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Electric dipole transitions in 88Sr and 90Zr
Volume 35B, number 2
PHYSICS
ELECTRIC
DIPOLE
LETTERS
TRANSITIONS
10May 1971
IN
88Sr
AND
90Zr
J. D. VERGADOS and T. T. S. KUO Physics Department *, State University of New York, Stony Brook, N,Y. 11790, USA Received 1 April 1971
Electric dipole transitions in 88Sr and 90Zr have been studied using realistic two-body interactions. Shell model wave functions of definite isospin were used and both the T-upper and the T-lower transitions were calculated. In the case of 90Zr the pairing effect in its ground state was taken into account and, in addition to the ground state E 1 widths Fy0 , the first excited state widths F y 1 were obtained. The results are in good agreement with experiment. Within the f r a m e w o r k of the s h e l l m o d e l the d i p o l e s t a t e s a r e w e l l d e s c r i b e d a s p - h e x c i t a t i o n s on the g r o u n d and l o w - l y i n g e x c i t e d s t a t e s . F o r c l o s e d s h e l l n u c l e i the d i p o l e s t a t e s w e r e t r a d i t i o n a l l y c o n s i d e r e d a s l p - l h s t a t e s . H o w e v e r it h a s b e e n shown that the l p - l h b a s i s is not a d e q u a t e to g u a r a n t e e good i s o s p i n f o r n u c l e i that h a v e e x c e s s n e u t r o n s (N > Z). S i n c e the d i s c o v e r y of i s o b a r i c a n a l o g s t a t e s r e v i v e d i s o s p i n a s a m e a n i n g f u l q u a n t u m n u m b e r [1-6], s p e c i a l a t t e n t i o n m u s t be p a i d to i s o s p i n . It has b e e n shown that an a p p r o p r i a t e s e t of 2 p - 2 h s t a t e s is n e c e s s a r y in o r d e r to g u a r a n t e e good i s o s p i n [1]. Such e x c i t a t i o n s w i l l be i n c l u d e d w o r k i n g with a b a s i s of good i s o s p i n in the s p i r i t of r e f s . [1] and [2]. In the p r e s e n t i n v e s t i g a t i o n we w i l l f o c u s o u r a t t e n t i o n on the d i p o l e s t a t e s of 88Sr and 9~0Zr. T h e s e n u c l e i h a v e b e e n the s u b j e c t of v e r y e x t e n s i v e e x p e r i m e n t a l r e s e a r c h in r e c e n t y e a r s [6-10]. T h e o r e t i cal c a l c u l a t i o n s [1,8] a r e a l s o a v a i l a b l e f o r t h e s e n u c l e i that a r e in r e a s o n a b l y good a g r e e m e n t with e x p e r i m e n t . H o w e v e r t h e s e c a l c u l a t i o n s u s e d p h e n o m e n o l o g i c a l i n t e r a c t i o n s . T h e r e f o r e it is i n t e r e s t i n g to i n v e s t i g a t e w h e t h e r r e a s o n b l y r e a l i s t i c i n t e r a c t i o n s can r e p r o d u c e the d i p o l e s p e c t r u m . F o r t h i s p u r p o s e we u s e d the K u o - B r o w n m a t r i x e l e m e n t s d e r i v e d f r o m the H a m a d a - J o h n s t o n p o t e n t i a l . The c h o i c e of the 1 - b o d y H a m i l t o n i a n is not unique. If one i n c l u d e s e x p l i c i t l y the e x c e s s n e u t r o n s into the c a l c u l a t i o n s one can s t a r t f r o m a s e t of s i n g l e p a r t i c l e e n e r g i e s i d e n t i c a l f o r n e u t r o n s and p r o t o n s . H o w e v e r if the i n t e r a c t i o n of the " a c t i v e " p a r t i c l e s w i t h the e x c e s s n e u t r o n s is i g n o r e d , which g i v e s r i s e to the s y m m e t r y p o t e n t i a l , one b u i l d s the s y m m e t r y e n e r g y into the p r o b l e m f r o m the s t a r t by a s s i g n i n g s i n g l e p a r t i c l e e n e r g i e s w h i c h a r e d i f f e r e n t f o r n e u t r o n s and p r o t o n s . H e r e we adopted the l a t t e r a p p r o a c h and o u r 1 - b o d y H a m i l t o n i a n can be w r i t t e n a s f o l l o w s : h=~{[en(j)-
j
,
u(j)
~U(j)] +--t(j)"
To
To}+h c
w h e r e j r u n s o v e r a l l o r b i t a l s h c is the a v e r a g e 1 - b o d y c o u l o m b p o t e n t i a l , T o is the i s o s p i n of the c l o s e d s h e l l , U(j) i s the s y m m e t r y e n e r g y f o r o r b i t j and E n ( j ) a r e the s i n g l e p a r t i c l e e n e r g i e s f o r n e u t r o n s . A l t h o u g h the a b o v e H a m i l t o n i a n i s e a s y to handle [2], in p r a c t i c e we m a d e the a p p r o x i m a t i o n [1]
U(j) = U = 6
MeV
The s i n g l e p a r t i c l e e n e r g i e s f o r n e u t r o n s w e r e t a k e n e s s e n t i a l l y f r o m e x p e r i m e n t [1,2]. T h e y w e r e c h o s e n to be - 12.2, - 8.5, - 7.7, - 6.2, - 5.4, 0, 1, 2.5, 3.0, 5.0 M e V f o r the l f 7 / 2 , l f 5 / 2 , 2P3/2 , 2 P l / 2 , l g 9 / 2 , 2d5/2, 3 S l / 2 , l d 3 / 2 , l g 7 / 2 , lh11/2 o r b i t a l s r e s p e c t i v e l y . The d e t a i l s about i s o s p i n and p a r t i c l e - h o l e c o n v e n t i o n s can be found e l s e w h e r e [2]. F o r the c l o s e d s h e l l n u c l e u s we d i s t i n g u i s h t h r e e t y p e s of s t a t e s (i) P r o t o n p a r t i c l e in the e x c e s s r e g i o n (e.g. h e r e 2 P l / 2 , lgg/2 o r b i t a l s i n d i c a t e d by a ~ on the top of the s i n g l e p a r t i c l e q u a n t u m n u m b e r )
* Work supported by the U.S. Atomic Energy Commission. 93
V o l u m e 35B. n u m b e r 2
PHYSICS
LETTERS
10 May 1971
0 (ii) N e u t r o n h o l e in the e x c e s s r e g i o n X
¢2 :~-0:
I(Jl ~'l)J
it is e a s y to v e r i f y that the a b o v e s t a t e s h a v e d e f i n i t e i s o s p i n T = T< = T O = ½(N- Z) by o p e r a t i n g with the o p e r a t o r T+ (iii) Both the " a c t i v e " p a r t i c l e and the " a c t i v e " hole a r e o u t s i d e the e x c e s s r e g i o n . T h e n we apply the r u l e s of c o n s t r u c t i n g good i s o s p i n w a v e f u n c t i o n s w h i c h a r e known f r o m the t h e o r y of N = Z n u c l e i ( r e m e m b e r i n g that the c o r e has i s o s p i n d i f f e r e n t f r o m z e r o ) .
3,T
In the a b o v e d i a g r a m both the i s o s p i n and the a n g u l a r m o m e n t u m c o u p l i n g i s i n d i c a t e d . ( T = T< = T O o r T = T .... T0 + 1.) It m u s t be noted t h a t f o r a g i v e n s e t of s p a c e q u a n t u m n u m b e r s of c l a s s (iii) t h e r e a r e t w i c e a s m a n y T . s t a t e s ( c o r r e s p o n d i n g to t = 0, 1). T h a n t h e r e a r e T> s t a t e s (which a r i s e only f r o m ! = 1). It is e a s y to s e e that the s t a t e s c o r r e s p o n d i n g to t = 0 a r e of l p - l h n a t u r e . H o w e v e r in the c a s e of / = 1 the w a v e f u n c t i o n c o n t a i n s 2 p - 2 h c o m p o n e n t s . It is e a s y to v e r i f y that:
i(JlJ21) JI;TO;T
= T O + 1) = ~
1
/-To,
l(jlj~l)jtmt = 0) 1TOTO) + v-T~l(jlJ21)Jtmt
= 1)j T O T O - 1)
and
I(j]j~I)JI;To;T : where
TO):-
T/T~ I(jlj~l)Jtm/=O)] ToTo ) + ~
[ T o T 0) r e p r e s e n t s t h e c l o s e d s h e l l n u c l e u s and the s t a t e 1~ by o p e r a t i n g w i t h the o p e r a t o r T_, i.e.
ToT0 -
ToT 0 -
1)
-
T V2T0 I T o T o )
=~.~gi[]i(p)];l(n)]J=
](jlj~l)Jtmt = 1 ) ] T o T 0 IToTo-
1)
1) i s o b t a i n e d [2] f r o m the
O)
w h e r e i r u n s o v e r all the e x c e s s o r b i t a l s and gi = V ~ O " F r o m t h i s d i s c u s s i o n it i s c l e a r that the s t a t e s c l a s s (iii) with l = 1 c o n t a i n 2 p - 2 h c o m p o n e n t s . If one p r o c e e d s one s t e p f u r t h e r and e x p a n d s t h e s e s t a t e s in the p - n r e p r e s e n t a t i o n it is e a s y to s e e that the above s t a t e s a r e i d e n t i c a l w i t h t h o s e d i s c u s s e d in r e f . [1]. It is w o r t h noting that f o r h e a v y n u c l e i with m a n y e x c e s s n e u t r o n s s t a t e s of c l a s s (iii) b e c o m e l e s s i m p o r t a n t and the l p - 2 h m o d e l is a d e q u a t e to e x p l a i n the d i p o l e s t a t e s (e.g. 208pb) so t h a t in t h i s c a s e one can u s e d i r e c t l y the p - n r e p r e s e n t a t i o n . The m a t r i x e l e m e n t s of the H a m i l t o n i a n b e t w e e n s t a t e s that a r e of l p - l h n a t u r e is t r i v i a l . The r e s t of the m a t r i x e l e m e n t s can be c a l c u l a t e d m o s t s i m p l y by. m a k i n g u s e of t h e f a c t that the n u c l e a r H a m i l t o n i a n c o m m u t e s w i t h the i s o s p i n o p e r a t o r s , i.e. [T±, H N] = 0. H e n c e [2]
<,~,1IHN}~)
=
/t+ TO
(_)r v ~ ,
((~j~l)jt] VI ( j j
j~-l)
Jr)
/t+T^ _ ( ~ 2 1 H N I ¢ ~ ) : (_)t+l / ~ T 0 u ( ( j l j ~ l ) j t t VI ( j i j~.-1)jt)
(4~31HN]~) 94
=( (j lJ 21)jt]HN I (ji j~- l )Jt )
Volume 35B, number 2
PHYSICS
LETTERS
10 May 1971
T h e a b o v e f o r m a l i s m s t r i c t l y s p e a k i n g a p p l i e s only w h e n the s t a t e iToT O} is a c l o s e d s h e l l n u c l e u s . The f o r m a l i s m a p p l i e s a l s o in the c a s e when the s t a t e I TO T0} is not a c l o s e d s h e l l n u c l e u s p r o v i d e d that one t a k e s P a u l i p r i n c i p l e c o r r e c t i o n s into a c c o u n t f o r c l a s s e s (i) and (ii). In the p r e s e n t i n v e s t i g a t i o n we f o c u s o u r a t t e n t i o n on 88Sr and 9 0 Z r . In the c a s e of 88Sr the g r o u n d s t a t e iT0 TO } w a s a s s u m e d to be c l o s e d s h e l l with the P3/2 p r o t o n s h e l l and the g9/2 n e u t r o n s h e l l c o m p l e t e l y f i l l e d . T h e n the c a l c u l a t i o n p r o c e e d e d a s it was i n d i c a t e d above. F o r the c a s e of 9 0 Z r two a p p r o a c h e s w e r e a d o p t e d . In the f i r s t a p p r o a c h (I) we a s s u m e d that the g r o u n d s t a t e I TO T0} of 9 0 Z r is a c l o s e d s h e l l with the P l / 2 p r o t o n o r b i t a l b e i n g f i l l e d and the c a l c u l a t i o n p r o c e e d e d a s in the c a s e of 88Sr. A l t h o u g h , a s we s h a l l s e e b e l o w , the a g r e e m e n t with the e x p e r i m e n t is r a t h e r good, we know that t h i s d e s c r i p t i o n is i n c o m p l e t e . The ground s t a t e o f 90 Z r is " co r r e 1ate d . It i s r a t h e r w e l l d e s c r i b e d as ap2/2 + bg2/2 with a2:b 2 = 2 . 2 " . T h e r e f o r e in the c a s e of T = T~ : 6 we p e r f o r m e d a c a l c u l a t i o n (II) w h i c h t a k e s t h i s a f f e c t into a c c o u n t . The b a s i s w a s e x p a n d e d to i n c l u d e the following: T X Xl = ~ / ~ 0 I q)..};~ 4.~ = [[jl (p) J2(n)]J} = ~X[
X2 = (j12(p)) 0 I[J2(n)y31(n)] J)
Yl ¢ J 3
× 3 =)'2(P) 0 [[;2(P)j31(p)] J )
71 ¢ ~ 2
×4 In the a b o v e e x p r e s s i o n s ) ' i n d i c a t e s an o r b i t in the e x c e s s r e ~ ! o n , (p) and (n) stand f o r p r o t o n and n e u t r o n r e s p e c t i v e l y , T O i s the i s o s p i n of the c l o s e d s h e l l ( h e r e ° 8 S r ) and 4'3 is the s t a t e of c l a s s (iii) d e s c r i b e d a b o v e . T h e a b o v e b a s i s m e a n s that e f f e c t i v e l y we c a l c u l a t e the T ~ d i p o l e s t a t e s o b t a i n e d by e x c i t i n g both the g r o u n d and f i r s t e x c i t e d 0 + s t a t e s . So we a r e in a p o s i t i o n to c a l c u l a t e the d i p o l e t r a n s i t i o n s both to the g r o u n d (Y0) and f i r s t e x c i t e d s t a t e (~'1). S i n c e , a s we s h a l l s e e b e l o w , the d i p o l e e m i s s i o n s p e c t r u m f o r 70 t r a n s i t i o n s d o e s not i m p r o v e c o m p a r e d with e x p e r i m e n t by u s i n g the e x p a n d e d b a s i s (II), we did not a t t e m p t to p e r f o r m t h i s c a l c u l a t i o n in the c a s e of T = T< = 5. On the o t h e r hand in the c a s e of T = T> = 6 we p e r f o r m e d two t y p e s of c a l c u l a t i o n . In the f i r s t c a l c u l a t i o n (II1) we u s e d the b a r e G - m a t r i x e l e m e n t s e v e r y w h e r e . H o w e v e r the c o r e p o l a r i z a t i o n c o n t r i b u t i o n f r o m the d i a g r a m :
i s v e r y i m p o r t a n t f o r the p r o t o n - p r o t o n i n t e r a c t i o n . H e n c e in the s e c o n d c a l c u l a t i o n (II2) s u c h c o r e p o l a r i z a t i o n s w e r e i n c l u d e d f o r the p r o t o n - p r o t o n i n t e r a c t i o n in a d d i t i o n to the b a r e G - m a t r i x e l e m e n t s . T h e r e s u l t s of the c a l c u l a t i o n can be s u m m a r i z e d a s f o l l o w s : A. 88Sr. In the c a s e of 8 8 S r the Giant D i p o l e R e s o n a n c e (G.D.R.), c h a r a c t e r i z e d by T = T< = T 0 = 6, i s s o m e w h a t f r a g m e n t e d ( s e e fig. 1). The m a i n p e a k is p r e d i c t e d at 15.61 MeV with width F v ; 15.46 keV. T w o s e c o n d a r y p e a k s a r e p r e d i c t e d at 12.18 M e V with width F.y = 3.47 k e V and at 18.00 M e V with width 5.58 keV. The a v e r a g e of t h e s e p e a k s is l o c a t e d at 15.7 MeV and c o i n c i d e s w i t h the p o s i t i o n of the m a i n p e a k . T h i s a g r e e s q u i t e w e l l with a p r e v i o u s c a l c u l a t i o n [1]. In (7,n) r e a c t i o n s [10] a good L o r e n t z i a n f i t w a s o b t a i n e d f o r the c r o s s - s e c t i o n w i t h E x = 16.7 M e V and width F = 4.3 MeV. T h e s e r e s u l t s a r e in a g r e e m e n t with t h o s e of 8 7 R b ( p , v ) 8 8 S r r e a c t i o n [6]. In the c a s e of T = T> = T 0 + 1 = 7 t r a n s i t i o n s s t r o n g p e a k s a r e p r e d i c t e d at 15.49 M e V w i t h width I'.g = 1.10 keV, at 19.25 M e V w i t h width F~/ = 3.65 k e V and at 23.43 M e V w i t h width r ~ = 1.31 keV. E x p e r i m e n t a l l y [6] in 87Rb(p,'y)88Sr r e a c t i o n s one s e e s two r e s o n a n c e s at E x = 15.7 and 16,0 MeV, a s t r o n g r e s o n a n c e at E x = 17.2 MeV, a s h a r p r e s o n a n c e at E x = 17.8 MeV, a b r o a d s t r u c t u r e w i t h s o m e f i n e s t r u c t u r e at 19.2 M e V and a v e r y p r o n o u n c e d s t r u c t u r e a r o u n d 21 MeV. On the o t h e r hand (7',p) e x p e r i m e n t s [7] s e e m to i n d i c a t e the e x i s t e n c e of 4 p e a k s at 16.5, 17.5, 21.0, and 23.0 MeV. A l t h o u g h a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n c a l c u l a t i o n and e x p e r i m e n t i s not p o s s i b l e , the a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t is good. In p a r t i c u l a r we note the p r e s e n c e of a p e a k at a r o u n d 23 MeV. T h i s i s in a g r e e m e n t with the f a c t that a p e a k at r o u g h l y t h i s * The first excited 0+ state is - b p 2 / 2
+ ag2/2. 95
Volume
35B, number
2
PHYSICS
LETTERS
10 M a y 1 9 7 1
6'
Zre°( i-..~0 + ) (T)
Sr~(l=.,~O +)
G.D,R. X I/~
5
4
>=3
3
2
2
II IOI
15 Ex
i
L
2O
MeV
Fig. 1. Calculated widths for dipole transitions 1- ~ 0+ in 88Sr. Widths corresponding to T> transitions are indicated by dotted lines with the letter A on the top. G.D.R. stands for Giant Dipole Resonance.
,I I0
Ex
J~ MeY
20
' ~-
Fig. 2. Calculated widths for dipole transitions 1 - ~ 0+ in 90Zr (calculation (I) as explained in main text). The notation is the same as in fig..1.
e n e r g y has a l s o b e e n p r e d i c t e d [2] and o b s e r v e d [17] in 89y. The s t r e n g t h of t h e s e r e s o n a n c e s will p r o b a b l y be s u p p r e s s e d in (p,~) r e a c t i o n s at t h e s e e n e r g i e s since the e x p e c t e d Fp0 i s e s t i m a t e d to be s m a l l . In d e t a i l the analog s t a t e s a r e p r e d i c t e d at e x c i t a t i o n e n e r g i e s 15.07, 15.49, 16.12, 17.25, 17.91, 19.25, 20.15, and 23.43 MeV with widths 47, 1097. 38, 16, 126, 3652, 20, 1311 eV r e s p e c t i v e l y . The strong9 '~dip°le s t a t e at 19.25 MeV i s m a i n l y d 5 / 9 2f~ 2 (22%) and g T / 9 2f 5 2 (66%). B. u Z r . The T< G.D.R. i s p r e d i c t e d at E x = 15.6 MeV with width F. = 15.6 keV and i s somewhat f r a g m e n t e d (see fig. 2). This is in good a g r e e m e n t with e x p e r i m e n t [6].~ In the c a s e of T = T~ = 6 s t a t e s the c a l c u l a t i o n (I) p r e d i c t s t h r e e s t r o n g dipole s t a t e s at 15.77, 19.30 and 23.50 MeV with widths 1.49, 4.73, and 1.99 keV r e s p e c t i v e l y . E x p e r i m e n t a l l y [6] m o s t of the T> dipole s t r e n g t h is found in t h r e e r e g i o n s : Multiple fine s t r u c t u r e between E x = 16.4 and 17.2 MeV, s h a r p fine s t r u c t u r e at 19.2 MeV and a b r o a d s t r u c t u r e with evidence of fine s t r u c t u r e in the r e g i o n Ex = 20.8 MeV. On the o t h e r hand 9 0 Z r ( v , p ) 8 9 y e x p e r i m e n t s [12] indicate a p e a k at around 22 MeV. With the p o s s i b l e e x c e p t i o n of the p e a k p r e d i c t e d at 23.50 MeV the a g r e e m e n t between t h e o r y (I) and e x p e r i ment is v e r y good. Although the a g r e e m e n t between t h e o r y (I) and e x p e r i m e n t is good, the above c a l c u l a t i o n i g n o r e s the f a c t that the ground s t a t e of 9 0 Z r is c o r r e l a t e d . So a c a l c u l a t i o n was p e r f o r m e d that t a k e s into account such c o r r e l a t i o n s in the m a n n e r d e s c r i b e d above. Using the b a r e G - m a t r i x e l e m e n t s (HI) we obtained a n a l o g s t a t e s at e x c i t a t i o n e n e r g i e s 12.75, 13.31, 14.41, 15.39, 16.18, 16.52, 17.00, 17.27, 17.35, 17.47, 18.16, 18.37, 18.61, 19.45, 19.52, 21.10, 22.20, 22.93 and 23.95 with F T 0 w i d t h s 28, 22, 106, 1067, 84, 1, 2, 18, 7, 42, 194, 2027, 1538, 450, 26, 22, 80, 1165, and 407 eV r e s p e c t i v e l y . The c o r r e s p o n d i n g FT1 widths a r e 46, 1, 158, 0, 0, 1026, 16, 0, 122, 2, 844, 366, 105, 535, 7, 2888, 395, 290, 842 eV. On the o t h e r hand u s i n g r e n o r m a l i z e d G - m a t r i x e l e m e n t s (II2) we obtained analog s t a t e s at e x c i t a t i o n e n e r g i e s 13.14, 14.00, 15.08, 15.72, 16.25, 17.03, 17.19, 17.31, 17.71, 18.05, 18.72, 19.09, 19.41, 19.69, 19.88, 20.76, 21.61, 22.72, and 24.90 MeV. The c o r r e s p o n d i n g ground s t a t e widths a r e 71, 61, 338, 999, 44, 25, 5, 2, 27, 3, 4019, 690, 37, 128, 5, 17, 245, 1452, and 224 eV r e s p e c t i v e l y . The F y l w i d t h s a r e 30, 0, 90, 607, 9, 396, 161, 434, 3, 109, 737, 720, 10, 551, 87, 3200, 70, 519, 1115 eV. The p o s i t i o n of F~I widths in both c a s e s is shifted r e l a t i v e l y to the FV 0 widths by roughly the s e p a r a t i o n e n e r g y of the ground and f i r s t e x c i t e d 0 + s t a t e s (N 2 MeV). We a l s o note that in both c a l c u l a t i o n s (II1) and II2) the s t r o n g T> dipole state is l o w e r e d c o m p a r e d with the p r e d i c t i o n s of c a l c u l a t i o n (I). T h i s shift, which is about 1 MeV f o r c a l c u l a t i o n (II1) and 0.5 MeV f o r c a l c u l a t i o n (II2), m a k e s the a g r e e m e n t 96
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10 May 1971
Z r 9 ° (i- T>.-..O+ T<)
ry°
4-
(If2)
Zr gO ( I- T>--,,-0 +) (Ill)
>
>~
2-
i)
% '~ rr° r.
%1 ,~ ao Ex
, ,Ir'F I 2l
22
23
;'4
MeV
Fig. 3. Calculated widths for T> dipole transitions in 90Zr leading both to the ground (1~0) and to the first excited state ( I ~ 1 ) (calculation II1).
i
16
|
I'~
18
IT:
19
EX
20
i
21
22
23
24
MeV
Fig. 4. Calculated widths for T> dipole transitions in 90Zr leading both to the ground ( I~y0) and to the first excited state ( I ~ 1 ) (calculation II2).
b e t w e e n t h e o r y and e x p e r i m e n t w o r s e . (A s i m i l a r shift, but in the o p p o s i t e d i r e c t i o n , has a l s o b e e n found in the c a l c u l a t i o n s of r e f . [8].) T h i s p r o b a b l y i m p l i e s that o u r p - h i n t e r a c t i o n is r a t h e r weak. Our p r e s e n t c a l c u l a t i o n c o n f i r m s the e x p e c t e d [1,12] s p l i t t i n g of the G.D.R. into two g r o u p s c o r r e s p o n d i n g to the two i s o s p i n s T -- T< = T O and T = T> = T O + 1. Rough e s t i m a t e s b a s e d on the H a m i l t o n i a n of eq. (1) w i t h U ( j ) = U = cont. p r e d i c t the e n e r g y s e p a r a t i o n b e t w e e n T > and T < to be AZ = (To+ I)U/T 0
A l t h o u g h t h i s r e l a t i o n y i e l d s the c o r r e c t e n e r g y s e p a r a t i o n b e t w e e n the a n a l o g and i t s a n t i a n a l o g , it f a i l s to p r o v i d e the c o r r e c t s e p a r a t i o n e n e r g y b e t w e e n the c o l l e c t i v e T> and T< s t a t e s . It h a s b e e n s h o w n [12] t h a t in the l a t t e r c a s e t h i s r e l a t i o n s h i p m u s t be m o d i f i e d a s f o l l o w s : hE = (To+I)UD/T 0
UD -~ 0.6 U
T h e a b o v e f o r m u l a y i e l d s A E = 4.7 M e V f o r 88Sr and A E --~ 4.0 M e V f o r 9 0 Z r . Our c a l c u l a t i o n p r e d i c t s an a v e r a g e s e p a r a t i o n e n e r g y A E = 4.3 M e V and AE = 4.0 MeV r e s p e c t i v e l y , w h i c h is in r e m a r k a b l e a g r e e m e n t with the a b o v e e s t i m a t e s . In c o n c l u s i o n we can s a y that, in v i e w of the a p p r o x i m a t i o n s m a d e by the l i m i t e d s h e l l m o d e l s p a c e we c o n s i d e r e d and by the f a c t that we i g n o r e d c o m p l e t e l y the p r e s e n c e of the c o n t i n u u m s t a t e s , the r e a l i s t i c i n t e r a c t i o n s that we u s e d r e p r o d u c e r a t h e r w e l l the d i p o l e e m i s s i o n s p e c t r u m of n u c l e i n e a r closed shells. One of the a u t h o r s (J. D. V.) i s s p e c i a l l y i n d e b t e d to P r o f e s s o r S, F a l l i e r o s f o r e n l i g h t e n i n g d i s c u s s i o n s and to P r o f e s s o r G. E. B r o w n f o r his i n t e r e s t in the p r o b l e m .
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
98
B. Goulard, T. A. Hughes and S. F a l l i e r o s , Phys. Rev. 181 {1965) 134.% J. D. Vergados and T. T. S. Kuo, Nucl. P h y s . , to be published. H. Morinaga, Z. Physik 188 (1965) 182. S. F a l l i e r o s , B. Goulard and H. H. Venter, Phys. L e t t e r s 19 (1965) 398. J. L. Black and N. W. Tanner, Phys. L e t t e r s 11 (1965) 135. M. Hasinoff et al., Phys. L e t t e r s 30B {1969) 337. K. Shoda et al., Phys. Rev. L e t t e r s 23 (1969) 800. T. A. Hughes and S. F a l l i e r o s , E1 t r a n s i t i o n s from analog states in 90Zr, in nuclear isospin, eds. J. D. Anderson et al. {Academic Press~ 1969) p. 109. C. D. Kowaloski et al., Phys. Rev. 161 (1967) 1107. P. C a r l o s . private communication. N. Cue, J. Amann, P. Paul, E. M. Diener, private communication. S. F a l l i e r o s and B. Goulard, Nucl. Phys. A147 (1970) 3. T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40; A92 (1967) 481.
PHYSICS
ELECTRIC
DIPOLE
LETTERS
TRANSITIONS
10May 1971
IN
88Sr
AND
90Zr
J. D. VERGADOS and T. T. S. KUO Physics Department *, State University of New York, Stony Brook, N,Y. 11790, USA Received 1 April 1971
Electric dipole transitions in 88Sr and 90Zr have been studied using realistic two-body interactions. Shell model wave functions of definite isospin were used and both the T-upper and the T-lower transitions were calculated. In the case of 90Zr the pairing effect in its ground state was taken into account and, in addition to the ground state E 1 widths Fy0 , the first excited state widths F y 1 were obtained. The results are in good agreement with experiment. Within the f r a m e w o r k of the s h e l l m o d e l the d i p o l e s t a t e s a r e w e l l d e s c r i b e d a s p - h e x c i t a t i o n s on the g r o u n d and l o w - l y i n g e x c i t e d s t a t e s . F o r c l o s e d s h e l l n u c l e i the d i p o l e s t a t e s w e r e t r a d i t i o n a l l y c o n s i d e r e d a s l p - l h s t a t e s . H o w e v e r it h a s b e e n shown that the l p - l h b a s i s is not a d e q u a t e to g u a r a n t e e good i s o s p i n f o r n u c l e i that h a v e e x c e s s n e u t r o n s (N > Z). S i n c e the d i s c o v e r y of i s o b a r i c a n a l o g s t a t e s r e v i v e d i s o s p i n a s a m e a n i n g f u l q u a n t u m n u m b e r [1-6], s p e c i a l a t t e n t i o n m u s t be p a i d to i s o s p i n . It has b e e n shown that an a p p r o p r i a t e s e t of 2 p - 2 h s t a t e s is n e c e s s a r y in o r d e r to g u a r a n t e e good i s o s p i n [1]. Such e x c i t a t i o n s w i l l be i n c l u d e d w o r k i n g with a b a s i s of good i s o s p i n in the s p i r i t of r e f s . [1] and [2]. In the p r e s e n t i n v e s t i g a t i o n we w i l l f o c u s o u r a t t e n t i o n on the d i p o l e s t a t e s of 88Sr and 9~0Zr. T h e s e n u c l e i h a v e b e e n the s u b j e c t of v e r y e x t e n s i v e e x p e r i m e n t a l r e s e a r c h in r e c e n t y e a r s [6-10]. T h e o r e t i cal c a l c u l a t i o n s [1,8] a r e a l s o a v a i l a b l e f o r t h e s e n u c l e i that a r e in r e a s o n a b l y good a g r e e m e n t with e x p e r i m e n t . H o w e v e r t h e s e c a l c u l a t i o n s u s e d p h e n o m e n o l o g i c a l i n t e r a c t i o n s . T h e r e f o r e it is i n t e r e s t i n g to i n v e s t i g a t e w h e t h e r r e a s o n b l y r e a l i s t i c i n t e r a c t i o n s can r e p r o d u c e the d i p o l e s p e c t r u m . F o r t h i s p u r p o s e we u s e d the K u o - B r o w n m a t r i x e l e m e n t s d e r i v e d f r o m the H a m a d a - J o h n s t o n p o t e n t i a l . The c h o i c e of the 1 - b o d y H a m i l t o n i a n is not unique. If one i n c l u d e s e x p l i c i t l y the e x c e s s n e u t r o n s into the c a l c u l a t i o n s one can s t a r t f r o m a s e t of s i n g l e p a r t i c l e e n e r g i e s i d e n t i c a l f o r n e u t r o n s and p r o t o n s . H o w e v e r if the i n t e r a c t i o n of the " a c t i v e " p a r t i c l e s w i t h the e x c e s s n e u t r o n s is i g n o r e d , which g i v e s r i s e to the s y m m e t r y p o t e n t i a l , one b u i l d s the s y m m e t r y e n e r g y into the p r o b l e m f r o m the s t a r t by a s s i g n i n g s i n g l e p a r t i c l e e n e r g i e s w h i c h a r e d i f f e r e n t f o r n e u t r o n s and p r o t o n s . H e r e we adopted the l a t t e r a p p r o a c h and o u r 1 - b o d y H a m i l t o n i a n can be w r i t t e n a s f o l l o w s : h=~{[en(j)-
j
,
u(j)
~U(j)] +--t(j)"
To
To}+h c
w h e r e j r u n s o v e r a l l o r b i t a l s h c is the a v e r a g e 1 - b o d y c o u l o m b p o t e n t i a l , T o is the i s o s p i n of the c l o s e d s h e l l , U(j) i s the s y m m e t r y e n e r g y f o r o r b i t j and E n ( j ) a r e the s i n g l e p a r t i c l e e n e r g i e s f o r n e u t r o n s . A l t h o u g h the a b o v e H a m i l t o n i a n i s e a s y to handle [2], in p r a c t i c e we m a d e the a p p r o x i m a t i o n [1]
U(j) = U = 6
MeV
The s i n g l e p a r t i c l e e n e r g i e s f o r n e u t r o n s w e r e t a k e n e s s e n t i a l l y f r o m e x p e r i m e n t [1,2]. T h e y w e r e c h o s e n to be - 12.2, - 8.5, - 7.7, - 6.2, - 5.4, 0, 1, 2.5, 3.0, 5.0 M e V f o r the l f 7 / 2 , l f 5 / 2 , 2P3/2 , 2 P l / 2 , l g 9 / 2 , 2d5/2, 3 S l / 2 , l d 3 / 2 , l g 7 / 2 , lh11/2 o r b i t a l s r e s p e c t i v e l y . The d e t a i l s about i s o s p i n and p a r t i c l e - h o l e c o n v e n t i o n s can be found e l s e w h e r e [2]. F o r the c l o s e d s h e l l n u c l e u s we d i s t i n g u i s h t h r e e t y p e s of s t a t e s (i) P r o t o n p a r t i c l e in the e x c e s s r e g i o n (e.g. h e r e 2 P l / 2 , lgg/2 o r b i t a l s i n d i c a t e d by a ~ on the top of the s i n g l e p a r t i c l e q u a n t u m n u m b e r )
* Work supported by the U.S. Atomic Energy Commission. 93
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0 (ii) N e u t r o n h o l e in the e x c e s s r e g i o n X
¢2 :~-0:
I(Jl ~'l)J
it is e a s y to v e r i f y that the a b o v e s t a t e s h a v e d e f i n i t e i s o s p i n T = T< = T O = ½(N- Z) by o p e r a t i n g with the o p e r a t o r T+ (iii) Both the " a c t i v e " p a r t i c l e and the " a c t i v e " hole a r e o u t s i d e the e x c e s s r e g i o n . T h e n we apply the r u l e s of c o n s t r u c t i n g good i s o s p i n w a v e f u n c t i o n s w h i c h a r e known f r o m the t h e o r y of N = Z n u c l e i ( r e m e m b e r i n g that the c o r e has i s o s p i n d i f f e r e n t f r o m z e r o ) .
3,T
In the a b o v e d i a g r a m both the i s o s p i n and the a n g u l a r m o m e n t u m c o u p l i n g i s i n d i c a t e d . ( T = T< = T O o r T = T .... T0 + 1.) It m u s t be noted t h a t f o r a g i v e n s e t of s p a c e q u a n t u m n u m b e r s of c l a s s (iii) t h e r e a r e t w i c e a s m a n y T . s t a t e s ( c o r r e s p o n d i n g to t = 0, 1). T h a n t h e r e a r e T> s t a t e s (which a r i s e only f r o m ! = 1). It is e a s y to s e e that the s t a t e s c o r r e s p o n d i n g to t = 0 a r e of l p - l h n a t u r e . H o w e v e r in the c a s e of / = 1 the w a v e f u n c t i o n c o n t a i n s 2 p - 2 h c o m p o n e n t s . It is e a s y to v e r i f y that:
i(JlJ21) JI;TO;T
= T O + 1) = ~
1
/-To,
l(jlj~l)jtmt = 0) 1TOTO) + v-T~l(jlJ21)Jtmt
= 1)j T O T O - 1)
and
I(j]j~I)JI;To;T : where
TO):-
T/T~ I(jlj~l)Jtm/=O)] ToTo ) + ~
[ T o T 0) r e p r e s e n t s t h e c l o s e d s h e l l n u c l e u s and the s t a t e 1~ by o p e r a t i n g w i t h the o p e r a t o r T_, i.e.
ToT0 -
ToT 0 -
1)
-
T V2T0 I T o T o )
=~.~gi[]i(p)];l(n)]J=
](jlj~l)Jtmt = 1 ) ] T o T 0 IToTo-
1)
1) i s o b t a i n e d [2] f r o m the
O)
w h e r e i r u n s o v e r all the e x c e s s o r b i t a l s and gi = V ~ O " F r o m t h i s d i s c u s s i o n it i s c l e a r that the s t a t e s c l a s s (iii) with l = 1 c o n t a i n 2 p - 2 h c o m p o n e n t s . If one p r o c e e d s one s t e p f u r t h e r and e x p a n d s t h e s e s t a t e s in the p - n r e p r e s e n t a t i o n it is e a s y to s e e that the above s t a t e s a r e i d e n t i c a l w i t h t h o s e d i s c u s s e d in r e f . [1]. It is w o r t h noting that f o r h e a v y n u c l e i with m a n y e x c e s s n e u t r o n s s t a t e s of c l a s s (iii) b e c o m e l e s s i m p o r t a n t and the l p - 2 h m o d e l is a d e q u a t e to e x p l a i n the d i p o l e s t a t e s (e.g. 208pb) so t h a t in t h i s c a s e one can u s e d i r e c t l y the p - n r e p r e s e n t a t i o n . The m a t r i x e l e m e n t s of the H a m i l t o n i a n b e t w e e n s t a t e s that a r e of l p - l h n a t u r e is t r i v i a l . The r e s t of the m a t r i x e l e m e n t s can be c a l c u l a t e d m o s t s i m p l y by. m a k i n g u s e of t h e f a c t that the n u c l e a r H a m i l t o n i a n c o m m u t e s w i t h the i s o s p i n o p e r a t o r s , i.e. [T±, H N] = 0. H e n c e [2]
<,~,1IHN}~)
=
/t+ TO
(_)r v ~ ,
((~j~l)jt] VI ( j j
j~-l)
Jr)
/t+T^ _ ( ~ 2 1 H N I ¢ ~ ) : (_)t+l / ~ T 0 u ( ( j l j ~ l ) j t t VI ( j i j~.-1)jt)
(4~31HN]~) 94
=( (j lJ 21)jt]HN I (ji j~- l )Jt )
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T h e a b o v e f o r m a l i s m s t r i c t l y s p e a k i n g a p p l i e s only w h e n the s t a t e iToT O} is a c l o s e d s h e l l n u c l e u s . The f o r m a l i s m a p p l i e s a l s o in the c a s e when the s t a t e I TO T0} is not a c l o s e d s h e l l n u c l e u s p r o v i d e d that one t a k e s P a u l i p r i n c i p l e c o r r e c t i o n s into a c c o u n t f o r c l a s s e s (i) and (ii). In the p r e s e n t i n v e s t i g a t i o n we f o c u s o u r a t t e n t i o n on 88Sr and 9 0 Z r . In the c a s e of 88Sr the g r o u n d s t a t e iT0 TO } w a s a s s u m e d to be c l o s e d s h e l l with the P3/2 p r o t o n s h e l l and the g9/2 n e u t r o n s h e l l c o m p l e t e l y f i l l e d . T h e n the c a l c u l a t i o n p r o c e e d e d a s it was i n d i c a t e d above. F o r the c a s e of 9 0 Z r two a p p r o a c h e s w e r e a d o p t e d . In the f i r s t a p p r o a c h (I) we a s s u m e d that the g r o u n d s t a t e I TO T0} of 9 0 Z r is a c l o s e d s h e l l with the P l / 2 p r o t o n o r b i t a l b e i n g f i l l e d and the c a l c u l a t i o n p r o c e e d e d a s in the c a s e of 88Sr. A l t h o u g h , a s we s h a l l s e e b e l o w , the a g r e e m e n t with the e x p e r i m e n t is r a t h e r good, we know that t h i s d e s c r i p t i o n is i n c o m p l e t e . The ground s t a t e o f 90 Z r is " co r r e 1ate d . It i s r a t h e r w e l l d e s c r i b e d as ap2/2 + bg2/2 with a2:b 2 = 2 . 2 " . T h e r e f o r e in the c a s e of T = T~ : 6 we p e r f o r m e d a c a l c u l a t i o n (II) w h i c h t a k e s t h i s a f f e c t into a c c o u n t . The b a s i s w a s e x p a n d e d to i n c l u d e the following: T X Xl = ~ / ~ 0 I q)..};~ 4.~ = [[jl (p) J2(n)]J} = ~X[
X2 = (j12(p)) 0 I[J2(n)y31(n)] J)
Yl ¢ J 3
× 3 =)'2(P) 0 [[;2(P)j31(p)] J )
71 ¢ ~ 2
×4 In the a b o v e e x p r e s s i o n s ) ' i n d i c a t e s an o r b i t in the e x c e s s r e ~ ! o n , (p) and (n) stand f o r p r o t o n and n e u t r o n r e s p e c t i v e l y , T O i s the i s o s p i n of the c l o s e d s h e l l ( h e r e ° 8 S r ) and 4'3 is the s t a t e of c l a s s (iii) d e s c r i b e d a b o v e . T h e a b o v e b a s i s m e a n s that e f f e c t i v e l y we c a l c u l a t e the T ~ d i p o l e s t a t e s o b t a i n e d by e x c i t i n g both the g r o u n d and f i r s t e x c i t e d 0 + s t a t e s . So we a r e in a p o s i t i o n to c a l c u l a t e the d i p o l e t r a n s i t i o n s both to the g r o u n d (Y0) and f i r s t e x c i t e d s t a t e (~'1). S i n c e , a s we s h a l l s e e b e l o w , the d i p o l e e m i s s i o n s p e c t r u m f o r 70 t r a n s i t i o n s d o e s not i m p r o v e c o m p a r e d with e x p e r i m e n t by u s i n g the e x p a n d e d b a s i s (II), we did not a t t e m p t to p e r f o r m t h i s c a l c u l a t i o n in the c a s e of T = T< = 5. On the o t h e r hand in the c a s e of T = T> = 6 we p e r f o r m e d two t y p e s of c a l c u l a t i o n . In the f i r s t c a l c u l a t i o n (II1) we u s e d the b a r e G - m a t r i x e l e m e n t s e v e r y w h e r e . H o w e v e r the c o r e p o l a r i z a t i o n c o n t r i b u t i o n f r o m the d i a g r a m :
i s v e r y i m p o r t a n t f o r the p r o t o n - p r o t o n i n t e r a c t i o n . H e n c e in the s e c o n d c a l c u l a t i o n (II2) s u c h c o r e p o l a r i z a t i o n s w e r e i n c l u d e d f o r the p r o t o n - p r o t o n i n t e r a c t i o n in a d d i t i o n to the b a r e G - m a t r i x e l e m e n t s . T h e r e s u l t s of the c a l c u l a t i o n can be s u m m a r i z e d a s f o l l o w s : A. 88Sr. In the c a s e of 8 8 S r the Giant D i p o l e R e s o n a n c e (G.D.R.), c h a r a c t e r i z e d by T = T< = T 0 = 6, i s s o m e w h a t f r a g m e n t e d ( s e e fig. 1). The m a i n p e a k is p r e d i c t e d at 15.61 MeV with width F v ; 15.46 keV. T w o s e c o n d a r y p e a k s a r e p r e d i c t e d at 12.18 M e V with width F.y = 3.47 k e V and at 18.00 M e V with width 5.58 keV. The a v e r a g e of t h e s e p e a k s is l o c a t e d at 15.7 MeV and c o i n c i d e s w i t h the p o s i t i o n of the m a i n p e a k . T h i s a g r e e s q u i t e w e l l with a p r e v i o u s c a l c u l a t i o n [1]. In (7,n) r e a c t i o n s [10] a good L o r e n t z i a n f i t w a s o b t a i n e d f o r the c r o s s - s e c t i o n w i t h E x = 16.7 M e V and width F = 4.3 MeV. T h e s e r e s u l t s a r e in a g r e e m e n t with t h o s e of 8 7 R b ( p , v ) 8 8 S r r e a c t i o n [6]. In the c a s e of T = T> = T 0 + 1 = 7 t r a n s i t i o n s s t r o n g p e a k s a r e p r e d i c t e d at 15.49 M e V w i t h width I'.g = 1.10 keV, at 19.25 M e V w i t h width F~/ = 3.65 k e V and at 23.43 M e V w i t h width r ~ = 1.31 keV. E x p e r i m e n t a l l y [6] in 87Rb(p,'y)88Sr r e a c t i o n s one s e e s two r e s o n a n c e s at E x = 15.7 and 16,0 MeV, a s t r o n g r e s o n a n c e at E x = 17.2 MeV, a s h a r p r e s o n a n c e at E x = 17.8 MeV, a b r o a d s t r u c t u r e w i t h s o m e f i n e s t r u c t u r e at 19.2 M e V and a v e r y p r o n o u n c e d s t r u c t u r e a r o u n d 21 MeV. On the o t h e r hand (7',p) e x p e r i m e n t s [7] s e e m to i n d i c a t e the e x i s t e n c e of 4 p e a k s at 16.5, 17.5, 21.0, and 23.0 MeV. A l t h o u g h a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n c a l c u l a t i o n and e x p e r i m e n t i s not p o s s i b l e , the a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t is good. In p a r t i c u l a r we note the p r e s e n c e of a p e a k at a r o u n d 23 MeV. T h i s i s in a g r e e m e n t with the f a c t that a p e a k at r o u g h l y t h i s * The first excited 0+ state is - b p 2 / 2
+ ag2/2. 95
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10 M a y 1 9 7 1
6'
Zre°( i-..~0 + ) (T)
Sr~(l=.,~O +)
G.D,R. X I/~
5
4
>=3
3
2
2
II IOI
15 Ex
i
L
2O
MeV
Fig. 1. Calculated widths for dipole transitions 1- ~ 0+ in 88Sr. Widths corresponding to T> transitions are indicated by dotted lines with the letter A on the top. G.D.R. stands for Giant Dipole Resonance.
,I I0
Ex
J~ MeY
20
' ~-
Fig. 2. Calculated widths for dipole transitions 1 - ~ 0+ in 90Zr (calculation (I) as explained in main text). The notation is the same as in fig..1.
e n e r g y has a l s o b e e n p r e d i c t e d [2] and o b s e r v e d [17] in 89y. The s t r e n g t h of t h e s e r e s o n a n c e s will p r o b a b l y be s u p p r e s s e d in (p,~) r e a c t i o n s at t h e s e e n e r g i e s since the e x p e c t e d Fp0 i s e s t i m a t e d to be s m a l l . In d e t a i l the analog s t a t e s a r e p r e d i c t e d at e x c i t a t i o n e n e r g i e s 15.07, 15.49, 16.12, 17.25, 17.91, 19.25, 20.15, and 23.43 MeV with widths 47, 1097. 38, 16, 126, 3652, 20, 1311 eV r e s p e c t i v e l y . The strong9 '~dip°le s t a t e at 19.25 MeV i s m a i n l y d 5 / 9 2f~ 2 (22%) and g T / 9 2f 5 2 (66%). B. u Z r . The T< G.D.R. i s p r e d i c t e d at E x = 15.6 MeV with width F. = 15.6 keV and i s somewhat f r a g m e n t e d (see fig. 2). This is in good a g r e e m e n t with e x p e r i m e n t [6].~ In the c a s e of T = T~ = 6 s t a t e s the c a l c u l a t i o n (I) p r e d i c t s t h r e e s t r o n g dipole s t a t e s at 15.77, 19.30 and 23.50 MeV with widths 1.49, 4.73, and 1.99 keV r e s p e c t i v e l y . E x p e r i m e n t a l l y [6] m o s t of the T> dipole s t r e n g t h is found in t h r e e r e g i o n s : Multiple fine s t r u c t u r e between E x = 16.4 and 17.2 MeV, s h a r p fine s t r u c t u r e at 19.2 MeV and a b r o a d s t r u c t u r e with evidence of fine s t r u c t u r e in the r e g i o n Ex = 20.8 MeV. On the o t h e r hand 9 0 Z r ( v , p ) 8 9 y e x p e r i m e n t s [12] indicate a p e a k at around 22 MeV. With the p o s s i b l e e x c e p t i o n of the p e a k p r e d i c t e d at 23.50 MeV the a g r e e m e n t between t h e o r y (I) and e x p e r i ment is v e r y good. Although the a g r e e m e n t between t h e o r y (I) and e x p e r i m e n t is good, the above c a l c u l a t i o n i g n o r e s the f a c t that the ground s t a t e of 9 0 Z r is c o r r e l a t e d . So a c a l c u l a t i o n was p e r f o r m e d that t a k e s into account such c o r r e l a t i o n s in the m a n n e r d e s c r i b e d above. Using the b a r e G - m a t r i x e l e m e n t s (HI) we obtained a n a l o g s t a t e s at e x c i t a t i o n e n e r g i e s 12.75, 13.31, 14.41, 15.39, 16.18, 16.52, 17.00, 17.27, 17.35, 17.47, 18.16, 18.37, 18.61, 19.45, 19.52, 21.10, 22.20, 22.93 and 23.95 with F T 0 w i d t h s 28, 22, 106, 1067, 84, 1, 2, 18, 7, 42, 194, 2027, 1538, 450, 26, 22, 80, 1165, and 407 eV r e s p e c t i v e l y . The c o r r e s p o n d i n g FT1 widths a r e 46, 1, 158, 0, 0, 1026, 16, 0, 122, 2, 844, 366, 105, 535, 7, 2888, 395, 290, 842 eV. On the o t h e r hand u s i n g r e n o r m a l i z e d G - m a t r i x e l e m e n t s (II2) we obtained analog s t a t e s at e x c i t a t i o n e n e r g i e s 13.14, 14.00, 15.08, 15.72, 16.25, 17.03, 17.19, 17.31, 17.71, 18.05, 18.72, 19.09, 19.41, 19.69, 19.88, 20.76, 21.61, 22.72, and 24.90 MeV. The c o r r e s p o n d i n g ground s t a t e widths a r e 71, 61, 338, 999, 44, 25, 5, 2, 27, 3, 4019, 690, 37, 128, 5, 17, 245, 1452, and 224 eV r e s p e c t i v e l y . The F y l w i d t h s a r e 30, 0, 90, 607, 9, 396, 161, 434, 3, 109, 737, 720, 10, 551, 87, 3200, 70, 519, 1115 eV. The p o s i t i o n of F~I widths in both c a s e s is shifted r e l a t i v e l y to the FV 0 widths by roughly the s e p a r a t i o n e n e r g y of the ground and f i r s t e x c i t e d 0 + s t a t e s (N 2 MeV). We a l s o note that in both c a l c u l a t i o n s (II1) and II2) the s t r o n g T> dipole state is l o w e r e d c o m p a r e d with the p r e d i c t i o n s of c a l c u l a t i o n (I). T h i s shift, which is about 1 MeV f o r c a l c u l a t i o n (II1) and 0.5 MeV f o r c a l c u l a t i o n (II2), m a k e s the a g r e e m e n t 96
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Z r 9 ° (i- T>.-..O+ T<)
ry°
4-
(If2)
Zr gO ( I- T>--,,-0 +) (Ill)
>
>~
2-
i)
% '~ rr° r.
%1 ,~ ao Ex
, ,Ir'F I 2l
22
23
;'4
MeV
Fig. 3. Calculated widths for T> dipole transitions in 90Zr leading both to the ground (1~0) and to the first excited state ( I ~ 1 ) (calculation II1).
i
16
|
I'~
18
IT:
19
EX
20
i
21
22
23
24
MeV
Fig. 4. Calculated widths for T> dipole transitions in 90Zr leading both to the ground ( I~y0) and to the first excited state ( I ~ 1 ) (calculation II2).
b e t w e e n t h e o r y and e x p e r i m e n t w o r s e . (A s i m i l a r shift, but in the o p p o s i t e d i r e c t i o n , has a l s o b e e n found in the c a l c u l a t i o n s of r e f . [8].) T h i s p r o b a b l y i m p l i e s that o u r p - h i n t e r a c t i o n is r a t h e r weak. Our p r e s e n t c a l c u l a t i o n c o n f i r m s the e x p e c t e d [1,12] s p l i t t i n g of the G.D.R. into two g r o u p s c o r r e s p o n d i n g to the two i s o s p i n s T -- T< = T O and T = T> = T O + 1. Rough e s t i m a t e s b a s e d on the H a m i l t o n i a n of eq. (1) w i t h U ( j ) = U = cont. p r e d i c t the e n e r g y s e p a r a t i o n b e t w e e n T > and T < to be AZ = (To+ I)U/T 0
A l t h o u g h t h i s r e l a t i o n y i e l d s the c o r r e c t e n e r g y s e p a r a t i o n b e t w e e n the a n a l o g and i t s a n t i a n a l o g , it f a i l s to p r o v i d e the c o r r e c t s e p a r a t i o n e n e r g y b e t w e e n the c o l l e c t i v e T> and T< s t a t e s . It h a s b e e n s h o w n [12] t h a t in the l a t t e r c a s e t h i s r e l a t i o n s h i p m u s t be m o d i f i e d a s f o l l o w s : hE = (To+I)UD/T 0
UD -~ 0.6 U
T h e a b o v e f o r m u l a y i e l d s A E = 4.7 M e V f o r 88Sr and A E --~ 4.0 M e V f o r 9 0 Z r . Our c a l c u l a t i o n p r e d i c t s an a v e r a g e s e p a r a t i o n e n e r g y A E = 4.3 M e V and AE = 4.0 MeV r e s p e c t i v e l y , w h i c h is in r e m a r k a b l e a g r e e m e n t with the a b o v e e s t i m a t e s . In c o n c l u s i o n we can s a y that, in v i e w of the a p p r o x i m a t i o n s m a d e by the l i m i t e d s h e l l m o d e l s p a c e we c o n s i d e r e d and by the f a c t that we i g n o r e d c o m p l e t e l y the p r e s e n c e of the c o n t i n u u m s t a t e s , the r e a l i s t i c i n t e r a c t i o n s that we u s e d r e p r o d u c e r a t h e r w e l l the d i p o l e e m i s s i o n s p e c t r u m of n u c l e i n e a r closed shells. One of the a u t h o r s (J. D. V.) i s s p e c i a l l y i n d e b t e d to P r o f e s s o r S, F a l l i e r o s f o r e n l i g h t e n i n g d i s c u s s i o n s and to P r o f e s s o r G. E. B r o w n f o r his i n t e r e s t in the p r o b l e m .
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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B. Goulard, T. A. Hughes and S. F a l l i e r o s , Phys. Rev. 181 {1965) 134.% J. D. Vergados and T. T. S. Kuo, Nucl. P h y s . , to be published. H. Morinaga, Z. Physik 188 (1965) 182. S. F a l l i e r o s , B. Goulard and H. H. Venter, Phys. L e t t e r s 19 (1965) 398. J. L. Black and N. W. Tanner, Phys. L e t t e r s 11 (1965) 135. M. Hasinoff et al., Phys. L e t t e r s 30B {1969) 337. K. Shoda et al., Phys. Rev. L e t t e r s 23 (1969) 800. T. A. Hughes and S. F a l l i e r o s , E1 t r a n s i t i o n s from analog states in 90Zr, in nuclear isospin, eds. J. D. Anderson et al. {Academic Press~ 1969) p. 109. C. D. Kowaloski et al., Phys. Rev. 161 (1967) 1107. P. C a r l o s . private communication. N. Cue, J. Amann, P. Paul, E. M. Diener, private communication. S. F a l l i e r o s and B. Goulard, Nucl. Phys. A147 (1970) 3. T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40; A92 (1967) 481.