Nucleon widths of isobaric analogue states in nuclei

Volume 22, number 4

NUCLEON

P H Y SI C S L E T T E R S

WIDTHS

OF ISOBARIC

ANALOGUE

1 September 1966

STATES

IN N U C L E I

V. V. BALASHOV and E. L. YADROVSKY

Research Institute of Nuclear Physics of Moscow State University, Moscow, USSR Received 20 July 1966

Widths of the 2- analogue states in 90Zr are calculated. These states are analysed in terms of their isospin components.

The d i s c o v e r y of i s o b a r i c analogue s t a t e s in the m e d i u m n u c l e i i n i t i a t e d the work on s t u d i e s of i s o s p i n a c c u r a c y in n u c l e i with N - Z >> 1. The t h e o r e t i c a l t r e a t m e n t of these s t a t e s i s c o m p l i cated by t h e i r b e i n g located at high excitation e n e r g i e s in the c o n t i n u u m of both the p r o t o n and n e u t r o n c h a n n e l s . An u r g e n t p r o b l e m of analogue state widths a r i s e s f r o m this. R e c e n t l y , i n t e r e s t i n g e x p e r i m e n t a l data on the p a r t i a l widths of the analogue s t a t e s , c o r r e s p o n ding to t h e i r decay not only to the ground state but to excited s t a t e s of the p r o d u c t a s well, have a p p e a r e d . T h i s e n a b l e s us to make some p r o g r e s s in e l u c i d a t i n g the s t r u c t u r e of these s t a t e s . In this p a p e r we p r o c e e d f r o m an a p p r o x i m a t e v a r i a n t of the unified theory of n u c l e a r r e a c t i o n s suggested in ref.1. A c c o r d i n g to ref. 1, open and closed c h a n n e l s c o n t r i b u t e to the f o r m a t i o n of the compound n u c l e u s on equal grounds. The p o s i t i o n s of l e v e l s of the compound n u c l e u s is d e t e r m i n e d by the " c h a n n e l - c o u p l i n g - m a t r i x " , the d i m e n s i o n a l i t y of which i s equal to the full n u m b e r of c h a n n e l s of the p r o b l e m .

Wij (E)

= [ E - (E i

- ½iri(E) ] 6 ij -Vii

Here E i i s the p o s i t i o n of the l e v e l (discrete or q u a s i s t a t i o n a r y ) in channel i before the o f f - d i a g - , onal i n t e r a c t i o n Vii i s t u r n e d on; F i ( E ) - a n f u n c Table 1 TV = TO - ½

TV = TO + ½

(n) -1

V 2~OO+"~

W~o¥1

(n) -I

~2To I+1

"/ 2To ~/ 2To7 1

Table 2

T=T o

T =T o

T.1 = T o + ½

T 1 = T o - ½

2To

~

Y 2 T-oo'~ 1

fl-

T=To+I T 1 = TO +

I 2To + 1) (2To + 2)

2V ;k 2

, 2T ° 1 1 - V (2To+ 1) (2To + 2) 2To+ 12 ~ o + 1 V~o+2

-

] / 2To V 2To'~2

-

( 1 2To+ 2

tion of the energ~ n u c l e o n w~dth in the c o r r e s ponding open channel (we d i s r e g a r d c l u s t e r c h a n n e l s ) . At s m a l l Fi(E) the position of r e s o n a n c e s i s d e t e r m i n e d by the e i g e n v a l u e s £~ of the r e a l p a r t of the coupling m a t r i x , and t h e i r widths a r e e x p r e s s e d by the a p p r o p r i a t e e i g e n v e c t o r s Xi(M):

:

Lx! )12 × ri(E .) i

T h i s l e a d s to the known r e d u c e d - w i d t h s method of t r e a t m e n t of the decay of h i g h l y - e x c i t e d nuclear states. In the f r a m e work of the method we have cons i d e r e d the analogue s t a t e s 2- in the n u c l e u s 90Zr, taking into account the m i x i n g of t r a n s i t i o n s f r o m different s h e l l s and not m a k i n g beforehand the a s s u m p t i o n of i s o s p i n p u r i t y . Each such t r a n sition (except for the t r a n s i t i o n s of n e u t r o n s f r o m the shell lg~ was r e p r e s e n t e d in the calculation by the following t h r e e configurations: (c~) n e u t r o n . 1 1 p a r t i c l e - h o l e . -of the type (2p~)n (2d~)n; (fl) "two p a r t i c l e - t w o hole" - of the type (2p½)-n1 {(lg~) - 1 (lg~)p}0+ and (~) proton " p a r t i c l e hole" ~- of the type (21~)-p1 (2d~)p. In p a r a l l e l , the

507

Volume 22. number 4

PHYSICS LETTERS

1 September 1966

Table 3 Structure of the states 2- in 90Zr - analogs of the ground and lower excited levels of 90y. Admixture of the configurations with T = 5

T 1 = I~

Contribution of the main configurations with T = 6 (adm. with ampl. less than 0.1 are not shown).

lf~ 2d~ 2p~ 2d i 2p~ 2d~ 2p~ 2d~ 2p~ 2d25

I ° 62 1 1 0.312 I

c a l c u l a t i o n of the s t a t e s ~1- , ~ , ~ and ~- of the n u c l e u s 89Zr with the m i x i n g of the configurations (n-1)7of the type (2p½) -1 and (ff-1)-of the type (2p½) -1 {(lg~)-I (lg~)o}0+ was c a r r i e d on. The " z e r o ~ a p p r o x i m a t i o n ' ~ l e v e l s for the c o n f i g u r a t i o n s were chosen a c c o r d i n g to the e m p i r i c a l and s e m i e m p i r i c a l s c h e m e s [2,3] with the Coulomb e n e r g y of proton E = 11.6 MeV taken into account [4]. The configuration mixing was calculated by the f o r m u l a e of ref.5 with the z e r o - r a n g e p o t e n tial. -

7-

V(1,2) = g [ ( 1 - a ) + ~ a l a 2 ] 5 ( r l - ~ 2) g = -1050 MeV, fm3; ~ = 0.13 The p r o j e c t i o n of the obtained wave functions on s t a t e s with definite i s o s p i n (the m a t r i c e s of the c o r r e s p o n d i n g u n i t a r y t r a n s f o r m a t i o n a r e shown in t a b l e s 1 and 2) shows that in the states u n d e r d i s c u s s i o n , i s o s p i n i s a good quantum n u m b e r with a v e r a g e a c c u r a c y higher than 95%. T h i s i s a p p r o x i m a t e l y the same degree of a c c u r a c y a s that c h a r a c t e r i s t i c , of the ground s t a t e s of nuclei, a c c o r d i n g to Sliv and Kharitonov [6]. It i s i m p o r t a n t , however, to point out that in the f r a m e work of the chosen approach, the effect of the i s o s p i n m i x i n g of s t a t e s by the Coulomb i n t e r a c t i o n i s i m i t a t e d by a n o t h e r f a c t o r - the i n a c c u r a c y of the i n i t i a l data on the position of the s i n g l e - p a r t i c l e n e u t r o n and proton l e v e l s and the lack of a s t r i c t a c c o r d a n c e between these data and the f o r m and p a r a m e t e r s of the twon u c l e o n potential. T h i s i s a very c o m p l i c a t e d p r o b l e m . P o s t p o n i n g its detailed d i s c u s s i o n until a s u b s e q u e n t p a p e r , we would like to r e p o r t the r e s u l t s of the c a l c u l a t i o n u n d e r the a s s u m p t i o n of the " m i n i m u m " mixing of T' S- due only to the Coulomb i n t e r a c t i o n of the proton and the proton "hole". F r o m table 3 it is seen that the m a i n factor, i n f l u e n c i n g the r e l a t i o n between the proton widths 508

Exper- Theory iment

T 1 = ~2

2p~2 2d2~ 2p½ 2d~ 2p~ 2d~ lg~ 2f½ 2p~ 2d2~ 2p½ 2d~2 3x10 -3 3×10-3 4×10-2 4×10-3 3×10 -3 2×10-2 3×i0 -3 3x10-3 2 × 1 0 - 3 . 11×10-211×10-3 2×10 -312 ×10 -211×10 -212 ×10 -4 6xi0-3 3x10-2 4×10-3

E exc.

I I °1 2 i I:-0 403

I I 1 I 1 0. 20 1-0.360 I

13.12 15.55

0 784 I 0 182

:_

13.32 15.28 15.45 16.58

of e l a s t i c and i n e l a s t i c s c a t t e r i n g i s the m i x i n g of v a r i o u s shell c o n f i g u r a t i o n s having the s a m e isotopic s t r u c t u r e (T = T o + 1). I n c l u s i o n of the "background" p a r t i c l e - h o l e e x i t a t i o n s with T = T o i s of no significance h e r e . T u r n i n g to details, we should like to point out that, c o n t r a r y to the individual p a r t i c l e model, the state 2- at 15.5 MeV has to be a s c r i b e d not to the doublet (2p½) -1 (2d~) but in the m a i n to the configuration (2p~)-I (2d~); the doublet r e p r e s e n tative i s situated h i g h e r - a r o u n d 16.5 MeV. It i s i n t e r e s t i n g also, that the state 2- at 15.5 MeV s e e m s not to be the second analogue. A c c o r d i n g to the c a l c u l a t i o n , u n d e r it, a r o u n d 15.3 MeV, t h e r e i s one m o r e level 2- - with a v e r y s m a l l r e d u c e d width with r e s p e c t to the ground state (and p o s s i b l y this i s why i s was not found e i t h e r in the e x p e r i m e n t s of Fox et al. or in the r e l e v a n t e x p e r i m e n t s on s t r i p p i n g [4]. The p r e s e n t calculation gives only a " f r a m e " of the actual p i c t u r e of the r e s o n a n c e s o b s e r v e d in the e x p e r i m e n t . We had cut off the b a s i s , having excludec: f r o m the coupling m a t r i x the s t a t e s of c o m p l i c a t e d n a t u r e (and f i r s t of a l l the "two p a r t i c l e - two hole" s t a t e s with T = To) over which the d i s t r i b u t i o n of the "pure" analogue s t a t e s o c c u r s . T h e r e f o r e , t h e r e i s no s e n s e in a detailed c o m p a r i s o n with e x p e r i m e n t of the a b s o lute width v a l u e s (they a r e 3-4 t i m e s l e s s than e x p e r i m e n t a l ones). F o r the lowest analogue state 2(~o)2

= l 0.245 t h e ° r y 0.238 e x p e r i m e n t

F o r the state 2- at 15.5 MeV this r a t i o i s a l r e a d y m o r e than unity (~ 2.15). The situation for the n e u t r o n widths i s much worse. While the n e u t r o n reduced widths for the level ½- of the n u c l e o n s 90Zr a g r e e quite well in o r d e r of magnitude with the e x p e r i m e n t [7] (02 ~ 10 -4) for the level ~" they a r e u n d e r e s t i m a -

Volume 22, number 4

t e d by two o r d e r s . H o w e v e r , it h a s to be taken into c o n s i d e r a t i o n that the n e u t r o n r e d u c e d - w i d t h v a l u e s , c h a r a c t e r i z i n g the d e g r e e of i s o s p i n p u r i t y v i o l a t i o n , a r e e x t r e m e l y s e n s i t i v e to a lot of v a r i o u s p a r a m e t e r s of the c a l c u l a t i o n . It i s s u f f i c i e n t to note that the b a c k g r o u n d of the s t a t e s with T = T o i s v e r y n o n u n i f o r m a c c o r d i n g to the c a l c u l a t i o n , t h e r e a r e a g r e a t n u m b e r of r e s o n a n c e s with T = T o in the r e g i o n 12.0 - 17.0 MeV: 12.49; 13.67; 14.08; 14.35; 14.47; 15.95 MeV etc.

A d e t a i l e d ru n of the c a l c u l a t i o n p a r a m e t e r s will be c a r r i e d out e l s e w h e r e . T h e a n a l y s i s of the analogue r e s o n a n c e s of 90Zr with o t h e r spin v a l u e s (0-, 1-, 3-) and a d i s c u s s i o n of the c o r r e s p o n d i n g s t a t e s of 9 0 y will be r e p o r t e d t h e r e a s well.

ON

(y,p)

1 September 1966

P H Y SI C S L E T T E R S

AND

In any c a s e , the r e s u l t s obtained by now on the nucleon widths m ay be c o n s i d e r e d a s a convincing a r g u m e n t f o r a high validity of i s o s p i n in the r e gion of n u c l e i width A ~ 100.

References 1. V.V.Balashov, P. Doleshal, G. Ya. Korotkih and V. N. Fetisov, Yadernaja Physika 2 (1965) 643. 2. J. Sehr~ler, Nuovo Cim. 60 (1959) 512. 3. V.A.Kravtsov, Massy atomov (Atomizdat 1965). 4. J.D. Fox et al,, Phys. Lett. 12 (1964) 198. 5. V.V. Batashov, K.A. Tumanov and Yu. M. Shirokov, Yaderniye reaktsiyi pri malih i srednih energiyah. (Moscow, 1958) p.548. 6. L.A. Sliv and Yu. T. Kharitonov, Ydernaya Physika 1 (1965) 1129. 7. G.S.Mani, G.C.Dutt, Phys. Letters 16 (1965) 50.

(y,n) REACTIONS MECHANISM IN AND HEAVY NUCLEI REGION

THE

MEDIUM

V. V. BALASHOV and E. L. YADROVSKY Research Institute of Nuclear Physics of Moscow State University, Moscow, USSR Received 20 July 1966

Coupling in additional configurations necessary to obtain levels of good isospin pushes the (y,p) cross section to higher energies than the (y,n) one. m6 M~,'b

R e c e n t l y , F a l l i e r o s e t al. [1] s u g g e s t e d that the known u p w a r d s shift of the c r o s s s e c t i o n m a x i m u m of the r e a c t i o n (y,p) with r e s p e c t to the (7,n) c r o s s s e c t i o n m a x i m u m i s c o n n e c te d with the i s o t o p i c s p l i t t i n g of the dipole giant r e s o n a n ce. The c a l c u l a t i o n of M a i l i n g et al. [2] f o r Ni have shown that in the l ig h t n u c l e i r e g i o n t h i s a s s u m p t i o n i s valid. How f a r t h i s e f f e c t p e r s i s t s when we go o v e r to the m e d i u m - w e i g h t and h e a v y n u c l e i depends on i s o t o p i c spin a c c u r a c y in t h i s n u c l e i r e g i o n and on the way of f o r m a t i o n of a c o l l e c t i v e state in e a c h of the two t r a n s i t i o n gr oups : T = ½ (N-Z); T = ½ (N-Z) + 1. In a p r e v i o u s work of the a u t h o r s [3] an a n a l y s i s of n u cl eo n widths of i s o b a r i c - analogue s t a t e s was c a r r i e d out, which d e m o n s t r a t e d high p u r i t y of i s o t o p i c spin in n u c l e i with N - Z >> 1. In the p r e s e n t p a p e r , in the f r a m e work of the m e t h o d d e s c r i b e d in r e f . 3 we c o n s i d e r e d the

r-] - - T~5

- - TJ,6

1604

,,otl $0-

~10-

0

8

~2

f6

2O

2~

E¢ (New)

Fig. 1. Dipole resonance in 9 0 Z r - "zero approximation". dipole a b s o r p t i o n s p e c t r u m in the n u c l e u s 9 0 Zr . A l l p a r a m e t e r s a r e the s a m e a s in r e f . 3. A s it

509