are
left undisturbed and a dineutron is removed from the group of
prolate orbitals, however, should not be blocked.
As Figure 7 ~ 10 shows, the strengths of the Q states vary
smoothly, including the odd-mass target case, although the strengths
transitions. At the top, the transition
to the ground state involves a coherent
removal of neutron pairs from the oblate
orbitals, lowering the Fermi surface. At
the bottom, a neutron pair is (coherently)
removed from the prolate levels, leaving
the Fermi surface undisturbed and hence
populating an excited state of the residual
n u c l e u s .
Figure 7~12. Figure similar to 7~11 but for an odd-neutron
18 ftnucleus such as W. The favored L = 0
transition shown at the top is reduced
in strength due to the blocking of an
oblate orbital by the odd nucleon. The
transition to the excited state formed by
coherently removing a neutron pair from
the prolate orbitals should not be blocked.
Figure 7 ~ 13- L = 0 strengths for favored transitions
(open dots) and transitions (closed dots)
for the tungsten nuclei. The value for
182W was obtained by summing the strengths
of the three close-lying 0+ states.
Figure 7-11. Schematic representat ion o f two L = 0
O B L A T EL E V E L S
P R O L A T EL E V E L S
( p , t ) t r a n s i t i o n t o g r o u n d s t a t e
( p , t ) t r a n s i t i o n t o e x c i t e d 0 + s t a t e
OB LA TEL E V E L S
PR O L A TEL E V E L S
a n a l o g to g r o u n d s t a t e t r an s i t io n
( b lock ing in c lu d e d )
a n a l o g to e x c i t e d O + s t a t e t r a n s i t i o n
(b lock ing h a s no e f f e c t )
J{p
,t
) (o
rb.
un
its
)
RESIDUAL MASS
1*5
of the ground states vary considerably. Thus, the L = 0 transi
tion to the (Qc (x) i[510]) state in the ' (p,t) ' W reaction
is in fact not blocked, in accordance with the model. The strength
of the Q c states does decrease with increasing mass in contrast
to the model which predicts a constant strength, but the reduction
in strength can, perhaps, be accounted for by assuming that as
their excitation energy increases, the Qc states mix with other
0+ states.
1*6
The strengths and 0+ to 2+ energy differences for all
J^K = 2+ 0 states located are listed in Table 7-2. The strengths of
the transitions are taken to be the sums of the differential cross
sections at the first six angles, from 5° to *25° in 74° steps,
divided by the appropriate Q-dependence factor from Figure 5-2.
Also shown are the ratios of the strengths of the 2+ states to
those of their associated 0+ states. The S(2+ )/S(0+ ) ratios
180 178for the 0j bands in W and W are seen to be very different
TT ^from the ratios for other K = 0 bands, being about a factor of
3 or * larger than those of the ground state bands and a factor
^ 1 Q|. 1 0 /
of 9 larger than the ratios for the 0^ bands in W and W.
Such a result can in fact be qualitatively understood in the present
178 180model for the Qc bands in W and W. As has been discussed
by Hanson £t_ aj_. (Ha 75, Ha 76) and as can be seen in Figure 6-*,
the L = 2 direct transfer strength is unusually weak in tungsten
due to both cancellations between contributions from various o r
bitals as well as due to the small quadrupole transfer strength
of each of the orbitals near the Fermi surface. In comparison,
the quadrupole transfer strength of the four prolate orbitals
in the group at 8 MeV single-particle binding energy is calculated
to be quite large, giving S(2+ )/S(0+ )Q = 6 S(2+ )/S(0 )r cC b . o .
for the direct route. Although the result is suggestive, it is not
possible to make a direct comparison with experiment due to the
effect of multistep inelastic routes also populating the 2+ 0 states.
7.* J = 2 STATES IN K = 0* BANDS
TABLE 7-2
J^K = 2*0 STRENGTHS
Ex(2+ )-Ex(0+ ) (a) I d l ' b) 2+ (c) . (d)Final A (keV)________ (pb/sr) (p,t) S /S
Ground State Bands
184 111 423 325 .96
182 100 623 515 .75
180 104 492 502 1.00
178 106 718 845 .96
First Excited 0+ Bands
184 119 6 9 ^ 5 8 ^ l . / e ^
182 136 41 42 .38
180 7 4 + 6 146 304 4.7
178 86 ± 5 (f) 176 289 3.0
(a) Excitation energy different between 2+ state and its associ
0+ state
(b) Sum over six laboratory angles 5° to 42?° in 75° steps
2+ y —(c) S^p = ( L dfi )/Q - dependence factor from Figure 5“2
2+(d) Ratio of S ^ p ^ to L = 0 strength from Table 7-1
(e) Doublet, upper limit given
(f) Consistent with value of 82 keV from Reference Ca 76.
SYSTEMATIC
OCTUPOLE AND
Chapter 8
OF COLLECTIVE STATES:
GAMMA VIBRATIONAL STATES
8.1 THE OCTUPOLE VIBRATIONAL BANDS
The 3 member of the K1' = 2 octupole band was located
1 78in each of the nuclei studied, except for W where it formed
an unresolved doublet with the 2* of the gamma vibrational
band. In each case, it was weakly excited, and followed no
simple pattern in strength or excitation energy as seen in *Table 8-1.
8.2 GAMMA VIBRATIONAL BANDS
The J^K = 2 2 gamma vibrational states represent a most
interesting but confusing study, especially with regards to
1 82 1 84W and W. The conclusions of Gunther et^ a_l_. (Gu 71) from
+ 182 18Atheir (d .d1) study were that the 2 2 states in W and W
were well developed gamma vibrations with little mixing between
them and the ground or 0* bands. In addition, in this study, the
^8 ^ W ( d ,p ) ^ a n d 'B ^ W ( d , t / B2W cross sections to the 2+ ,
3+ , 4+ , and 5+ members of the gamma bands were well reproduced
by DWBA calculations using R P A - c a 1c u 1ated structure for the
gamma vibrations with no mixing with K = 0 bands. In contrast,
the calculations of Kumar and Baranger (Ku 68), which were
successful in platinun and os mi um nuclei predict large mixings
+1between the ground, Ch, and gamma bands. The recent m e a s u r e
ment of the quadrupole moments of the 2^ in '88W and ^8AW
by O'Brien et al. (OB 77) showed a near-zero quadrupole moment
moment for the 2* in ^B ^W and a trend reasonably reproduced by
150
TABLE 8-1
L = 2,3 TRANSITIONS
Exc i tat ion c do Energy / dfi , . . ,
Final Mass (keV) (pb/sr) (p,t) S/S (o'!')
J^K = 2+ 2 GAMMA-r
18A 903 110 ± 20 9A .13
182 1219 175 179 .26
180 1118 76 106 .21
178 1111 7 A (d) 1 2 3 (d) . 1 5 (d)
JT K = 3"2 OCTUPOLE
18A 121A 22
182 1372 27
180 1083 68
178 1120 7A^(a) Sum over six laboratories, 5° to A2r° in 7i° steps.
2+ y —(b) S^ ^ = {L d^) / (Q - dependence factor from Figure 5-2)
2+(c) Ratio of S^ to L = 0 ground state strength from Table 7- l
(d) Doublet, upper limit given
To this morass of conflicting conclusions will now be
added the (p,t) systematics for populating the 2*. The shape and
population systematics of the 2+ gamma are displayed in Figure 7-2
and Table 8-1. The strength of the 2* is largest in the ^ w - ’- 182
transition and decreases as the residual mass changes. To investi
gate this changing strength, the L = 2, AK = 2 transfer strength
was calculated for every two-neutron configuration within the
space of the single-particle orbitals shown in Figure 2-2. The
spectroscopic amplitudes were calculated with Equation 3.11
and utilized the Nilsson coefficients from Davidson (Da 68) with
P g = 0.45 and 6 = 0.3. The relative strengths so calculated
are shown in Table 8-2 where the "sign" is the sign of the total
bound-state form factor at the nuclear surface. It is immediately
obvious that only four configurations have significant pickup
strength, the (*[510], 3/2 [512]), the (*[510], 5/2 [5 1 2 ]), the
(it521], 5/2[512]), and the (3/ 2 [5 1 2 ], 7/2[503]) configurations.
An estimate of the total available L, AK = 2,2 strength can be
obtained by summing these intrinsic strengths weighted with the
appropriate V factors (from the BCS calculation with parameters
taken from Table 6-2), as per the Os approximation, Equation
3.12, while neglecting the backward-going amplitudes b.^U.U^.
Figure 8-1 gives the relative cross sections so obtained and the
excitation energy for each of the four main configurations. At
the bottom of the figure are the experimental excitation energies
the Kumar-Baranger c a l c u l a t i o n s .
TABLE 8-2
J*K = 2+ 2 PICKUP STRENGTHS
da
^ A2.5°Configurat ion (yb/sr)
1/2[510] ® 3/2[512] 660® 5/2[512] 16*
1/2[521] ® 3/2[512] 10
® 5/2[512] 1*7
3/2[512] ® 7/2[503] 33*
® 7/2[51*] 3
5/ 2 [5 1 2 ] ® 9/ 2 [505] 15
1/ 2 [660] ® 5/2[6*2] 1
3/2[651] ® 7/2 [633] 1
5/2[6*2] ^ 7/2[62*] 87/2[633] g 1 I/2[6l5] 2
(a) Sign of form factor outside of nuclear surface
Figure 8-1. J^K = 2+ 2 strengths for the pickup to
various two-neutron-hole configurations.
The y-axis is the excitation energy of
the two-quasiparticle configuration while
the x-axis is the residual mass from the
(p,t) reaction. Shown above each point
is the relative pickup strength of the
configuration as described in the text.
The horizontal bars at the bottom of the
figure display the excitation energies of
the 2+ states while above the bars are the Y
relative total available pickup strengths
and below are the relative experimental
strengths (all arbitrarily normalized to
1 ft?the W re su lt ).
1 8 4 1 8 2 1 8 0 1 7 8RESIDUAL M A S S
155
of the 2^ states with the relative experimental strengths below
and the calculated total available strength above. The experi-
178mental value for W is an upper limit. The trend of decreasing
strength from residual nuclei of mass 182 to 178 is reproduced,
but the prediction of a relative strength of 1.7 for the 186 -+]8k transition is not in agreement with the experimental value of 0.5.
It should be remembered, however, that the V factors used in
these calculations have been taken from BCS calculations, the
validity of which are in doubt due to the gap in the si ng le
particle spectrum.
Hamamoto (Ha 73) has performed an RPA calculation which
shows the (*[510], 3/2 [512]), the (*[510], 5/2[512]), and the
(3/2[512], 7/2[514]) to be the major two-neutron configuration
involved in the gamma vibration, with proton configurations
constituting less than 20% of the wavefunction. Considering
these configurations to deposit their entire available L, AK = 2,2
strength in the 2+ state, the calculation predicts relative
strengths for these states in the residual nuclei 178:180:182:18A
of 0.1:0.3:1•0:1.9 in comparison with the experimental values
of 0.7 (upper 1 im it):0 .6:1.0:0.5. in no better agreement than the
previous calculation.
The question of which neutron configurations really take
part in the gamma vibration would be left in doubt but for two
observat io ns:
156
(1) The decreasing transfer strength to the 2* as the
residual mass goes from 182 to 178 indicates that the
configurations involved are primarily orbitals near the
center of the Fermi surface, where the V factors change
considerably with changing mass.
/o\ 183,,/ J81,,(2; The W(p,t) W experiment does not show two strong
bandheads of J^K = 3/2 - 3/2 and 5/2 - 5/2 near thelOo IQa ,
Q-value of the W(p,t) W(2 ) transition. These
two states could be formed by coupling the odd
+[510] particle to the gamma vibration.
These results lead to the conclusion that the (+[510] ,X)
configurations are of paramount importance in the population of the
2* states with (p,t) reactions on the even-mass tungsten isotopes,
in agreement with the conclusion of Hamamoto. The identification
of the specific orbital X which contributes to the (?[510],X ) 2+ 2] O 1 ft 1
configuration also comes from the W(p,t) W results where the
second strongest peak in the spectrum with S(p,t) = 219 pb/sr
(sum of the differential cross sections of the first six angles,
divided by the Q-dependence factor) is the 5/2 - 5/2[512] state
13 2 18 o ^whereas the strength for the W(p,t) W(2 ) transition is
179 ub/sr. Thus, this one L = 2 transition in the odd-mass
target case contains most, if not all, of the transfer strength
normally populating the gamma vibration in even-mass tungsten
nuclei. The conclusion is therefore that the pickup of the
( H 5 1 0 ] , 5/2[512])2+ 2 two-neutron configuration supplies almost
all o f the g a m m a v i b r a t i o n a l p i c k u p s t r e n g t h for the e v e n - m a s s
(p,t) r e a c t i o n s .
T h e fact that b a s i c a l l y o n l y o n e c o n f i g u r a t i o n Is i m p o r t a n t
the (p,t) p o p u l a t i o n of the 2+ s t a t e s in e v e n - m a s s t u n g s t e n
n u c l e i (even t h o u g h t h e s e s t a t e s a r e s t r o n g l y p o p u l a t e d ) s u g g e s t s
that b r o a d s y s t e m a t i c s t u d i e s o f the p o p u l a t i o n o f 2* s t a t e s w i t h
(p,t) a n d (t,p) r e a c t i o n s (Ca 73b) tell us l i t t l e a b o u t the
s t r u c t u r e o f g a m m a v i b r a t i o n s in g e n e r a l . T h i s is not to say,
h o w e v e r , that t he a n o m a l o u s l y s t r o n g (t,p) p o p u l a t i o n of the
2+ s t a t e in n o t e d by C a s t e n a n d G a r r e t t (Ca 73b) is trivial
to e x p l a i n , but o n l y t hat the d e t a i l e d s t r u c t u r e o f the 2*
s t a t e is v e r y i m p o r t a n t . O n e e x p e r i m e n t w h i c h w o u l d be usef u l
in d e t e r m i n i n g w h i c h c o n f i g u r a t i o n s a r e i m p o r t a n t in (t,p)
1 Q -j 1 O rt r a n s i t i o n s to t h e s e s t a t e s w o u l d be the W ( t , p ) W e x p e r i m e n t .
C h a p t e r 9
S U M M A R Y A N D C O N C L U S I O N S
158
159
In a n y e x p e r i m e n t a l s t u d y o f n u c l e a r s t r u c t u r e it is not
e n o u g h to p r e s e n t d a t a w i t h o u t a f r a m e w o r k w i t h i n w h i c h to u n d e r
s t a n d , at least q u a l i t a t i v e l y , t he r e s u l t s o b t a i n e d . T h i s s y s t e m
a t i c s t u d y o f a s e r i e s o f r e l a t e d n u c l e i has d e m o n s t r a t e d that,
w i t h o u t r e c o u r s e to d e t a i l e d c a l c u l a t i o n s , the p a i r i n g c h a r a c t e r
of s t a t e s c a n be u n d e r s t o o d in t e r m s of r e l a t i v e l y s i m p l y m o d e l s .
W h e n s u c h s i m p l e m o d e l s a r e a p p l i e d to a s y s t e m as c o m p l e x as
r a r e e a r t h n u c l e i , h o w e v e r , the l i m i t a t i o n s o f the m o d e l s m u s t be
c a r e f u l l y c o n s i d e r e d . In the p r e s e n t c a s e , o n l y th e n e u t r o n p a i r i n g
d e g r e e o f f r e e d o m is b e i n g p r o b e d ; w h e r e a s m a n y o t h e r c o l l e c t i v e
d e g r e e s o f f r e e d o m c e r t a i n l y e n t e r into the s t r u c t u r e o f m a n y of
the s t a t e s d e s c r i b e d here. F o r e x a m p l e , t he p r o t o n p a i r i n g d e g r e e
o f f r e e d o m ha s b e e n n e g l e c t e d as w ell as the b e t a - v i b r a t i o n a 1
( q u a d r u p o l e - q u a d r u p o l e ) i n t e r a c t i o n a m o n g q u a s i p a r t i c l e s . O t h e r
m o r e e x o t i c d e g r e e s o f f r e e d o m , s u c h as s p i n - q u a d r u p o l e , h a v e a l s o
b e e n s u g g e s t e d to e x p l a i n the e x i s t e n c e of m o r e t h a n o n e 0* s t a t e
w i t h i n the s u p e r c o n d u c t i n g p a i r i n g g a p of s o m e n u c l e i (e.g. R e f e r e n c e s .
A b 72, Be 69b, Pu 72). N o n e o f t h e s e h a v e b e e n c o n s i d e r e d here.
In v i e w of t h e s e o b v i o u s s h o r t c o m i n g s , the fac t that the p r e s e n t
m o d e l s g i v e r e a s o n a b l e a n d u n d e r s t a n d a b l e i n t e r p r e t a t i o n s to the
(p,t) a n d (t,p) d a t a m a y at f i r s t s e e m f o r t u i t o u s . U p o n r e f l e c t i o n ,
h o w e v e r , w e r e a l i z e that the e f f e c t o f o t h e r c o l l e c t i v e d e g r e e s
of f r e e d o m m a y m o d i f y t he s t r u c t u r e o f p a r t i c u l a r s t a t e s but that
t h e y w i l l no t d e s t r o y t he n e u t r o n p a i r i n g d e g r e e of f r e e d o m itself.
9 - 1 LIMITATIONS OF THE STUDY
T h e e f f e c t m a y t h e n be to m o d i f y the m o d e l r e s u l t s , p e r h a p s changi
the e n e r g y of a g i v e n s t a t e or f r a g m e n t i n g the L = 0 s t r e n g t h ,
b ut the o v e r a l l r e s u l t s m a y v e r y w e l l r e m a i n the same. W i t h t h ese
p o i n t s in m i n d , c o n c l u s i o n s w i l l n o w be d r awn.
161
T h e t u n g s t e n n u c l e i h a v e b e e n s e e n to be a s h o w c a s e for
p a i r i n g v i b r a t i o n s , w i t h b o t h t y p e s in e v i d e n c e . T h e g r o u n d s t a t e s
+ 1 82a n d f i r s t e x c i t e d 0 s t a t e s in W a n d the h e a v i e r i s o t o p e s fit
s u r p r i s i n g l y w e l l into a p a i r i n g v i b r a t i o n s c h e m e . T h i s m o del
p r e d i c t s s t r o n g (p,t) a n d (t,p) t r a n s i t i o n s to the 0* s t a t e in
184W, a n d a l m o s t v a n i s h i n g l y small (t , p) t r a n s i t i o n s to t h e s e
186 188s t a t e s in W a n d W. T h e m o d e l a l s o m a k e s a p r e d i c t i o n for
l 80 +W, w h e r e t h e r e s h o u l d e x i s t an e x c i t e d 0 s t a t e at 1.0 to 1.1
M e V w h i c h s h o u l d not be e x c i t e d in (p,t). S i n c e no s u c h s t a t e
w a s in e v i d e n c e in the (p,t) d a t a p r e s e n t e d h e re, the l o c a t i o n of
+a 0 s t a t e at thi s e x c i t a t i o n e n e r g y w o u l d lend c o n s i d e r a b l e w e i g h t
181 180to the m o d e l . S u c h a s t a t e m i g h t be l o c a t e d w i t h the T a ( p , 2 n y ) W
r e a c t i o n v i a the d e t e c t i o n of the c o i n c i d e n t g a m m a t r a n s i t i o n s
s u c h as 0* ->2g'> 9 g or p e r h a p s d i r e c t l y w i t h the ^ 8 Hf (8 L i ,d) ^
a l p h a t r a n s f e r r e a c t i o n .
T h e l o w - m a s s i s o t o p e s w e r e f o u n d to p o s s e s s e x c i t e d 0+
s t a t e s c h a r a c t e r i z e d by the s a m e e x p e r i m e n t a l (p,t) Q - v a l u e a nd w e r e
178 180h e n c e c a l l e d s t a t e s . In W a n d W, w h e r e t h e i r a s s o c i a t e d
2+ s t a t e s c o u l d be u n a m b i g u o u s l y l o c a t e d , the 0+ - t o - 2 + e n e r g y
s p a c i n g s w e r e f o u n d to be s i g n i f i c a n t l y d i f f e r e n t f r o m that of the
o j b a n d s in the h i g h e r - m a s s i s o t o p e s , a n d f r o m t h o s e of all of the
g r o u n d bands. T h e i r v e r y d i f f e r e n t s t r u c t u r e w a s f u r t h e r e m p h a
s i z e d by the l a r g e r r a t i o of S ( 2 + ) to S ( 0 + ). T h e s t r u c t u r e of t h e s e
Q c s t a t e s w a s i n t e r p r e t e d to be the g r o u n d - s t a t e of the t a r g e t
9 . 2 PAIRING VIBRATIONS
162
n u c l e u s c o u p l e d to t wo h o l e s in a g r o u p of p r o l a t e o r b i t a l s s o m e w h a t
b e l o w the Fermi s u r f a c e w h i c h a r e p a r t i a l l y d e c o u p l e d f r o m the
o b l a t e o r b i t a l s w h i c h c o n s t i t u t e the b u l k of the Fermi s u r f a c e .
T h e m o del p r o v i d e s an e x p l a n a t i o n for b o t h the c o n s t a n t Q - v a l u e
a n d the l a rge S (2 ) / S ( 0 ) r a t i o but d o e s no t p r o v i d e a s i m p l e
e x p l a n a t i o n o f the l a r g e r m o m e n t o f i n e r t i a of t h e s e b a n d s s i n c e the
r e m o v a l of two n e u t r o n s f r o m p r o l a t e l e v e l s s h o u l d d e c r e a s e the m o m e n t
o f inertia. T h e m o del d o e s f h o w e v e r , m a k e the s t r o n g p r e d i c t i o n
183 181tha t the e x c i t e d L = 0 t r a n s i t i o n in the W ( p , t ) W r e a c t i o n
s h o u l d n ot be b l o c k e d by the u n p a i r e d n u c l e o n , a p r e d i c t i o n that
is b o r n e o ut by e x p e r i m e n t . O n e f u r t h e r p r e d i c t i o n of the m o del
+ 1 82m u s t be m e n t i o n e d , the 2 . 7 2 5 M e V 0 s t a t e in W, w h i c h is o n e
o f the Q c s t a t e s , s h o u l d not be p o p u l a t e d in the (t,p) r e a c t i o n
184s i n c e it is t w o - h o l e c o u p l e d to the W g r o u n d stat e . It is
e x p e r i m e n t a l l y d i f f i c u l t to c h e c k this p a r t i c u l a r p r e d i c t i o n du e
to p r o t o n s f r o m the (t,p) r e a c t i o n o n light c o n t a m i n a n t s in the
1 80W t a r g e t w h i c h o b s c u r e the s p e c t r u m at f o r w a r d a n g l e s (Be 78).
T h e s e t wo t y pes of p a i r i n g v i b r a t i o n s , a l t h o u g h v e r y d i f f e r e n t
in s t r u c t u r e , h a v e o n e v e r y i m p o r t a n t c o m m o n c a u s e ; the d e c o u p l i n g
of o n e g r o u p of s i n g l e - p a r t i c l e o r b i t a l s f r o m a n o t h e r . T h i s d e
c o u p l i n g leads to the s y s t e m a t i c s in e x c i t a t i o n e n e r g y or Q - v a l u e
as well as s u p p l y i n g s u b s t a n t i a l L = 0 s t r e n g t h to the e x c i t e d
0+ state.
163
9 .3 T R A N S I T I O N T O J* = 2+ S T A T E S
T h e a n g u l a r d i s t r i b u t i o n s h a p e s fo r t r a n s i t i o n s to the 2+
s t a t e s in the g r o u n d s t a t e r o t a t i o n a l b a n d s w e r e f o u n d to c h a n g e
w i t h t a r g e t m a s s . T h e o r i g i n of this e f f e c t c a n be t r a c e d to the
c h a n g i n g l o c a t i o n of the Fermi s u r f a c e w h i c h c h a n g e s the i n d i v i d u a l
c o n t r i b u t i o n of s p e c i f i c l e v e l s to the total s t r e n g t h . T h e d i r e c t
q u a d r u p o l e t r a n s f e r s t r e n g t h w a s f o u n d to be w e a k in the h i g h -
m a s s i s o t o p e s a n d to i n c r e a s e as the t a r g e t m a s s d e c r e a s e s w h i l e the
m u l t i p l e - s t e p i n e l a s t i c t r a n s i t i o n s t r e n g t h r e m a i n s r a t h e r c o n s t a n t .
T h e c h a n g i n g a n g u l a r d i s t r i b u t i o n s h a p e s t h e n r e s u l t f r o m an i n t e r
f e r e n c e b e t w e e n the t w o r o u tes.
E x c i t e d J ^ K = 2 0 a n d 2 2 s t a t e s w e r e a l s o i n v e s t i g a t e d
a l t h o u g h f e w d e f i n i t e c o n c l u s i o n s c o u l d be d r a w n . In p a r t i c u l a r ,
the s t r o n g c o u p l i n g b e t w e e n t h e s e t wo s t a t e s r e p o r t e d in s e v e r a l
w o r k s a n d p r e d i c t e d by K u m a r a n d B a r a n g e r , c o u l d n ot be r e a s o n a b l y
c h e c k e d w i t h the p r e s e n t r e s u l t s , s i n c e the (p,t) r e a c t i o n is not in
g e n e r a l s e n s i t i v e to s u c h c o u p l i n g s . It is of s o m e i n t e r e s t , how-
182e v e r , that in W w h e r e t he t w o s t a t e s a r e n e a r l y d e g e n e r a t e and
s i g n i f i c a n t m i x i n g is e x p e c t e d , the 2+ 2 s t a t e w a s p o p u l a t e d w i t h
a l m o s t 10 t i mes the s t r e n g t h of the 2+ 0 w h i l e in '84 W the r a t i o
w a s f o u n d to be o n l y 2.
F i g u r e 9-1 s h o w s the s t r e n g t h s for the t r a n s i t i o n s to the
2+ 0 , 2+ 0. , a n d 2+ 2 s t a t e s as w ell as the s u m of the total s t r e n g t h , g 1
T h e v e r y s t r o n g m a s s d e p e n d e n c e is s t r i k i n g a l t h o u g h it m u s t be
r e a l i z e d that t he e f f e c t s of m u l t i p l e s t e p p r o c e s s e s a r e c o n s i d e r a b l e .
F i g u r e 9 _ 1. T r a n s f e r s t r e n g t h s to J 11 = 2+ s t a t e s for
the t u n g s t e n i s o t opes. T h e t r a n s f e r s t r e n g t h
is d e f i n e d as the s u m m e d c r o s s s e c t i o n o v e r
the f i r s t s ix a n g l e s a p p r o x i m a t e l y c o r r e c t e d
for k i n e m a t i c e f f e c t s . A l s o p l o t t e d a re
t he s ums o f the j \ = 2+ 2 a n d 2+ 0j r e s u l t s ,
a n d t he s u m total of all t h r e e o f the
t r a n s i t i o n s .
STRE
NGTH
(A
RBIT
RARY
U
NIT
S)i
T h e s t u d y o f t he s y s t e m a t i c s o f t he t u n g s t e n i s o t o p e s w i t h
the (p,t) r e a c t i o n h as i l l u m i n a t e d n e w a n d i n t e r e s t i n g f a c e t s of
the n e u t r o n p a i r i n g d e g r e e o f f r e e d o m w i t h o u t r e c o u r s e to d e t a i l e d
a n d c o m p l e x s t r u c t u r e c a l c u l a t i o n s . W h a t is s u r p r i s i n g , p e r h a p s ,
is that in the m i d s t of p o t e n t i a l l y i n c r e d i b l e c o m p l e x i t i e s , s e veral
c l e a r a n d b e a u t i f u l p a t t e r n s e m e r g e , a l l o w i n g a g l i m p s e o f the
s i m p l i c i t y of n a t u r e . S u c h g l i m p s e s a r e the t r u e r e w a r d s of
r e s e a r c h .
166
9 . k GENERAL COMMENTS
167
A p 70
A r 6 6
Ar 75
Ar 76
As 72a
As 72b
As 72c
As 7*
As 75
Au 65
Au 69
Ba 67
Ba 69
Ab 72 . K. A b d u 1v a g a b o v a , S. P. Ivanova, a n d N. I. P y a t o v ,
Phys. Lett. 3 8B (1972) 215.
. A. A p o n i c k , C. M. C h e s t e r f i e l d , D. A. B r o m l e y , and
N. K. G l e n d e n n i n g , Nucl. Phys. A l 59 (1970) 367-
. A r t n a , Nucl. D a t a S h e e t s J_ (1966) B 1 -1 -37-
. A r i m a and F. lachello, Phys. Rev. Lett. 35. (1975) IO6 9 .
. A r i m a a nd F. la c h e l l o , Ann. Phys. (N.Y.) 99. (1976) 253-
. J. A s c u i t t o and Bent S o r e n s o n , Nucl. Phys. A 1 9 0
(1972) 297.
. J. A s c u i t t o an d Ben t S o r e n s o n , Nucl. Phys. A l 90
(1972) 309.
. J. A s c u i t t o , C. H. King, a nd L. J. M c V a y , Phys.
Rev. Lett. 29. (1972) 1106.
. J. A s c u i t t o , C. H. King, L. J. M c V a y , a nd Bent S 0 r e n s o n ,
Nucl. Phys. A 2 2 6 (197*) *5*.
. J. A s c u i t t o , J. S. V a a g e n , K. A. Erb, D. L. H a n s o n ,
D. A. B r o m l e y , an d J. J. K o l a t a , Phys. Lett. 5 5B
(1975) 289.
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. A u s t e r n , D i rec t N u c l e a r R e a c t i on T h e o r i es (Wiley,
N e w Y o r k , 1969)•
. F. B a y m a n a nd A. K a l l i o , Phys. Rev. 156 (1967) 1121.
\. F. B a y m a n , D. R. Bes, and R. A. B r o g l i a , Phys. Rev.
BIBLIOGRAPHY
168
Le tt. 2 3 (1969) 1299.
Ba 70 J. B a n g an d S. W o l l e s e n , P h ys. L e t t . 3 3B (1970) 395.
Ba 73 R. C. B a r b e r , J. 0. M e r e d i t h , F. C. G. S o u t h o n , P.
W i l l i a m s , J. W. B a r n a r d , K. S h a r m a , a nd H. E.
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Be 66 D. R. Bes and R. A. B r o g l i a , Nucl. Phys. 80 (1966) 2 8 9 .
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(1969) 1190.
Be 6 9 b S. T. B e l y a e v and B. A. R u m i a n t s e v , Phys. Lett. 3 0B
(1969) ***•
Be 71 F. D. B e c c h e t t i a n d G. W. G r e e n l e e s in P o l a r i zat i on
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Be 72 D. R. Bes, R. A. B r o g l i a , a nd B. N i l s s o n , Phys. Lett.
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Be 7* F. B e r n t h a l , B. D. J e l t e m a , J. S. B o y n o , T. L. Khoo,
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Be 78 R. R. B e t t s a nd M. H. M o r t e n s e n , (1978) to be p u b l i s h e d .
Bo 52 A. Bohr, Dan. V i d . Selsk. Mat. Fys. M e d d . , no. 1*
(1952).
Bo 53 A. B o h r and B. M o t t l e s o n , Dan. V i d . S e l s k . Mat. Fys. M e d d . ,
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169
Bo 6 k
Bj 66
Br 68
Br 69
Br 71
Br 72
Br 73
Ca 71
Ca 72
A. Bohr, C o m p t e s R e n d u s du C o n g r e s I n t e r n a t i o n a l de
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S c i e n t i f i q u e (Paris, 1964) k 8 j .
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A 1 3 5 (1969) 561.
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( A r g o n n e Nat'l L a b o r a t o r y , A r g o n n e , M l , 19 7 2 )
( A r g o n n e P h y s i c s Div. Informal R e pt. P H Y - 1 9 7 2 H ) 1 1 3 .
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R. F. C a s t e n , P. K l e i n h e i n z , P. J. D a ly, a n d B. Elbek,
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N i l s s o n , Phys. Lett. 40B (1972) 333.
Ca 73 R. F. C a s t e n a nd O l e H a n s e n , Nucl. Phys. A 2 1 0 (1973)
489-
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Ch 72 R. R. C h a s m a n , Phys. Rev. Lett. 2j3 (1972) 1275.
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Da 68 J. P. D a v i d s o n , C o l l e c t i v e M o d e l s o f the N u c l e u s ( A c a d e m i c
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Da 71 P. J. D a ly, K. A h l g r e n , K. J. H o f s t e t t e r , a nd R. H o c h e l ,
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D a ly, and B. E l bek, Phys. Rev. £ 3 (1971) 1271-
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44.
De 72 P. D e b e n h a m and N. M. H i n t z , N u c l . Phys. A 195 (1972)
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3 8 5 .
171
D e S 6 3
El 72
Er I k
FI 71
FI 73
FI 77
Fr 7k
Ga 71
G a 77
G o 70
G r 66
A. d e - S h a l i t and Igal T a l m i , N u c l e a r Shell M o d e l
( A c a d e m i c P r ess, N e w Y o r k , 1963)-
Th. W. E l z e , J. S. Boyno, a n d J. R. H u i z e n g a , Nucl.
Phys. A i 87 (1972) **73-
K. A. Erb, D. L. H a n s o n , R. J. A s c u i t t o , B. S o r e n s o n ,
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D. G. F l e m i n g , C. G u n t h e r , G. H a g e m a n , B. H e r s k i n d ,
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D. G. F l e m i n g , C. G u n t h e r , G. H a g e m a n , B. H e r s k i n d , a nd
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E. R. F l y n n , G. L. S t r u b l e , R. G. L a n i e r , a nd L. G.
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A. M. F r i e d m a n , K. K a t o r e , D. A l b r i g h t , an d J. P.
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J. D. G a r r e t t , R. M i d d l e t o n , D. J. P u l l e n , S. A. A n d e r s e n ,
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93-
P. F. A. G o u d s m i t , J. K o n i j n , a n d F. W. N. d e B o e r ,
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R. G r a e t z n e r , G. B.
E l b e k , N u c l . Phys. 76 (1966) 1 .
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172
Gr 74
G r 75
Gu 71
Ha 73
Ha 75
Ha 76
Hi 65
Hi 72
Gr 71
Hi 75
lg 72
R. E. G r i f f i n , A. D. J a c k s o n , a n d A. B. V o l k o v , Phys.
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D. L. H a n s o n , D o c t o r a l D i s s e r t a t i o n , Y a l e U n i v e r s i t y
(1 9 7 5) u n p u b l i s h e d .
D. L. H a n s o n , F. J. A s c u i t t o , J. S. V a a g e n , K. A. Erb,
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S. H i n d s , J. H. B j e r r e g a r d , 0. H a n s e n , and 0. N a t h a n ,
Phys. Lett. _1_4 (1965) 48.
N. M. H i n t z in P r o c e e d i n g s o f the S y m p o s i u m on Two -
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Nat'l Lab., A r g o n n e , I l l i n o i s , 1973) ( A r g o n n e
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(1972) 323.
BIBLIOGRAPHY ( c o n t ' d )
173
J e 77
Ki 72
Ki 73
Ki 74
KI 73
Ko 69
Ku 68
Ku 69
Le 75
Li 73
B. D. J e l t e m a , F. M. B e r n t h a l , T. L. Khoo, and C. L.
Do rs, N u c l . Phys. A 2 8 0 (1977) 21.
C. H. King, R. J. A s c u i t t o , N. Stei n , a nd Bent S o r e n s e n ,
Phys. Rev. Lett. 29 (1972) 71.
C. H. K i ng, D o c t o r a l D i s s e r t a t i o n , Y a l e U n i v e r s i t y
(1973) u n p u b l i s h e d .
C. H. King, W. D. C a l l e n d e r , L. J. M c V a y , N. Stein,
a nd D. A. B r o m l e y , p r e p r i n t Y A L E 3 0 7 4 - 3 3 0 (1974)
u n p u b l i s h e d .
P. K l e i n h e i n z , P. J. Daly, a n d R. F. C a s t e n , Nucl.
Phys. A 2 0 8 (1973) 93-
D. K o v a r , D o c t o r a l D i s s e r t a t i o n , Y a l e U n i v e r s i t y (1969)
u n p u b l i s h e d .
K. K u m a r and M. B a r a n g e r , Nucl. Phys. A l 22 (1968) 273-
P. D. Kunz, c o m p u t e r c o d e D W U C K (1969) u n p u b l i s h e d .
I. Y. Lee, J. X. S a l a d i n , J. H o l d e n , J. O ' B r i e n , C.
B a k t a s h , C. B e m i s , P. H. S t e l s o n , F. K. M c G o w a n ,
W. T. M i l n e r , J. L. C. F o rd, R. L. R o b i n s o n , and
W. T u t t l e , Phys. Rev. C_l_2. (1975) 1483-
Th. L i n d b l a d , H. Ryde, a n d P. K l e i n h e i n z , Nucl. Phys.
A 2 1 0 (1973) 253-
BIBLIOGRAPHY ( c o n t ' d )
M a 65 J. H. E. M a t t a u c h , VI. T h i e l e , a n d A. A. W a p s t r a ,
N u c l . P h y s . 67 (1965) 1.
Ma 66 J. R. M a x w e l l , G. M. R e y n o l d s , a nd N. M. H i n t z , Phys.
R e v . j 5 1 (1966) 1000.
M a 70 J. V. M a h e r , J. R. E r s k i n e , A. M. F r i e d m a n , J. P.
S c h i f f e r , a n d R. H. S i e m s s e n , Phys. Rev. Lett.
25 (1970) 3 0 2 .
Ma 72a J. V. M a h e r , J. R. E r s k i n e , A. M. F r i e d m a n , R. H. S i e m s s e n ,
a n d J. P. S c h i f f e r , Phys. Rev. C 5 (1972) 1380.
M a 7 2 b J. V. M a h e r , J. J. K o l a t a , a n d R. W. M i l l e r , Phys.
Rev. £ 6 (1972) 358.
M a 73 C. M a g u i r e , D o c t o r a l D i s s e r t a t i o n , Y a l e U n i v e r s i t y
( 1 9 7 3) u n p u b l i shed.
Me 70 W. M c L a t c h i e , W. D a r c e y , and J. E. K i t c h i n g , Nucl.
Phys. A l 59 (1970) 615-
Mi 67 0. M i k o s h i b a , R. K. S h e l i n e , T. U d a g a w a , a n d S. Y o s h i d a ,
N u c l . Phys. A l 01 (19 6 7 ) 202.
M o 77 M. H. M o r t e n s e n , R. R. B e tts, a n d C. K. B o c k e l m a n ,
Phys. Lett. 70B_ (1977) 17.
Ni 55 S. G. N i l s s o n , M at. Fys. M e dd. Dan. V i d . Selsk. 23_
no. 16 (1955).
17*
BIBLIOGRAPHY ( c o n t ' o )
175
Ni 61 S. G. N i l s s o n a n d 0. P r ior, Dan. Vid. Mat. Fys. Medd.
3 2 , no. 16 (1961).
Ob 77 J- J- O ' B r i e n , J. X. S a l a d i n , C. B a k t a s h , a n d J. G.
A l e s s i , Nucl. Phys. A 2 9 1 (1977) 510.
Og 71 W. O g l e , S. W a h l b o r n , R. P i e p e n b r i n g , a n d S. F r e d r i c k s s o n ,
Rev. o f Mod. Phys. b3. (1971) *2*.
O o 70 M. O o t h o u d t , N. M. H i n t z , an d P. V e d e l s b y , Phy s Lett.
32B (1970) 270.
O o 73 M. O o t h o u d t a nd N. M. H i n t z , N u cl. Phys. A 2 1 3 (1973) 221.
Ri 68 G. R i p k a , in Advances in Nuc1 e a r Ph ys ics Vo1 . 1, ed.
M. B a r a n g e r and E. V ogt ( P l e n u m P r e s s , N e w York,
1968) 183.
Ri 72 W. I . v a n R i j and S. H. K a h a n a , Phys. Rev. Lett. £ 8
(1972) 50.
Ru 72 B. A. R u m y a n t s e v a n d U. B. T e l i t s y n , Sov. J. o f Nucl.
Phys. J_5 (1972) 3 8 7 .
Se 75 H. S e g o w a , K.-l. Kubo, a n d A. A r i m a , Phys. Rev. Lett.
35 (1975) 357.
Sh 76 H. L. S h a r m a an d N. M. H i ntz, Phys. Rev. C 13 (1976) 2288.
Sh 78 G. S h e r w o o d , D o c t o r a l D i s s e r t a t i o n , Y a l e U n i v e r s i t y
(1978) u n p u b l i shed.
So 65 V. G. S o l o v i e v , Nucl. Phys. 69 (1965) 1-
BIBLIOGRAPHY ( c o n t ' d )
176
So 75 P, C, S o o d a n d P. C. J o s h i , J. Phys. G J_ (1975) 2k2.
T u 78 P. P. T u n g , D o c t o r a l D i s s e r t a t i o n , Y a l e U n i v e r s i t y
(1978) u n p u b l i s h e d , a nd P. P. T u n g , K. A. Erb,
M. W. S a c h s , G. B. S h e r w o o d , R. J. A s c u i t t o , and
D. A. B r o m l e y , Phys. Rev. C (1978) in press.
Va 73 D. V a u t h e r i n , Proc. of t he Intl. C o n f on N u c l . Phys. (Munich)
ed. J. de B o e r an d H. J. M a n g ( N o r t h - H o l l a n d ,
A m s t e r d a m , 1973) 107.
BIBLIOGRAPHY ( c o n t ' d )
APPENDIX A
177
S i gn C o n v e n t i ons for R ad ia 1 F o r m F a c t o r s
T w o s i g n c o n v e n t i o n s a r e c u r r e n t l y in u s e for the radial p a r t of
s i n g l e - p a r t i c l e b o u n d - s t a t e w a v e f u n c t i o n s e x p a n d e d o n a s p h e r i c a l b a s i s
set. O n e c o n v e n t i o n s p e c i f i e s that the v a l u e of the w a v e f u n c t i o n at
i n f i n i t y be g r e a t e r t h a n zero, the o t h e r c o n v e n t i o n that the w a v e f u n c t i o n
n e a r z e r o be p o s i t i v e . S i n c e t w o - n u c l e o n t r a n s f e r c a l c u l a t i o n s d e p e n d
u p o n the r e l a t i v e m a g n i t u d e s an d p h a s e s of the v a r i o u s s p h e r i c a l c o n f i g u
r a t i o n s , e x p r e s s i o n s an d f u n c t i o n v a l u e s m u s t a l w a y s be kept in the s ame
c o n v e n t i o n . T h i s a p p e n d i x lists s e v eral s t a n d a r d p a p e r s in the f i eld
a n d g i v e s t h e i r s i g n c o n v e n t i o n s .
W a v e f u n c t ion Pos i t i ve N e a r O r i g i n
P.M. M o r s e a n d H.F. F e s h b a c k , M e t h o d s of T h e o r e t i c a l P h y s i c s
( M c G r a w - H i 11, N.Y. , 1953).
A. D e S h a l i t a n d I. T a l m i , N u c l e a r Shell T h e o r y ( A c a d e m i c P r e s s ,
N.Y., 1963).
B.E. Chi, N u cl. Phys. 83 (1966) 97.
W a v e f u n c t ion Pos i t i ve at Infini ty
S.G. N i l s s o n , Mat. Fys. Medd. Dan. Vid. Selsk. 29^ (1955) 35-
J.P. D a v i d s o n , C o l l e c t i v e M o d e l s of the N u c l e u s ( A c a d e m i c Pres s ,
N . Y., 1963).
Program DWUCK, wri t ten by P.D. Kunz, unpublished.
178APPENDIX B
Spli t t i n g o f (p,t) T r a n s f e r S t r e n g t h on O d d - M a s s N u c l e i
T h e t r a n s i t i o n m a t r i x e l e m e n t for the (p,t) r e a c t i o n
(J. K. a.) - (J K f a f ) is g i v e n by
M (J. K. a., J f K f o f ;R) 5 (2J. + I ) ' 1 x (B-l)
x I |<J. M. K. a. 1A j M (R) | J f M f K f a f >
J M. M M.I 1
w h e r e a r e p r e s e n t s a n y a d d i t i o n a l q u a n t u m n u m b e r s n e c e s s a r y to s p e c i f y
the stat e . U p o n a p p l i c a t i o n o f the W i g n e r - E c k h a r t t h e o r e m , this m a y
be w r i t ten
H - (2 J . ♦ I ) ’ 2 H I <J, H. K c. | [A* |J Kf 0 f >] | 2J M. 1 1 1 1 i i (B-2)
w h e r e t he b r a c k e t s d e n o t e v e c t o r c o u p l i n g
[Aj | J f K f nf >] j I < J f Hf J M | J. M. > AJH | J f Mf Kf a f
" "f (B-3)R e f e r e n c e A s 7 2 a g i v e s the e x p r e s s i o n for the r e q u i r e d m a t r i x e l e m e n t
in e q u a t i o n B-2 w i t h i n the a d i a b a t i c a p p r o x i m a t i o n .
f a ' f = < J. M. K. a. | C A j | J f K f a f >Dj M _' j _ i i i > J 1 i i (B-4)
= ( - ) J + K [ (1 + «K>Q) (1 + 6k 0) T " {< J i ' Ki J K 1 J f ‘ Kf > X 1 f
x f ,'n t r (a K. a, K,) 6 (K = K.- K,) + (-) J f + K f < J. -K. J K | J f K f > xJ l i l t " I T I I T T
r in t r / „ ,x fj (a. K. Qf - K f ) fi(K = K. + K f )
F or t r a n s i t i o n w i t h i n a s u p e r f l u i d b a n d ( (a^ = a. = a) a nd
(K. = Kj. = K)) the s e c o n d t e r m (K = 0) c a n be n e g l e c t e d
M = l \ H J <J- -K J 0 | J , -K > f ,ntr (aK aK) I2 J ' r j
(2 J f + 1 ) . 2
= (2J. + 1 ) I 1 (_) < J f K J 0 | j . K . > f ( a K aK)|i J
For J. = 5 = K o n l y o n e v a l u e o f J c an c o n t r i b u t e d u e to a n g u l a r
m o m e n t u m c o u p l i n g a n d p a r i t y c o n s i d e r a t i o n s . In p a r t i c u l a r ,
180
A P P E N D I X C
C o m p i 1 at ion o f A n g u 1ar Di s t r i b u t i o n s
T h e a n g u l a r d i s t r i b u t i o n s for s t a t e s p o p u l a t e d in the v a r i o u s
(p,t) e x p e r i m e n t s a r e show n . E a c h a n g u l a r d i s t r i b u t i o n is l a b e l l e d
"A - B" w h e r e A is the m a s s n u m b e r of the r e s i d u a l n u c l e u s a nd
B is a p e a k n u m b e r ( c o n s i s t e n t w i t h th e n u m b e r s in T a b l e s
t h r o u g h 5- 8 a n d F i g u r e s A-l t h r o u g h k ~ 5 ) • T h e e r r o r b a r s s h o w n
a r e s t a t i s t i c a l w i t h e s t i m a t e d f i t t i n g e r r o r s , w h e r e a p p r o p r i a t e .
W h e n not s h o w n , the e r r o r bars a r e s m a l l e r t h a n the p o i n t s .
(JS/q rl) (W
30)
©c.m (degrees)
10°
I I I I I I ' 182-
• •
J I I I L _ l L.
£-O
6©b ioP
"I—I—'—I—I—I—r—r 182-6
i J i I l! I i l_
-i—|—i—|—i—|—r 182-
• *- ♦
_i 1 i L_i I L20 4 0 6 0 80 20 4 0 6 0 8 0 2 0 4 0 6 0 8 0
© c m ( d e g r e e s )20 4 0 6 0 8 0 20 4 0 6 0 8 0
“i i 1 i 1 i ' r 182*16
_i I i I i I i L
JO
b
10'
1—1—i—'—i—1—r 182-21
J i I i I i L
10°
-i—i—'—i— '—r 182-26
20 40 60 80 20 40 60 80 20 4 0 60 8 0© c m ( d e g r e e s )
20 4 0 60 80 20 4 0 60 80
cr(©
cm)(
/xb/
sr)
© c m ( d e g r e e s )
-I—I—I—I— I—I—I—181-4
• •
kP
"i i 1 i 1 i 1 r 181-9
J I I L.
10°
->—i—1—i—'—i—1—r 181-14
* *4
i i i i i i -20 4 0 6 0 80 20 4 0 6 0 80
© c m ( d e g r e e s )
rt. 20 40 60 80
20 4 0 6 0 80©Cm (degrees)
p i n - i - r t i i ■ i.
180-1 '
_ ♦_Z i I 1 I 1 I I L =
.Qj l
'~EU
b
—I—'—I—1—I—1 TI 180-6 * ♦ ♦
ol
i ■ i ■ i . i
p-1 i 1—t -1—i—1 m180-11
♦
20 4 0 6 0 8 0 2 0 4 0 6 0 8 0© c m ( d e g r e e s )
2 0 4 0 6 0 80 20 4 0 60 80® cm ( d e g r e e s )
© c m ( d e g r e e s )
20 40 60 80
10*
o'
pn i i i i i i n178-1
-i I i I i i i r
cn\_Q
©b
CZl I I I I I I TP
I— * » ♦1 7 8 -6 _
I—i I i I i I i I-
l_i i i i r r i “i.
178-11 '
o°
4
)~i I I I I I I U20 4 0 60 80 20 4 0 6 0 80
J i i , i ■ i
pr—r
o'
1— '— r
-i—I_i I . i . i
_1 I I I l I I 1 I • 173-8♦ * « ,
L_i I I l I I I !_□
10°b i b Vi 1 7 8 -9
bu I i i i i i a bu l i I i i i nd
20 4 0 6 0 ■ 8 0® crtl ( d e g r e e s )
20 4 0 6 0 80 20 4 0 6 0 80
cr(®
c J (y
u.b/
sr)
© c m ( d e g r e e s )
APPENDIX D
C a l c u l a t i o n s o f the (p,t) S t r e n g t h of S t a t e s~ " ■ 1 1 1 C - — — -
W i t h i n the Os a p p r o x i m a t i o n the (p,t) d i f f e r e n t i a l c r o s s
s e c t i o n f or the g r o u n d s t a t e is g i v e n by (Be 72)
a (0+ ) = ( I Sa~ U V 1x q' L v v v
2
w h e r e is th 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 for p i c k u p of a d i
n e u t r o n f r o m the s i n g l e - p a r t i c l e o r b i t a l v a n d V a n d U a r e thev v
f u l l n e s s a n d e m p t i n e s s f a c t o r s for the level v. 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 for an e x c i t e d 0 + s t a t e (0*) g i v e n by
|0n > ' <5 + I B" % 1 0 >
w h e r e A + is the q u a s i p a r t i c l e c r e a t i o n o p e r a t o r for the s t a t e v,
I 0 > is the g r o u n d s t a t e w a v e f u n c t i o n a n d A n a n d B n a r e n u m e r i -' v v
cal c o e f f i c i e n t s w h i c h g i v e the s t r u c t u r e of the s t a t e , is
o ( 0+) = [) o (An V 2 + B n U 2 ) ] 2n i £ v v v v v
v
A s s u m i n g that the Q s t a t e s c o n s i s t of t wo ( q u a s i p a r t i c l e )
h o l e s in the f o u r p r o l a t e l e v e l s w h i c h a r e n e a r l y d e g e n e r a t e
in e n e r g y , a r e a s o n a b l e a s s u m p t i o n w o u l d be that the d i n e u t r o n
h o l e is s h a r e d e q u a l l y a m o n g t he f our l e v e l s , i m p l y i n g A ^ = 4
for v = (one of the f o u r p r o l a t e levels). If the b a c k w a r d -
g o i n g a m p l i t u d e s B^ a r e n e g l e c t e d , then
° (Q c } * ( 1 ° (4) V 2 )2
v
T h e v a l u e s of a c a n be t a k e n f r o m f i g u r e 5 - * a n d the V - f a c t o r sv
182f r o m t he BCS c a l c u l a t i o n for W (p,t), g i v i n g the result:
o(Q ) £_ = 0. 1o(0*>
182
T h i s v a l u e c o u l d be s o m e w h a t h i g h e r o r l o wer d e p e n d i n g u p o n the
v a l u e s o f A n u s ed, but r e p r e s e n t s a r e a s o n a b l e e s t i m a t e o f the
(p,t) s t r e n g t h o f the Q s t ates.c
183
C a l c u l a t i o n of B r o k e n - P a i r C o n f i g u r a t ion (p,t) S t r e n g t h o f 0 +
S t a t e s
W i t h i n t he s p a c e o f t he 13 o r b i t a l s s h o w n in F i g u r e 2-2a,
o n l y two s e n i o r i t y n o n - z e r o ( b r o k e n - p a i r ) t w o - n e u t r o n c o n f i g u r a t i o n s
c a n c o u p l e to g i v e J ^ K = 0 + 0; t h e s e a r e the ( 1 / 2 [ 5 1 0 ] , 1/ 2 [521])
an d ( 7 / 2 [ 5 0 3 ] , 7 / 2 [ 5 l 4 ] ) c o n f i g u r a t i o n s . T h e i r i n t r i n s i c (p,t)
s t r e n g t h s as c a l c u l a t e d w i t h E q u a t i o n 3.11 u s i n g N i l s s o n c o
e f f i c i e n t s f r o m D a v i d s o n (Da 68) a r e 1.1 a n d 4 . 8 a r b i t r a r y u n i t s ,
r e s p e c t i v e l y , as o p p o s e d to a v a l u e o f 22 for t he a v e r a g e i n t r i n s i c
d i n e u t r o n p i c k u p s t r e n g t h of the 13 o r b i t a l s n e a r the Fermi s u r
f a c e c a l c u l a t e d u s i n g t he s a m e p r e s c r i p t i o n . It is thus c o n
c l u d e d t h a t b r o k e n - p a i r t w o - n e u t r o n p i c k u p is n o t i m p o r t a n t in
p o p u l a t i n g the l o w - l y i n g J 17 = 0 + s t a t e s in t u n g s t e n , e s p e c i a l l y
w h e n it is c o n s i d e r e d tha t the 1/ 2 [5 2 1 ] a n d the 7/ 2 [ 503] a r e
d e e p lying h o l e s t a t e s w h i l e the 1/2 [ 510] a n d 7/ 2 [ 503] a r e a b o v e
the Fermi s u r f a c e .
T h e a b o v e a r g u m e n t m a k e s no m e n t i o n o f the p r o t o n d e g r e e
o f f r e e d o m w h i c h m ay, or m a y not, p l a y a rol e in the s t r u c t u r e
o f a n y g i v e n e x c i t e d s t a t e .
APPENDIX E
184
A P P E N D I X F_
T h e F a v o r e d L = 2 T rans i t ions i n the 48lW ( p , t ) R e a c t ion
W h i l e the D W B A p r e d i c t s t h a t the s h a p e s o f the a n g u l a r
d i s t r i b u t i o n s for t r a n s i t i o n s to the 3/2 1/2 - [510] a n d
5/2 1/2 - [510] s t a t e s wil l be the s a me, e x p e r i m e n t a l l y a l a rge
d i f f e r e n c e at f o r w a r d a n g l e s is a p p a r e n t . S u c h a d i f f e r e n c e c a n be
t r a c e d to the d i f f e r e n c e s b e t w e e n the m u l t i p l e - s t e p r o u t e s w h i c h
c o n t r i b u t e to the p o p u l a t i o n o f t h e s e s t a t e s . T h e v a r i o u s r o u t e s
wil l be d i s c u s s e d in this a p p e n d i x a f t e r a b r i e f d e s c r i p t i o n o f the
f o r m a l i s m n e c e s s a r y to e x p l a i n the d i f f e r e n c e s . In the f o l l o w i n g
d i s c u s s i o n , o n l y t w o - s t e p p r o c e s s e s will be c o n s i d e r e d .
For an o p e r a t o r
= [aL 1 x b L 2 ] 8
k ] k.
w h e r e a 8 a n d b^ a r e i r r e d u c i b l e t e n s o r o p e r a t o r s w h i c h o p e r a t e o n
the s a m e c o o r d i n a t e s , it c a n be s h o w n t h a t (Des63) ,
r(J K a | | T L | | J 1 K 1 a 1) = (-)J + L + L ‘ (2L + 1 ) 1/2 \ L 1 L 2 L
• b ,1 J 1 J jJ k a"[_
x (J K a | [ a E 1 | j j k a " ) (j k a " | | b * " 2 | | j ' K' a') (F.l)
W i t h this e x p r e s s i o n , a n d e x p r e s s i o n for the tw o i n d i v d i u a l r e
d u c e d m a t r i x e l e m e n t s , the t w o - s t e p c o n t r i b u t i o n s to the p o p u
l a t i o n o f the s t a t e s c a n be i n v e s t i g a t e d .
T h e i n e l a s t i c f o r m f a c t o r for a t r a n s i t i o n o f m u l t i p o l a r i t y
L is (Ki 73)
185
(J K a | | Q L | | J 1 K' a') = (2J + 1 ) 1/2 (-)J ' J ' < J K L 0|J' * ' > > (F.2)
w h e r e th e A K = 0 c o n t r i b u t i o n (the s i n g l e - p a r t i c l e c o n t r i b u t i o n )
has b e e n n e g l e c t e d a n d w h e r e V|_q » d e f i n e d in R e f e r e n c e Ki 73.
c o n t a i n s the radial i n f o r m a t i o n o f the r e d u c e d m a t r i x e l e m e n t .
For the c a s e o f t w o - n u c l e o n t r a n s f e r , n e g l e c t i n g the
K = 0 c o n t r i b u t i o n g i v e s
(J K a | | A * | |J ' K' a 1) = (2J + 1 ) 1/2 ( - ) J " J '
i ntr
x < J K L 0 | J ' K ' > f L ( a K a 1K 1) (F/3)
(see E q u a t i o n B .4).
C o n s i d e r i n g o n l y the t w o - s t e p r o u t e s which i n v o l v e an
i n e l a s t i c e x c i t a t i o n o f m u l t i p o l a r i t y t wo a n d a t r a n s f e r of
m u l t i p o l a r i t y z e r o ( s ince f g 0 * 1" >> f ^ ^ i s e e *'* 73) leads to
the f o l l o w i n g total r e d u c e d m a t r i x e l e m e n t s for t r a n s i t i o n s to the
3/2 a n d 5/2 s t a t e s :
R M E (3/2-) = - ( 4 / 5 ) 1 7 2 (f2ntr + V 2Q + • • •
(F.4)
R M E (5/2-) * + ( 6 / 5 ) 1 / 2 (f2Pt r + V 20 f ^ t r ) + . . •
So that in thi s a p p r o x i m a t i o n , the s h a p e s a r e still the same,
the o n l y d i f f e r e n c e b e i n g in the m a g n i t u d e s o f the c r o s s s e c t i o n s .
W h e n the n e x t h i g h e r o r d e r t e rms a r e i n c l u d e d , h o w e v e r ,
the s i t u a t i o n c h a n g e s :
R M E (3/2-) = - ( 4 / 5 ) , / 2 ( f’ntr + V 2 Q f ' ntr + a V 2Q f * n t r ) + . .
(F.
RME (5/2-) = + ( 6 / 5 ) 1 / 2 ( fJntr + V 2 Q f ’ntr + b V 2Q f ’n t r ) + . .
If a = b t hen the t wo a n g u l a r d i s t r i b u t i o n s w o u l d h a v e s i m i l a r
s h a p e s , h o w e v e r this is not the c a se. I n s t e a d it is f o u n d
a = 1 3 / (5/P*)
b = - ( 6 / 3 j / ( 5 / 7 )
so that for o n e s t a t e , t he last t e r m wil l a d d c o n s t r u c t i v e l y to
the rest of the f o r m f a c t o r w h i l e for the o t h e r s t a t e , it w ill
g i v e d e s t r u c t i v e i n t e r f e r e n c e .
T h e o b s e r v a t i o n o f d i f f e r e n t a n g u l a r d i s t r i b u t i o n s h a p e s
for w h a t in the D W B A w o u l d be c o n s i d e r e d s i m p l e L = 2 t r a n s f e r s ,
then, s h o u l d a l l o w the d i r e c t i n v e s t i g a t i o n o f mu 1 1 i p l e - s t e p
i n e l a s t i c r o u t e s in (p,t) r e a c t i o n s . S u c h a s t u d y , a l t h o u g h
not u n d e r t a k e n h e r e , s h o u l d c o n s t i t u t e a v e r y i n t e r e s t i n g a n d
r e w a r d i n g p r o j e c t . It s h o u l d be n o t e d that the c a s e c o n s i d e r e d
here, w h e r e the g r o u n d s t a t e o f the t a r g e t n u c l e u s h a s J = 1/2,
is o p t i m u m for s u c h a s t u d y s i n c e t he d i r e c t L = 2 t r a n s f e r
s t r e n g t h s p l i t s b e t w e e n o n l y t wo s t a t e s .
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