ISOSPIN STRUCTURE OF GIANT RESONANCES

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ISOSPIN STRUCTURE OF GIANT RESONANCES

R. Leonardi

To cite this version:

R. Leonardi. ISOSPIN STRUCTURE OF GIANT RESONANCES. Journal de Physique Colloques, 1984, 45 (C4), pp.C4-319-C4-326. �10.1051/jphyscol:1984424�. �jpa-00224090�

Colloque C4, supplkment a u n03, Tome 45, m a r s 1984 page C4-319

ISOSPIN STRUCTURE OF G I A N T RESONANCES

R. Leonardi

Dipartimento d i Fisica deZZ'Universit6 d i T ~ e n t o , Trento, I t a l y

Resume - Nous examinons l e probl6me de l a s t r u c t u r e en i s o s p i n d e s resonances ge'antes i s o v e c t o r i e l l e s . Nous proposons une t h e o r i e fondee s u r des r P g l e s de somme pour l a d e t e r m i n a t i o n d e s c e n t r o i d e s d e s d i f f s r e n t s fragments d p a r t i r d ' u n Hamiltonien donne. P l u s i e u r s o p e r a t e u r s d ' e x c i t a t i o n s o n t examines e t l e s formules e x p l i c i t e s d e r i v e e s .

A b s t r a c t - We analyze t h e i s o s p i n s t r u c t u r e of i s o v e c t o r q i a n t r e s o n a n c e s . We set-up a t h e o r y , based on sum r u l e s , t o p r e d i c t t h e c e n t r o i d ener- g i e s and t h e s t r e n g t h s of t h e v a r i o u s f r a g m e n t s , f o r a given H a m i l - t o n i a n . D i f f e r e n t o p e r a t o r s a r e d i s c u s s e d and e x p l i c i t examples a r e worked o u t .

I

-

An i n t e r e s t i n g f e a t u r e of t h e i s o v e c t o r g i a n t resonances i s t h a t t h e v a r i o u s i s o s p i n fragments s p l i t . T h i s phenomenon was p r e d i c t e d l o n g ago / 1 / and many experimental ef f o r t s were done i n t h e l a t e s i x t l e s and i n t h e e a r l y s e v e n t i e s i n o r d e r t o d e t e c t t h e l s o s p i n splitting mainly of t h e g i a n t d i p o l e resonance / 2 / . Theories were a l s o proposed t o p r e d i c t t h a t s p l i t t i n g / 3 / . A t t h a t time, however, r a t h e r few d a t a ( i f any) on the lowest fragment were a v a i l a b l e and t h e i d e n t i f i c a t i o n of t h e v a r i o u s components was somewhat ambiguous.Today, because o f t h e b e a u t i f u l measurements on charge exchange r e a c t i o n s /4/ and t h e p i o n e e r experiments on o t h e r e x c i t a t i o n s t h a n t h e e l e c t r i c d i p o l e ones / 5 / , t h e f i e l d i s again r a t h e r promising and t h e s i t u a t i o n can b e reexamined. The aim of t h ~ s t a l k i s n o t t o review t h e v a r i o u s d e t a i l e d works on t h e s u b j e c t o r t h e r e s u l t s of many experiments, b u t t o summarize t h e many a s p e c t s t h a t t h e d i f f e r e n t i s o v e c t o r e x c i t a t i o n s have i n common and t o s k e t c h o u t t h e main c h a r a c t e r i s t i c s of t h e i n g r e d i e n t s which f o r c e t h e v a r i o u s l s o s p i n fragments of an i s o v e c t o r e x c i t a t i o n t o s p l i t .

I1 - THEORY

A t t h e very beginning l e t us i n t r o d u c e t h e necessary q u a n t i t i e s , n o t a t i o n s and con- v e n t i o n s . When d e a l i n g with an i s o v e c t o r e x c i t a t i o n i t i s now t r a d i t i o n a l t o d i s t i n guishbetween i t s v a r i o u s charge exchange components AT = O,-+I. L e t us suppose, a s

t h e most common c a s e , t h a t t h e v a r i o u s i s o v e c t o r e x c i t a t i o n o p e r a t o r s a c t on a T3 = T > 6 n u c l e a r t a r g e t . Then a schematic s i t u a t i o n o f t h e v a r i o u s e x c i t e d i s o s p i n

* The unpublished m a t t e r p r e s e n t e d i n t h i s t a l k i s p a r t o f a d e t a i l e d paper t o be p u b l i s h e d i n c o l l a b o r a t i o n w i t h E. L i p p a r i n i and S. S t r i n g a r i .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984424

C4-320 JOURNAL DE PHYSIQUE

fragments i s summarized by t h e f a m i l i a r scheme of F i g . ( I ) .

Fig. 1 - The d i f f e r e n t fragments e x c i t e d by an i s o v e c t o r e x c i t a t i o n o p e r a t o r a c t i n g on a t a r g e t T 3 = T. The i s o g e o m e t r i c a l f a c t o r s a r e i n d i c a t e d .

Once t h e Coulomb energy i s removed, t h e fragments with t h e same i s o s p i n a r e essen- t i a l l y degenerate ( F i g . (2) ) .

Fig. 2 - Once t h e Coulomb energy i s removed t h e fragments w i t h t h e same i s o s p i n a r e degenerate. We have l a b e l l e d the fragments w i t h v a r i o u s energy c e n t r o i d s .

I n t h e following we w i l l i n d i c a t e w i t h u + , Oo, 0- t h e t o t a l s t r e n g t h s of t h e AT= + 1,0,-1 e x c i t a t i o n o p e r a t o r -+ P = $ ^{f i} Ti r e s p e c t i v e l y and w i t h E + , Eo, E- t h e c e n t r o i d s of t h e s t r e n g t h s . Owing t o t h e i s o s p i n geometry, 5- i s composed of t h r e e p a r t s , having i s o s p i n s T + ^{1 , T , T} - ^1. a. i s composed of two fragments w i t h T + 1

geometry it i s convenient t o i n t r o d u c e t h r e e "reduced" ( v i a Wigner-Eckart) q u a n t i - t i e s O T + l , oT, OT-l connected t o 0-, a,, a+, through t h e i s o g e o m e t r i c a l f a c t o r s . A convenient d e f i n i t i o n of t h e s e "reduced" s t r e n g t h s l e a d s t o t h e following expres- s i o n s / 6 / .

Note t h a t t h i s procedure f a c t o r i z e s o u t only t h e geometrical dependence on t h e is02 p i n . A f u r t h e r (dynamical) dependence of a T , on T' h a s t o be expected. (TI= T+1,T, T-1). The g o a l o f a f u l l y d e t a i l e d s t u d y of t h e i s o s p i n fragmentation of an isovec- t o r e x c i t a t i o n i s t h e p r e d i c t i o n ( o r t h e measure) of t h e s t r e n g t h d i s t r i b u t i o n of t h e v a r i o u s fragments. I n t h e following we w i l l r e s t r i c t o u r s e l v e s t o a schematic s t u d y of t h e total i s o s p i n s t r e n g t h and of t h e c e n t r o i d s e n e r g i e s ET+l, ET, ET-1 of t h e i s o s p i n fragments, w i t h i n t h e assumption t h a t t h e s t r e n g t h s a r e mainly concen- t r a t e d i n some narrow energy r e g i o n s . Furthermore we w i l l focus on l a r g e T n u c l e i . For t h e s e n u c l e i , f r o m r e l . ( l ) t h e bulk of t h e e x c i t a t i o n of 0- i s a T-1 fragment;

whereas t h e b u l k of 0, i s a T fragment. For t h e s e n u c l e i one e x p e c t s t h a t E-=' ET-1, Eo=' ET ( E + Z E ~ + ~ ) , s o t h a t we can f o c u s o u r a t t e n t i o n on E+, Eo, E-. I n o r d e r t o e x p r e s s E+, Eo, E- some convenient q u a n t i t i e s can be i n t r o d u c e d a s f o l l o w s : l e t AT be a q u a n t i t y such t h a t

and l e t AV be such t h a t

Then

E+ = Eo + A v + A T ,E- = Eo - A, + A T . ( 2 ) One e x p e c t s t h a t t h e l e a d i n g c o n t r i b u t i o n t o OV i s a " f i r s t o r d e r e f f e c t " i n t h e i n - t e r a c t i o n r e s p o n s i b l e f o r t h e s p l i t t i n g and t h e l e a d i n g c o n t r i b u t i o n t o AT i s a "se- cond o r d e r e f f e c t " , /7/ s o t h a t

Furthermore, a s f a r a s t h e dynamical dependence of AT and AV on t h e i s o s p i n i s ra- t h e r smooth we can s t u d y ET-1 , ET , ET+l , u s i n g t h e a p p r o p r i a t e geometry.

I n p a r t i c u l a r

ET+l - ET = T+1

(%

T 2T

I n t h e f o l l o w i n g we w i l l i l l u s t r a t e a simple t h e o r y , based on sum r u l e s , t o study

C4-322 JOURNAL DE PHYSIQUE

t h e g r o s s behaviour of AV, AT, u+ and 0-.

I n o r d e r t o g e t e q u a t i o n s i n v o l v i n g A,, AT, O + , 0- e t c . we w i l l use t h e f o l l o w i n g i m p o r t a n t sum r u l e s /8/:

where U ( l ) and u ( 2 ) a r e energy weighted and square-energy-weighted t o t a l s t r e n g t h s r e s p e c t i v e l y . S i n c e we assume t h a t t h e s t r e n g t h i s c o n c e n t r a t e d i n a narrow energy r e g i o n , 0 ( 2 ) = Eb ( 1 ) where E is t h e c e n t r o i d o f t h e s t r e n g t h and

I n t h i s case our sum r u l e s become

< O ~ C F - , F + I / O > = IS- - U+

<O[CF-,CF+,HIIIO> = E+ a + + E-0-

2 2

<oI

w i t h

E+ = Eo + A, + A, , ^{E- = Eo} - A, + AT

.

L a t e r we w i l l comment on t h e p o s s i b i l i t y of e v a l u a t i n g t h e s e sum r u l e s : l e t ' s j u s t now d i s t i n g u i s h " f i r s t o r d e r " and "second o r d e r " c o n t r i b u t i o n s i n t h e sum rules:

we have

The f i r s t sum r u l e does n o t i n v o l v e t h e Hamiltonian s o t h a t we have o n l y a z e r o o r - d e r c o n t r i b u t i o n fo. The second one has a f i r s t o r d e r ( i f any) c o n t r i b u t i o n 0 (V) ₁ .

The t h i r d sum r u l e i n v o l v i n g twice t h e Hamiltonian h a s i n p r i n c i p l e b o t h a f i r s t o r d e r ( 0 (V)) and a second o r d e r ( 0 2 ( T ) ) c o n t r i b u t i o n , t h e l a t t e r o r i g i n a t i n g from terms c o n t a i n i n g t h e i n t e r a c t i o n twice. The p r a c t i c a l p r e s c r i p t i o n t o s e p a r a t e o u t 2 t h e v a r i o u s c o n t r i b u t i o n s i s t h e following:

Where K i s t h e k i n e t i c energy p a r t of H = K + V.

I n s e r t i n g E+and E- from ( 2 ) i n ( 7 ) one can use an i t e r a t i n g procedure i n o r d e r t o e x t r a c t AV ( f i r s t o r d e r e f f e c t s from r e 1 . ( 4 ) ) and AT(second o r d e r ) and f i n a l l y U+ , 0- up t o second o r d e r c o n t r i b u t i o n s . This simple e x e r c i s e l e a d s t o t h e f o l l o - wing r e s u l t s :

E: f o - f2 - 02 ( V ) A =

v _{2 f l} ⁽⁹⁾

0, = < O ( F O F O ~ O> by d e f i n i t i o n of Eo ( c e n t r o i d ) one h a s

From formulas ( 9 , 1 0 , 1 1 , 1 2 ) some important p r o p e r t i e s become e v i d e n t : a ) t h e t e n s o r c o n t r i b u t i o n could be r a t h e r l a r g e : n o t e t h a t AT c o n t a i n s a c o n t r i b u t i o n of t h e k i n d A (-) Av s o t h a t a l a r g e s p l i t t i n g between E+ and E-, i . e . a l a r g e AV, g i v e s a

V 2E0

l a r g e c o n t r i b u t i o n t o AT. The s o c a l l e d b l m k i n g e f f e c t , which f a v o u r s a - w i t h r e s - p e c t t o u+, i s d i s p l a y e d i n formulas 11 and 12 i n a c l e a r way. These e x p r e s s i o n s can be t h e n compared with formula 13. Furthermore t h e s e g e n e r a l r e l a t i o n s can be e a s i l y worked o u t f o r simple o p e r a t o r s F and simple Hamiltonians H. I n o r d e r t o il- l u s t r a t e how q u i c k l y one can e n t e r t h e h e a r t o f a p r a c t i c a l problem l e t ' s j u s t work o u t some simple c a s e s , s t a r t i n g from schematic f o r c e s of t h e k i n d suggested by Bohr and Mottelson /9/ and L i p p a r i n i and S t r i n g a r i / l o / .

I11 - APPLICATIONS: DIPOLE

+ +

For t h e d i p o l e e x c i t a t i o n we use F = C T . zi and t h e schematic f o r c e of r e f . / 9 / , which i g n o r e s exchange e f f e c t s . I n t h i s c a s e one o b t a i n s

f = I O ~ C D - , D + I ~ O > = - T < r 2 4 >

3 v

2 ~ < r ~ > n - z < r 2 > D 2

where r = 2

i s t h e i s o v e c t o r m . s . r a d i u s and < r >, and < r >

V N - Z P are

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

0 ( V ) = 0 0 ( V ) = 0 0 (TI = 0.

1 2 2

Where VT i s t h e symmetry energy p o t e n t i a l ( 100

-

These r e s u l t s a r e i d e n t i c a l t o t h o s e o b t a i n e d w i t h i n an e x a c t d i a g o n a l i z a t i o n of our schematic Hamiltonian i n a R.P.A. b a s i s /11/.

From t h e s e formulas one can e a s i l y work o u t numerical e s t i m a t e s f o r AV , AT and u+ and a _ . Using VT = 120 MeV,

ri

A-4

P

JOURNAL DE PHYSIQUE

A Av - _T ⁼ 60 MeV, A 5/3

5

Note t h a t t h e s p l i t t i n g

60 2 3

E - E = ( T + l ) - + (T+l) (2T-1) - ^(MeV)

T+1 T A A5/3

has c o n t r i b u t i o n s n o t only from t h e " f i r s t o r d e r " terms ( t h e famous - 60 MeV law) b u t a l s o from "second o r d e r " terms which a r e s i g n i f i c a n t i n s i z e and s y s t e m a t i c . The A l a t t e r h a s been r e f e r r e d i n t h e p a s t a s t h e i s o t e n s o r energy /7/. Data /4a/ a r e now a v a i l a b l e f o r s e v e r a l n u c l e i and t h e y show-up c l e a r l y t h e " i s o t e n s o r " energy; our schematic p r e d i c t i o n s a r e i n l i n e w i t h t h e d a t a , a s t h e following t a b l e shows:

90Zr 3.9(4.3) 2.2 (2.9) 51 (60) 36 (23)

116

'n 4 . 0 ( 5 . 3 ) 2.5 ( 3 . 5 40 (60) 22 (23)

120 S 5 . 5 ( 6 . 3 ) 3.6 ( 4 . 1 ) 53 (60) 19 (23)

124 S 6.3 (7.4) 4.6 ( 4 . 6 ) 54 (60) 13 (23)

208

Pb 11.2(8.1) 4.5 ( 4 . 8 ) 73 (60) 23 (23)

l-&,le 1. Data and T h e o r e t i c a l P r e d i c t i o n s ( i n p a r e n t h e s i s ) : e v e r y t h i n g i s expressed i n MeV.

As t o t h e r e s u l t s on a+ and a _ , d a t a a r e r a t h e r s c a r c e ; we can compare our r e s u l t s with t h o s e o b t a i n e d w i t h i n an R.P.A. framework by Auerbach and Klein /12/ u s i n g a Skyrme I11 f o r c e . For example f o r 2 0 8 ~ b We o b t a i n :

2 2

Where t h e f i r s t row corresponds t o VT = 120 r = 5.55, r, = 1.15 r and t h e second t o vT = 100 r: = r2 = (5.55)'.

To be compared with 854 and 9.6 r e s p e c t i v e l y of r e f . / l 2 / . I n comparing t h e r e s u l t s one should keep i n mind t h a t whereas our formulas 11,12 p r e s e r v e t h e sum r u l e auto- m a t i c a l l y : a-

-

-

Data on u+/D- have been r e c e n t l y r e p o r t e d /4b/ f o r 120 S,. We o b t a i n o - / ~ + = 5.3 t o be compared w i t h t h e measured value c7-/ut = 4.

I V

-

The quadrupole o p e r a t o r can be t r e a t e d i n a s i m i l a r way, s t a r t i n g from t h e a p p r o p r i a t e schematic quadrupole-quadrupole i n t e r a c t i o n . We use 3 = C 7 xi y i . I n t h i s c a s e one h a s

<r4> = 1.3 < r 2 > 2 = 1252 f$i and <rv> 4 = 1551 f$

4 4

a+ = 1250 fm and 0- = 10350 fm

t o be compared w i t h a = 1283 and 0- = 8072 of r e f . 12; i n t h i s l a s t c a s e , because of numerical l i m i t a t i o n s , 20 % of t h e s t r e n g t h i s missing i n t h e d i f f e r e n c e a- - a+.

A s a f i n a l example we work o u t t h e case of t h e d i p o l e s p i n f l i p o p e r a t o r $ = c ? ~ ~ u ~ ~ .

One s t a r t s from t h e a p p r o p r i a t e schematic Hamiltonian c o n t a i n i n g a D i p o l e - s p i n f l i p - Dipole s p i n - f l i p s e p a r a b l e i n t e r a c t i o n ( w e d i s r e g a r d s m a l l s p i n - o r b i t e f f e c t s ) . Let us c a l l VaT t h e coupling c o n s t a n t r e l e v a n t t o t h i s i n t e r a c t i o n : w i t h i n t h i s Hamilto- n i a n t h e imput parameters f o r our formulas 9,10,11,12 a r e :

Note t h a t f o r t h i s case 0 ( V ) and 0 ( V ) a r e d i f f e r e n t from z e r o and p r o p o r t i o n a l t o

1 2 2

(VT - V a T ) , ( f i r s t o r d e r c o n t r i b u t i o n ) and 0 (T) t o (V - V ) (second o r d e r con-

2 f 5T

t r i b u t i o n ) . ^AS a consequence

I n t h i s case t h e asymmetry term AT depends on VUT and AV i s t h e n e t r e s u l t emerging from t h e e f f e c t of t h e symmetry energy T ^VT and t h e c o l l e c t i v e ( o p p o s i t e ) s h i f t i n - duced by t h e d i p o l e - d i p o l e i n t e r a c t i o n A VoT - T . Once more t h e s e r e s u l t s a r e t h e

2 A

same a s t h o s e deduced w i t h a d i r e c t d i a g o n a l i z a z i o n of t h e schematic Hamiltonian w i t h i n R.P.A. /13/.

V - CONCLUSIONS

We have i l l u s t r a t e d w i t h some examples t h e u s e f u l n e s s of formulas (9,10,11,12) u s i n g simple schematic Hamiltonians and e x c i t a t i o n o p e r a t o r s . To s t u d y t h e p r e d i c t i o n s given on t h i s m a t t e r by more s o p h i s t i c a t e d Hamiltonians one h a s t o work-out t h e com mutators e n t e r i n g i n ( 6 ) . Skyrme f o r c e s o r more conventional Hamiltonians of t h e type H = V ( r i j ) (W + MPX - ^{H P ~} + BP. U . ) can b e p r o f i t a b l y used t o t h i s aim. For

I j i j 1 3

t h i s l a s t i n t e r a c t i o n and a d i p o l e o p e r a t o r e x p l i c i t e x p r e s s i o n s f o r t h e v a r i o u s commutators can be found i n r e f . / l 4 / . Dynamical Coulomb e f f e c t s can a l s o b e included

C4-326 JOURNAL DE PHYSIQUE

i n ( 6 ) and t h e a n a l y s i s can be extended t o o t h e r i n t e r e s t i n g o p e r a t o r s , a s t h e mo- nopole and t h e magnetic d i p o l e o p e r a t o r s . For t h e f i r s t , i f one d i s r e g a r d s Coulomb e f f e c t s and exchanges, t h e formal r e s u l t s would be t h e same a s t h o s e o b t a i n e d f o r t h e quadrupole (formulas 16) where now Eo i s t h e g i a n t resonance energy of t h e mo- nopole e x c i t a t i o n . F o r t h e second t h e r e s u l t s would be s i m i l a r t o t h o s e expressed i n formulas 17 with t h e f a c t o r rz / r2 r e p l a c e d by u n i t y and Eo i n t e r p r e t e d a s t h e energy of t h e magnetic g i a n t resonance.

T h i s m a t t e r w i l l be t h e s u b j e c t of a more d e t a i l e d work.

References

a ) H. Morinaga, Phys. Rev. 97, 444 (1955) and

b) M.H. Mac F a r l a n e , i n I s o b a r i c Spin i n Nuclear P h y s i c s , e d i t e d by J.D.Fox and D. Robson (A.P. I n c . , New York 1966), p.383.

For a Review s e e P . P a u l , i n Photonuclear Reactions and A p p l i c a t i o n s 1973, e d i t e d by B.L. Berman (Lawrence Livermore Laboratory, Univ. of C a l i f o r n i a , 1973),Vol 1 p. 401.

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C ) R.;. Alky;z and S. F a l l i e r o s , Phys. Rev. L e t t . 7, 1016 (1971)

a ) W.A. S t e r r e n b u r g , S.M. A u s t i n , R.P. De V i t o and A. Galonsky, Phys.Rev.Lett.45, 1839 (1980)

b ) H.W. Baer, R. Bolton, J . D . Bowman, M.D. Cooper, F. Cverna, J - A l s t e r , A.Doron, A. E r e l l , M.A. MoinesterF.Blackmore and E.R. S i c i l i a n o , Proceedings of t h e I n t e r n a t i o n a l Conference on Nuclear S t r u c t u r e , e d i t e d by A. Van d e r Woude and B.J. Verhaar, North-Holland,Amsterdam. See a l s o c o n t r i b u t i o n 5.2 of t h i s conference.

a ) N.Marty, C. D j a l a l i , M. Morlet, A. W i l l i s , J . C . J o u r d a i n , N-Tnantaraman, G.M. Crawley, and A. Galonsky I.P.N.O. DRE 82-21.

b ) A-Richter, Proceedings of t h e I n t e r n a t i o n a l Conference on Nuclear P h y s i c s , F l o r e n c e , August 1983.

R. Leonardi, L e c t u r e s Notes i n Physics Vo1.61 pag. 501 E d i t e d by J . E h l e r s e t a l . , S p r i n g e r Verlag B e r l i n

R. Leonardi,Phys. Rev., 1214, 385 (1976)

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R. Leonardi, proceeding of t h e Topical Meeting on Nuclear F l u i d Dynamics, T r i e s t e 1983

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