EFFECTIVE INTERACTIONS AND THE EXCITATION OF GIANT RESONANCES

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EFFECTIVE INTERACTIONS AND THE EXCITATION OF GIANT RESONANCES

W. Love, M. Franey

To cite this version:

W. Love, M. Franey. EFFECTIVE INTERACTIONS AND THE EXCITATION OF GI- ANT RESONANCES. Journal de Physique Colloques, 1984, 45 (C4), pp.C4-231-C4-249.

�10.1051/jphyscol:1984418�. �jpa-00224083�

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JOURNAL DE PHYSIQUE

Colloque C4, suppl6ment au n03, Tome 45, mars 1984 page C4-231

EFFECTIVE INTERACTIONS AND T H E EXCITATION OF GIANT RESONANCES

W.G. Love and M.A. Franey*

Department of Physics and Astronomy, The U n i v e r s i t y o f Georgia, Athens, GA 30602, U.S.A.

*Department o f physics, The U n i v e r s i t y of Minnesota, Minneapolis, MN 55455, U.S.A.

Rdsumd - On 6 t u d i e l e r 6 l e d e s i n t e r a c t i o n s e f f e c t i v e s e n t r e p r o j e c t i l e s e t n u c l d o n s - c i b l e d a n s l ' e x c i t a t i o n d e s r d s o n a n c e s g d a n t e s . On s e l i m i t e a u c a s d e s e x c i t a t i o n s i n d u i t e s p a r d e s p r o t o n s d ' e n e r g i e s i n t e r m d d i a i r e s , oh l e s c a r a c t e r i s t i q u e s p r i n c i p a l e s d e l a sonde peuvent s e comprendre 2 p a r t i r d e l a m a t r i c e t nucldon-nucldon l i b r e . On C t u d i e l a f i a b i l i t d q u a n t i t a t i v e d e l ' a p p r o c h e m i c r o s c o p i q u e d e s r e s o n a n c e s g d a n t e s d a n s l a q u e l l e l e s i n t e r - a c t i o n s e f f e c t i v e s s o n t b a s e e s s u r d e s m a t r i c e s G ou d e s m a t r i c e s t l i b r e s , en m e t t a n t l ' a c c e n t s u r l e s i n c e r t i t u d e s p r o v e n a n t d e s p a r a m s t r e s d e s poten- t i e l s o p t i q u e s .

A b s t r a c t - The r o l e o f p r o j e c t i l e - n u c l e o n e f f e c t i v e i n t e r a c t i o n s i n t h e e x c i t a - t i o n of g i a n t r e s o n a n c e s i s s t u d i e d . E x p l i c i t c o n s i d e r a t i o n s a r e l i m i t e d t o proton-induced e x c i t a t i o n s a t i n t e r m e d i a t e e n e r g i e s where t h e main f e a t u r e s o f t h e p r o b e c a n b e u n d e r s t o o d i n t e r m s of t h e f r e e nucleon-nucleon t - m a t r i x . The q u a n t i t a t i v e r e l i a b i l i t y of a m i c r o s c o p i c approach t o t h e s t u d y o f g i a n t r e s o n a n c e s u s i n g e f f e c t i v e i n t e r a c t i o n s b a s e d on G-matrices and f r e e t - m a t r i c e s i s s t u d i e d and o p t i c a l model u n c e r t a i n t i e s a r e s t r e s s e d .

1. INTRODUCTION

The f i e l d of g i a n t r e s o n a n c e s i n n u c l e i h a s b e e n s u b s t a n t i a l l y e n r i c h e d 11-51 d u r i n g t h e p a s t s e v e r a l y e a r s . I n a d d i t i o n t o o b t a i n i n g more s y s t e m a t i c s f o r t h e more " c l a s s i c a l " modes o f e x c i t a t i o n s u c h a s t h e g i a n t d i p o l e and g i a n t q u a d r u p o l e r e s o n a n c e s , new v a r i e t i e s o f s p i n and i s o s p i n modes o f v a r i o u s m u l t i p o l a r i t i e s have been i d e n t i f i e d . The newer developments have b e e n made p o s s i b l e l a r g e l y by t h e o p e r a t i o n o f a c c e l e r a t o r s a t i n t e r m e d i a t e e n e r g i e s (100 5Eprot (MeV) 5 1000) where t h e c h a r a c t e r i s t i c s o f t h e p r o j e c t i l e - n u c l e u s c o u p l i n g s a r e somewhat d i f f e r e n t from t h o s e a t l o w e r e n e r g i e s .

At any energy a n i m p o r t a n t p a r t of i n t e r p r e t i n g t h e e x c i t a t i o n o f g i a n t r e s o - n a n c e s ( o r any o t h e r s t a t e s ) by h a d r o n s c a t t e r i n g is an u n d e r s t a n d i n g of t h e i n t e r - a c t i o n between t h e p r o j e c t i l e and t h e t a r g e t n u c l e u s . I n a m i c r o s c o p i c approach t h i s means knowing t h e p r o j e c t i l e - n u c l e o n c o u p l i n g a t t h e e n e r g y o f i n t e r e s t . Loosely s p e a k i n g , two t y p e s of m i c r o s c o p i c a p p r o a c h e s have been pursued. The f i r s t 161 i s p u r e l y e m p i r i c a l and c o n s i s t s of c a l i b r a t i n g some s i m p l e i n t e r a c t i o n by i n - s i s t i n g t h a t i t d e s c r i b e t h e e x c i t a t i o n o f well-known low-lying s t a t e s of t h e same symmetry ( s p i n , i s o s p i n , e t c . ) a s t h e g i a n t r e s o n a n c e t o b e s t u d i e d . With t h e pro- l i f e r a t i o n of g i a n t r e s o n a n c e s of d i f f e r e n t m u l t i p o l a r i t y , s p i n and i s o s p i n , t h i s p u r e l y e m p i r i c a l approach becomes awkward and t o some e x t e n t i m p r a c t i c a l . I n t h e second a p p r o a c h one a t t e m p t s t o d e r i v e t h e p r o j e c t i l e - n u c l e o n c o u p l i n g from more

" b a s i c " p r i n c i p l e s o r d a t a . I n t h e c a s e o f p r o t o n s c a t t e r i n g , w h i c h i s emphasized h e r e , t h i s c o u p l i n g is based on nucleon-nucleon (ii-N) d a t a and is o b t a i n e d from e i t h e r t h e f r e e N-N t - m a t r i x / 7 , 8 / o r a d e n s i t y - d e p e n d e n t G-matrix / 9 , 1 0 / . I n p r i n c i p l e t h i s second approach i s much more s a t i s f a c t o r y ; i n p r a c t i c e i t i s a l s o m o n i t o r e d by comparison w i t h t h e e x c i t a t i o n of l o w - l y i n g s t a t e s h a v i n g r e l a t i v e l y well-known t r a n s i t i o n d e n s i t i e s 151. Only t h e second approach w i l l b e c o n s i d e r e d h e r e .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984418

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C4-232 JOURNAL DE PHYSIQUE

I n s e c t i o n 2 some of t h e primary elements f o r d e s c r i b i n g nucleon-nucleus s c a t - t e r i n g a r e d i s c u s s e d b r i e f l y . I n s e c t i o n 3 e f f e c t i v e i n t e r a c t i o n s based on t h e f r e e t-matrix and a density-dependent G-matrix / l o / a r e considered and some of t h e i r most important c h a r a c t e r i s t i c s a r e examined. I n s e c t i o n 4 mostly i s o s c a l a r spin-independent e x c i t a t i o n s a r e s t u d i e d . I n s e c t i o n 5 charge exchange r e a c t i o n s a r e considered and i n s e c t i o n 6 s p i n e x c i t a t i o n s a r e d i s c u s s e d b r i e f l y .

2. SOME GENERAL CONSIDERATIONS

The i n t e r p r e t a t i o n of n u c l e a r e x c i t a t i o n s r e q u i r e s t h e c a l c u l a t i o n of a p r o j e c t i l e - n u c l e u s t-matrix which i n t h e s i n g l e - s c a t t e r i n g distorted-wave approximation considered h e r e is given by

where t h e x(') denote t h e d i s t o r t e d waves and V i p i s t h e e f f e c t i v e i n t e r a c t i o n between t h e p r o j e c t i l e and each t a r g e t nucleon. The i n i t i a l a & f i n a l t a r g e t and p r o j e c t i l e s p i n p r o j e c t i o n s a r e denoted by (M,IY') and ( v , v f ) ; k ( k ' ) i s t h e i n i t i a l ( f i n a l ) p r o j e c t i l e - n u c l e u s momentum and r e l a t i v i s t i c kinematics a r e used. Eq. (5) i s o v e r s i m p l i f i e d i n t h a t no e x p l i c i t allowance has been made f o r exchange e f f e c t s ; i n a c t u a l c a l c u l a t i o n s of nucleon-nucleus s c a t t e r i n g , t h e knock-on exchange terms a r e included by r e p l a c i n g Vi by vTE ⁼Vip[l-Pip] where P i p i s t h e o p e r a t o r which exchanges nucleons i a n d p. !t h a s %een demonstrated elsewhere /11/ t h a t i t is u s u a l l y adequate f o r Ep 2 75 MeV t o regard

vEE

a s a l o c a l o p e r a t o r of t h e same form as Vip and t h i s i s assumed i n t h e formalism presented h e r e .

2.1. T r a n s i t i o n p o t e n t i a l s and t r a n s i t i o n d e n s i t i e s

To i l l u s t r a t e more e x p h p i t l y t h e r a t h e r g e n e r a l way i n which n u c l e a r s t r u c t u r e e n t e r s t h e c a l c u l a t i o n of TFI i t i s convenient t o i n t r o d u c e a t r a n s i t i o n p o t e n t i a l which i s t h e t a r g e t m a t r i x element i n Eq. ( 1 ) . I n u a r t i c u l a r ,

which may be r e w r i t t e n a s :

where p , ( r ) i s t h e n u c l e a r t r a n s i t i o n d e n s i t y d e f i n e d by -+

I I

The s p i n and i s o s p i n dependence of V i e i n t r o d u c e s o t h e r d e n s i t i e s . A s l o n g a s t h e probe a c t s a s a one-body o p e r a t o r i n t h e t a r g e t ' s p a c e t h e t r a n s i t i o n d e n s i t y con- t a i n s a l l of t h e n u c l e a r s t r u c t u r e information which e n t e r s t h e t r a n s i t i o n 1'1'. Of course d i f f e r e n t probes sample t h e t r a n s i t i o n d e n s i t y d i f f e r e n t l y and t h i s makes them complementary. This same complementarity r e q u i r e s t h a t c a r e must be e x e r c i s e d when comparing s t r e n g t h s obtained from t h e use of d i f f e r e n t probes.

2.2. Momentum space c o n s i d e r a t i o n s

Some of t h e most important f e a t u ~ e s ~ o f t h e t r a n s i t i o n amplitude a r e b e t t e r i l l u s t r a t e d i n momentum space. I f V(rp-r) i n Eq. (3) is expressed i n terms of i t s F o u r i e r t r a n s f o r m ? ( q ) Eq. (1) becomes

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where D i s a d i s t o r t i o n f u n c t i o n g i v e n f o r m a l l y by:

+ + +

and becomes G(q-k+kf) i n t h e p l a n e wave l i m i t ; p" i s t h e F o u r i e r t r a n s f o r m of P ,

and q + i s t h e l o c a l momentum t r a n s f e r . I n a d d i t i o n t o i m p o r t a n t d i f f e r e n c e s i n t h e d i s t o r t i o n f u n c t i o n f o r d i f f e r e n t p r o j e c t i l e s V(q) w i l l a l s o b e d i f f e r e n t . I n p a r t i c u l a r , w i t h i n a f o l d i n g model c o n t e x t 1121 t h e F o u r i e r t r a n s f o r m of t h e n u c l e o n c o m p o s i t e - p a r t i c l e i n t e r a c t i o n i s r e l a t e d t o t h a t of t h e N-N i n t e r a c t i o n by

where PC(q) i s t h e F o u r i e r t r a n s f o r m o f t h e composite p r o j e c t i l e d e n s i t y . From Eq.

( 8 ) , t h e f i n i t e s i z e of P C ( r ) i s s e e n t o s u p p r e s s t h e h i g h e r momentum t r a n s f e r s i n t h e s i n g l e s c a t t e r i n g o f a composite p r o j e c t i l e r e l a t i v e t o t h o s e allowed i n nucleon s c a t t e r i n g . By i n s e r t i n g Eq. ( 7 ) i n t o Eq. (8) we may i d e n t i f y t h e probe f u n c t i o n P of r e f . 2 d e f i n e d by

+ + + _J

I t i s t h e e x t e n t t o which P ( k f , k ; r ) r e s e m b l e s t h e m u l t i p o l e o p e r a t o r r Y ~ ( ? ) n e a r t h e peak v a l u e o f t h e d i f f e r e n t i a l c r o s s s e c t i o n t h a t d e t e r m i n e s , f o r example, how r e l i a b l y ( p , p f ) a n d e l e c t r o m a g n e t i c t r a n s i t i o n r a t e s may be compared and r e l a t e d t o m u l t i p o l e sum r u l e s 12,131. A few comparisons of t h e probe f u n c t i o n P w i t h t h e m u l t i p o l e o p e r a t o r rJyJ have a l r e a d y been r e p o r t e d 1141 which show s i g n i f i c a n t d i f - f e r e n c e s f o r p r o t o n e n e r g i e s n e a r 150 MeV.

3. TYPES OF EFFECTIVE N-N INTERACTIONS AND THEIR PROPERTIES

E f f e c t i v e N-N i n t e r a c t i o n s f o r n u c l e o n s c a t t e r i n g a r e u s u a l l y assumed and con- s t r u c t e d t o b e o f t h e form:

Coul

vC, vLS, vT ^{and V} z t a n d f o r c e n t r a l , s p i n - o r b i t , t e n s o r . and Coulomb terms r e - s p e c t i v e l y , r=rip=l;?i-rpland S i p is t h e u s u a l t e n s o r o p e r a t o r . vC, vLS ^andvT

t y p i c a l l y depend on t h e i n c i d e n t p r o j e c t i l e energy / 8 / and, i n t h e c a s e of G-matrix

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C4-234 JOURNAL DE PHYSIQUE

interactions may also depend on the local density /lo/. When Vip of Eq. (11) is re- garded as including the knock-on exchange terms, each column the contributing parts of Vip to excitations of specific S and T transfers to the target. Note that in general vLS contributes to both S=O and S=1 excitations 1151; v C O ~ ~ has been included for completeness. (It should be stressed that the interaction strengths given in refs. 8 and 10 assume that the knock-on exchange terms will be included explicitly.)

For future reference it is convenient to write Vip in terms of the linear (q), orbital (L), spin (S), and total (J) angular momentum which it can transfer to the target in a single step. To do this we introduce the elementary tensors

and for simplicity restrict consideration to the central parts of Vip; the spin- orbit and tensor parts of Vip are discussed in more detail in ref. 5. In terms of these tensors

JiS -C

q dq 1 (-) VS(q) MLSJ(i)-MLSJ(p) LSJ

where isospin indices have been suppressed for brevity. For a given L, S, J and (local) momentum transfer q, E~sj(i) is the operator to which the nuclear structure is sensitive, and for small q this operator is proportional to the multipole operator rL(yL @ 0 ~occuring in long-wavelength electromagnetic processes. ) ~

3.1. Free N-N t-matrix interactions

Where applicable an effective interaction based directly on the free N-N t- matrix provides one of the simplest approaches for obtaining the projectile-nucleus coupling. This approach is especially appealing for the study of giant resonances where the primary objective is to extract nuclear structure information. This approach bypasses in an approximate way many of the difficulties /9,10/ associated with constructing and applying a N-N potential. Unfortunately, the impulse approximation (IA) is not expected to be reliable for proton bombarding energies (Ep) below -100 MeV and for certain classes of transitions, namely S=T=O, a number of correc- tions to the IA have been found necessary /10,16/ at even higher energies (see sect.

4). Despite their limited realm of quantitative validity, effective interactions based on the free N-N t-matrix provide useful insights into trends which may be expected in nucleon-nucleus scattering.

The steps in obtaining an effective interaction like that in Eq. (11) from free N-N scattering amplitudes are described in detail in ref. 8. The net result / 8 / at each energy is a sum of Yukawa terms for representing VC and vLS and r2 times a sum of Yukawa terms for vT. Sorne of the most important characteristics of the t-matrix interaction can be illustrated by plotting the moduli of its

anti-symmetrized matrix elements (tNN) in momentum space in the nucleon-nucleus system /8/ against momentum transfer ( q ) , projectile kinetic energy (Ep) and spin and isospin transfers (S and T).

Figure 1 shows a plot for the central parts of the force at q=O for

100 5 Ep (MeV) 5 800. The subscripts a and T refer to spin and isospin transfers of one unit. T'ne most striking feature of these curves is the strong dominance of the

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'0 200 400 600 800 Ep (MeV)

0.0 1 Î ₂₀₀^l ^l ₄₀₀¹ ¹ ₆₀₀¹ ¹ ₈₀₀Î Î ₁₀₀₀

E

,,

^(MeV)

Fig. 1. Energy dependence of t h e magnitudes of t h e c e n t r a l ( d i r e c t + exchange) p a r t s of t h e N-N t-matrix. On t h e r i g h t i s shown t h e energy dependence of t h e r a t i o

ItC /t:l2 a t z e r o momentum t r a n s f e r .

U T

s c a l a r - i s o s c a l a r p a r t of ^tNNa t a l l e n e r g i e s considered. The very s m a l l (and poorly determined) to s u g g e s t s t h a t t h e c e n t r a l p a r t of t h e f o r c e i s i n e f f e c t i v e f o r ex- c i t i n g i s o s c a l a r s p i n modes. I n s t r i k i n g c o n t r a s t , we s e e t h a t f o r i s o v e c t o r e x c i t a t i o n s t dominates over t, a t s m a l l i n t h i s energy regime. This is emphasized i n

FYI.

1 where t h e r a t i o 1 t / t T l q i s p l o t t e d v e r s u s Ep a t q-0. It i s now w e l l e s t a b l i s h e d /4,5,8,17-131 t h a t tgls ^{S = l}dominance of ~ N N r e n d e r s proton s c a t -

t e r i n g an e s p e c i a l l y s e n s i t i v e probe of i s o v e c t o r s p i n modes a t i n t e r m e d i a t e

e n e r g i e s . This dominance, t o g e t h e r w i t h t h e f a c t t h a t 1 ^toT¹⁵¹^tT1 below about 60 MeV h a s been e s p e c i a l l y important f o r t h e i d e n t i f i c a t i o n and i n t e r p r e t a t i o n of Gamow- T e l l e r (GT) resonances using t h e (p,n) r e a c t i o n /4,5,17,18/.

Fig. 2. Energy dependence of t h e s p i n - o r b i t and t e n s o r p a r t s of t h e t-matrix i n t e r - a c t i o n (t) a t q = 1.5 fm-'.

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F i g u r e 2 i l l u s t r a t e s t h e r e l a t i v e importance o f t h e s p i n - o r b i t and t e n s o r p a r t s of t h e i n t e r a c t i o n a s a f u n c t i o n o f bombarding energy a t a momentum t r a n s f e r of q=1.5 fm-l. T h i s v a l u e of q was chosen f o r i l l u s t r a t i o n s i n c e i t r e p r e s e n t s a com- promise of t h e peak p o s i t i o n s of t h e s p i n - o r b i t and t e n s o r terms which, from r e f . 8 , peak roughly n e a r 2.0 and 1 . 0 fm-I r e s p e c t i v e l y . Nore i m p o r t a n t l y , t h e s e non- c e n t r a l terms t e n d t o dominate i n t h i s r e g i o n of q. Most a p p a r e n t from F i g . 2 is t h e n e a r n e g l i g i b l e s i z e o f t h e i s o v e c t o r s p i n - o r b i t term s o t h a t i s o v e c t o r s p i n modes a t l a r g e q a r e dominated by t h e t e n s o r f o r c e a t a l l e n e r g i e s . The s m a l l n e s s of tLSr a l s o i m p l i e s a s t r o n g s e n s i t i v i t y of i s o v e c t o r S = l a n a l y z i n g powers t o o p t i c a l model p a r a m e t e r s s i n c e t h e r e i s t y p i c a l l y l i t t l e s o u r c e o f asymmetry i n t h e i n e l a s t i c r e a c t i o n mechanism. I s o s c a l a r S = l e x c i t a t i o n s a r e e x p e c t e d t o b e e x c i t e d c o m p e t i t i v e l y by t h e s p i n - o r b i t and t e n s o r p a r t s of tNN and t h i s i s b o r n e o u t i n d e t a i l e d c a l c u l a t i o n s / 8 , 1 7 , 1 9 , 2 0 / .

F i g u r e 3 shows t h e S = l and S=O t r a n s f e r p a r t s o f t~~ a t Ep=140 MeV a s a func- t i o n o f momentun t r a n s f e r q. A s d e s c r i b e d i n r e f . 8 , t o and t,, s h o u l d b e m u l t i p l i e d by -2.0 i n o r d e r t o compare d i r e c t l y w i t h f T and t: f o r nucleon-nucleus s c a t t e r i n g . These l a t t e r q u a n t i t i e s ( f T ) have a l r e a d y been normalized t o r e p r e s e n t t h e s t r e n g t h of t h e t e n s o r f o r c e i n nucleon-nucleus c o l l i s i o n s a s d e s c r i b e d i n t h e appendix of r e f . 8. (The l e f t - h a n d s i d e s of e q u a t i o n s A5a, A5b and A5c of r e f . 8 s h o u l d b e t T B , tTy and tT" r e s p e c t i v e l y . ) For b o t h i s o s c a l a r and i s o v e c t o r t r a n s i t i o n s , i t i s s e e n t o b e i m p o r t a n t t o i n c l u d e t h e t e n s o r f o r c e i n c a l c u l a t i o n s o f nucleon-nucleus s c a t t e r i n g f o r e s s e n t i a l l y a l l v a l u e s o f momentum t r a n s f e r . A s i g n i f i c a n t e x c e p t i o n i s s e e n t o o c c u r f o r t h e e x c i t a t i o n o f S=T=1 modes f o r q 5 0 . 5 fm-I and t h i s i s e s p e c i a l l y i m p o r t a n t / 4 , 8 , 1 7 , 1 8 / f o r i n t e r p r e t i n g GT e x c i t a t i o n s .

F i g . 3 . Moduli o f S = l and S=O p a r t s o f t

I I ^I I ^I

E= 140 MeV, AS=O -

a s a f u n c t i o n o f q a t E = 140 MeV.

P

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3.2. D e n s i t y dependent G-matrix i n t e r a c t i o n s

The most s o p h i s t i c a t e d (and i n p r i n c i p l e t h e most r e l i a b l e ) e f f e c t i v e i n t e r a c - t i o n s c o n s t r u c t e d t o d a t e a r e t h o s e d e r i v e d from G-matrices b a s e d on r e a l i s t i c N-N p o t e n t i a l s and c a l c u l a t e d i n t h e p r e s e n c e o f n u c l e a r m a t t e r where one of t h e two i n t e r a c t i n g n u c l e o n s h a s e n e r g y g r e a t e r t h a n t h e Fermi e n e r g y . These G-matrices a r e o b t a i n e d by s o l v i n g t h e Bethe-Goldstone e q u a t i o n a t a number o f d i f f e r e n t v a l u e s of t h e n u c l e o n d e n s i t y o r Fermi momentum (p = 2k$/3v2). The d e n s i t y dependence a r i s e s from t h e l i m i t a t i o n of s c a t t e r i n g t o unoccupied i n t e r m e d i a t e s t a t e s and from t h e s i n g l e p a r t i c l e e n e r g i e s which depend on t h e Fermi e n e r g y t h r o u g h t h e s i n g l e par- t i c l e p o t e n t i a l / 9 , 1 0 / i n which t h e n u c l e o n s move. The d e n s i t y dependence g e t s weaker w i t h i n c r e a s i n g bombarding e n e r g y where t h e f r a c t i o n o f i n t e r m e d i a t e s t a t e s e x c l u d e d by t h e P a u l i p r i n c i p l e d e c r e a s e s and t h e k i n e t i c energy dominates t h e s i n g l e p a r t i c l e e n e r g i e s .

The c a l c u l a t e d G-matrix i s much t o o c o m p l i c a t e d f o r nucleon-nucleus c a l c u l a - t i o n s s o t h a t a l o c a l e f f e c t i v e i n t e r a c t i o n (Vip) i s i n t r o d u c e d which r e p r o d u c e s some of t h e most i m p o r t a n t m a t r i x e l e m e n t s of G. What f i n a l l y emerges from t h i s p r o c e d u r e i s a n e f f e c t i v e i n t e r a c t i o n l i k e t h a t i n Eq. (11) which depends n o t o n l y on rip b u t a l s o on t h e l o c a l d e n s i t y a t which t h e p a r t i c l e s i n t e r a c t . + T h i s i s c a l l e d t h e l o c a l d e n s i t y a p p r o x i m a t i o n / 9 , 1 0 / (LDA); s c a t t e r i n g programs now e x i s t which can t r e a t d e n s i t y dependence i n t h i s way.

F i g u r e 4 shows a comparison / 1 6 / of e f f e c t i v e i n t e r a c t i o n s a t 140 MeV d e r i v e d from t h e f r e e t - m a t r i x o f r e f . 8 and from d e n s i t y - d e p e n d e n t G-matrices / l o / b a s e d on t h e P a r i s 1211 and Hamada-Johnston 1211 (HJ) p o t e n t i a l s . T h i s f i g u r e s u g g e s t s t h a t d e n s i t y dependent e f f e c t s s h o u l d b e q u i t e s t r o n g o v e r a l a r g e r a n g e o f momentum t r a n s f e r s f o r S=S, T=O e x c i t a t i o n s e s p e c i a l l y w h e n . t h e H J b a s e d e f f e c t i v e i n t e r - a c t i o n i s used. Even a t low d e n s i t y t h e r e a r e l a r g e d i f f e r e n c e s between t h e HJ e f f e c t i v e i n t e r a c t i o n and t h e f r e e t - m a t r i x a t t h i s e n e r g y ; i t h a s n o t b e e n d e t e r - x i n e d t o what e x t e n t t h i s can b e a t t r i b u t e d t o changes i n N-N p h a s e s h i f t s o v e r t h e p a s t twenty y e a r s . The t e n s o r and s p i n - o r b i t p a r t s of t h e e f f e c t i v e i n t e r a c t i o n a r e c a l c u l a t e d t o b e c o n s i d e r a b l y l e s s d e n s i t y dependent /16/.

L

- - _-

_Paris b

_- - -

_Hamada-

- -

_Johnston

-

_I1

-

I I I I I I

1 2 3 1 2 3

q (f m-I)

C C

F i g . 4. Comparison o f ( t o ( q ) = t o o ( q ) ( a t 140 MeV f o r t h e t - m a t r i x i n t e r a c t i o n from r e f . 8 ( s o l i d c u r v e on b o t h s i d e s ) w i t h t h e e f f e c t i v e i n t e r a c t i o n s from r e f . 1 0 b a s e d upon t h e P a r i s and H J p o t e n t i a l s . Long ( s h o r t ) dashed c u r v e s show low ( h i g h ) d e n s i t y l i m i t s .

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4. NATURAL PARITY EXCITATIONS: (S=O, T=O)

With a few notable exceptions (such as the isovector 2 excitation in 12c at +

16.1 MeV) natural parity states ( A T = (-)J) of spin J in+nuclei with O+ ground states are excited by the central (S=O) and spin-orbit (sp term) parts of

VY

through the spin-independent and current-independent transition density of Eq. (4).

There are, however, both proton and neutron transition densities pp and pn. For N=Z nuclei these are usually assumed to be related by pn = !or collective modes in heavier nuclei one often assumes pn = (N/Z)pp so that if pp is known, say from electron scattering, pn can be fixed.

To implement the above ideas it is convenient to rewrite the transition densities in Eqs. (4,7) in terms of their radial parts. Using the multipole expan- sion of we find:

is the radial transition density and the reduced matrix elements are in the conven- tion of ref. 22. The corresponding "radial" part of bItI(<) in Eq. (7) is

For purposes of testing or calibrating an effective interaction it is useful (if not essential) to compare hadron scattering with results using electromagnetic probes for rhe same transition. For natural parity transitions the common ingredient is the proton transition density. For inelastic electron scattering the square of the longitudinal form factor is to a good approximation given by

where P J , ~ ( ~ ) is the proton point density of Eq. (16) and fp(fn) is the intrinsic charge form factor of the proton (neutron) /12/. Similarly, the B(EJ) is given by

Eqs. (17,18) provide important constraints on P J , ~ ; when supplemented by isospin considerations such as discussed earlier p~,, is also determined.

Two different models /2/ which require explicit knowledge of the effective interaction are often used. The first is a fully microscopic treatment in which pJ(r) is obtained by a shell model diagonalization or an RPA calculation and is given by

.,<?ere R is the full radial part of the single particle (or hole) wave function and A+ creates a particle(j2)-hole(jl) pair coupled to angular momentum J. For very

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c o l l e c t i v e s t a t e s , i n c l u d i n g g i a n t r e s o n a n c e s , where f u l l y m i c r o s c o p i c t r a n s i t i o n d e n s i t i e s may b e u n a v a i l a b l e , a deformed d e n s i t y model / 2 / i s o f t e n used. I n one v e r s i o n of t h i s model t h e t r a n s i t i o n d e n s i t y p j , i ( i = p o r n) i s t a k e n t o b e

where pgS i s t h e ground s t a t e d e n s i t y and SJ ( t h e d e f o r m a t i o n l e n g t h ) is o b t a i n e d from t h e B(EJ) o r from t h e l o n g i t u d i n a l e l e c t r o n s c a t t e r i n g form f a c t o r v i a Eqs.

( 1 7 , 1 8 ) . The p r o t o n p o i n t t r a n s i t i o n d e n s i t y may b e d e t e r m i n e d even more d i r e c t l y when t h e c h a r g e t r a n s i t i o n d e n s i t y i s a v a i l a b l e by u n f o l d i n g t h e n u c l e o n ' s c h a r g e

d i s t r i b u t i o n a s i m p l i e d i n Eq. ( 1 7 ) .

I n t h e n e x t two s e c t i o n s we a p p l y t h e above i d e a s t o t h e e x c i t a t i o n of low- l y i n g n a t u r a l p a r i t y s t a t e s i n 12c and 2 0 8 ~ b where r e l a t i v e l y r e l i a b l e t r a n s i t i o n d e n s i t i e s a r e a v a i l a b l e from i n e l a s t i c e l e c t r o n s c a t t e r i n g . T h i s i s done w i t h b o t h t - m a t r i x and G-matrix e f f e c t i v e i n t e r a c t i o n s a t a number of e n e r g i e s i n a n a t t e m p t t o a s s e s s t h e i r r e l i a b i l i t y f o r s t u d y i n g t h e e x c i t a t i o n of l e s s well-known e x c i t a - t i o n s such a s g i a n t r e s o n a n c e s .

4.1. The 4.44 MeV, 2 T=O e x c i t a t i o n i n + 12c

Although "C i s q u i t e removed from i n f i n i t e n u c l e a r m a t t e r t h i s t r a n s i t i o n was chosen b e c a u s e i t s p r o t o n t r a n s i t i o n d e n s i t y and ground s t a t e d e n s i t y a r e w e l l e s t a b l i s h e d o v e r a wide r a n g e of momentum t r a n s f e r and pn = $ f o r N=Z n u c l e i . I n a d d i t i o n , ( p , p V ) d a t a 119,231 f o r t h i s t a n s i t i o n a r e a v a i l a b l e a t s e v e r a l e n e r g i e s between 100 and 800 MeV. The u n p u b l i s h e i d a t a of K. J o n e s was u s e d a t 398 MeV.

The t r a n s i t i o n d e n s i t y was c o n s t r u c t e d from t h e l p s h e l l wave f u n c t i o n s 1241 of Cohen and K u r a t h and r e n o r m a l i z e d (upward) t o a g r e e w i t h t h e l o n g i t u d i n a l ( e , e ' ) form f a c t o r o u t t o q-2-2.5 fm-l. T h i s c a l i b r a t e d t r a n s i t i o n d e n s i t y was t h e n u s e d t h r o u g h o u t . Comparisons between c a l c u l a t e d and measured c r o s s s e c t i o n s a r e shown i n F i g . 5 f o r p r o t o n beam e n e r g i e s of 1 2 0 , 200, 398 and 800 MeV. The phenomenological (PH) o p t i c a l model p a r a m e t e r s a t 120 and 200 MeV a r e t h o s e u s e d i n r e f . 1 9 ; t h o s e a t 398 and 800 MeV a r e u n p u b l i s h e d . The l a b e l I A d e n o t e s t h a t a n u p d a t e d v e r s i o n o f t n e f r e e t - m a t r i x i n t e r a c t i o n d e s c r i b e d i n r e f . 8 h a s b e e n u s e d ; t h e l a b e l GV d e n o t e s t h e d e n s i t y - d e p e n d e n t G-matrix o f r e f . 1 0 o f t h e a p p r o p r i a t e energy. The l a b e l FC ( f o l d e d - c o n s i s t e n t ) d e n o t e s t h a t t h e o p t i c a l p o t e n t i a l h a s been c a l c u l a t e d i n a f o l d i n g model from t h e same e f f e c t i v e i n t e r a c t i o n a s t h a t m e d i a t i n g t h e t r a n s i - t i o n . The c a l c u l a t e d d i f f e r e n t i a l c r o s s s e c t i o n s h a v e b e e n r e n o r m a l i z e d a s shown i n F i g . 5. R e n o r m a l i z a t i o n of t h e c a l c u l a t e d peak c r o s s s e c t i o n s n e a r 1 fm-1 i s n e c e s s a r y a t e a c h beam energy o t h e r t h a n 800 MeV f o r b o t h t- and G-matrices. At 120 and 200 MeV t h e G-matrix c a l c u l a t i o n s u s i n g phenomenological d i s t o r t e d waves which d e s c r i b e e l a s t i c s c a t t e r i n g g e n e r a t e d i f f e r e n t i a l c r o s s s e c t i o n s whose s h a p e s a r e c l e a r l y s u p e r i o r t o t h o s e c a l c u l a t e d w i t h t h e f r e e t - m a t r i x i n t e r a c t i o n . The b r o a d e r peak p r e d i c t e d by t h e G-matrix c a l c u l a t i o n s may b e t r a c e d t o t h e enhancement ( s u p p r e s s i o n ) of l a r g e ( s m a l l ) momentum t r a n s f e r s r e l a t i v e t o t h e t - m a t r i x c a l c u l a - t i o n s n o t e d by K e l l y 1161. (See F i g . 4.) T h i s i s i l l u s t r a t e d e x p l i c i t l y i n F i g . 5 a t 200 MeV where t h e c e n t r a l c o n t r i b u t i o n a l o n e (GVPHC) i s shown. T h i s same c u r v e a l s o i l l u s t r a t e s t h e i m p o r t a n c e of t h e N-N s p i n - o r b i t i n t e r a c t i o n a t t h i s e n e r g y ; even a t t h e peak c r o s s s e c t i o n i n c l u s i o n o f vLS r o u g h l y d o u b l e s t h e c a l c u l a t e d c r o s s s e c t i o n and t h i s h a s i m p o r t a n t i m p l i c a t i o n s f o r t h e s t u d y of g i a n t r e s o n a n c e s . The N-N s p i n - o r b i t t e r m s i n t h e t - m a t r i x and G-matrix i n t e r a c t i o n s a r e q u i t e s i m i l a r . As e x p e c t e d , t h e s e s h a p e d i f f e r e n c e s between t h e IAPH and GWH c a l c u l a t i o n s a r e r e l a t i v e l y s m a l l n e a r 400 MeV where t h e s h a p e and magnitude o f t h e IAPH c r o s s s e c t i o n a r e i n s l i g h t l y b e t t e r agreement w i t h t h e d a t a .

It h a s been n o t e d / 1 6 / t h a t b e t t e r agreement between c a l c u l a t e d and measured o b s e r v a b l e s a r e o f t e n o b t a i n e d when c o n s i s t e n t (FC) d i s t o r t e d waves a r e used i n t h e i n e l a s t i c c a l c u l a t i o n . The r e s u l t s o f u s i n g FC d i s t o r t e d waves a r e shown i n F i g . 5 f o r t h e c a s e o f t h e G-matrix a t Ep = 1 2 0 , 200 and 398 MeV. The s h a p e s o f t h e c a l c u - l a t e d c r o s s s e c t i o n s a r e u n i f o r m l y p o o r e r t h a n w i t h phenomenological waves; t h e r e - q u i r e d r e n o r m a l i z a t i o n s a r e s l i g h t l y c l o s e r t o u n i t y . I n p a r t i c u l a r , i f t h e c a l c u -

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F i g . 5 . C r o s s s e c t i o n s f o r t h e "C ( p , p l ) r e a c t i o n t o t h e 4 . 4 4 MeV 2 s t a t e a t p r o - +

t o n e n e r g i e s of 1 2 0 , 200, 398 and 800 MeV. The r e n o r m a l i z a t i o n f a c t o r s N(N') a r e d i s c u s s e d i n t h e t e x t a s a r e t h e d i f f e r e n t t y p e s of c a l c u l a t i o n IAPH e t c . N ' i s o n l y u s e d a t 398 MeV. The c u r v e s ^(-a- ) a t 120 and 200 MeV c o r r e s p o n d t o a f i x e d - d e n s i t y ( k ~ = 0 . 5 fm-l) G-matrix c a l c u l a t i o n and t o a c e n t r a l - o n l y d e n s i t y - d e p e n d e n t G-matrix c a l c u l a t i o n (GVPHC) r e s p e c t i v e l y .

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l a t e d c r o s s s e c t i o n s a r e s c a l e d t o match t h e peak e x p e r i m e n t a l c r o s s s e c t i o n s , t h e r e n o r m a l i z a t i o n f a c t o r f o r t h e GVPH and GVFC c a l c u l a t i o n s may b e summarized a p p r o x i m a t e l y by ( s e e T a b l e 1 ) :

Over t h i s same e n e r g y r a n g e N(1AE-I) = 0.72 2 0.12. It i s r e a s o n a b l e t o a s k how w e l l t h e s e FC d i s t o r t e d waves d e s c r i b e e l a s t i c s c a t t e r i n g . T h i s h a s b e e n examined f o r q 5 2.5 fm-I a t 120 and 200 MeV where t h e G-matrix f o l d e d p o t e n t i a l s o v e r e s t i m a t e 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 s i n a v e r y s i m i l a r way a t each e n e r g y , e s p e c i a l l y f o r q 5 0.8 fm-1 and q 1 . 5 fm-1. At each energy t h e c a l c u l a t e d e l a s t i c a n a l y z i n g powers a r e s i g n i f i c a n t l y improved r e l a t i v e t o t h o s e o b t a i n e d w i t h t h e ( s e a r c h e d ) phenomenological o p t i c a l p a r a m e t e r s , e s p e c i a l l y f o r q $ 0 . 8 fm-I.

The c o n s i d e r a t i o n s o f t h i s s e c t i o n s u g g e s t t h a t e i t h e r t h e t - m a t r i x o r G-matrix i n t e r a c t i o n may b e used t o c a l c u l a t e peak c r o s s s e c t i o n s i n l i g h t n u c l e i a f t e r r e - n o r m a l i z a t i o n ; when s h a p e c o n s i d e r a t i o n s a r e i m p o r t a n t f o r m u l t i p o l a r i t y i d e n t i f i c a - t i o n s a y , t h e G-matrix, is more r e l i a b l e between 100 and 200 MeV. The n e c e s s i t y t o r e n o r m a l i z e t h e G-matrix i s n o t understood. Also, from Fig. 5, t h e low-density ( k F = 0.5 fm-I) G-matrix c a l c u l a t i o n f o r t h e 12c e x c i t a t i o n a t 120 MeV d i f f e r s from t n e f r e e t - m a t r i x c a l c u l a t i o n a t l a r g e q by a s much a s i t d i f f e r s from t h e f u l l density-dependent G-matrix c a l c u l a t i o n . T h i s r e s u l t indicates t h a t t h e n o t e d i m - provement o b t a i n e d u s i n g t h e G-matrix i s o n l y p a r t i a l l y due t o i t s d e n s i t y dependence.

4.2. The 2.61 MeV, 3- e x c i t a t i o n i n 2 0 8 ~ b

T h i s c l a s s i c c o l l e c t i v e e x c i t a t i o n was chosen f o r s t u d y b e c a u s e of what we know about i t a s a r e s u l t of t h e v a s t amount o f a t t e n t i o n i t h a s r e c e i v e d , b o t h t h e o r e t - i c a l l y and e x p e r i m e n t a l l y . I n p a r t i c u l a r , t h e p r o t o n ground s t a t e and t r a n s i t i o n /25/ d e n s i t i e s have been e x t r a c t e d from e l e c t r o n s c a t t e r i n g measurements; i n a d d i t i o n , c a l c u l a t e d (RPA) p r o t o n a n d n e u t r o n t r a n s i t i o n d e n s i t i e s a r e a v a i l a b l e which p r o v i d e a good d e s c r i p t i o n o f t h e l o n g i t u d i n a l c h a r g e form f a c t o r . I n e l a s t i c p r o t o n d a t a a r e a l s o a v a i l a b l e f o r t h i s t r a n s i t i o n a t a number o f e n e r g i e s . Here we c o n s i d e r ( p , p t ) d a t a a t 135 /26/, 200 (ORNL - Oreg. c o l l a b o r a t i o n and r e f . 27),

334 (F. B e r t r a n d e t a l . , unpublished) and 800 I 2 8 1 NeV.

The phenomenological o p t i c a l model p a r a m e t e r s a t 135 MeV were o b t a i n e d by r e - f i t t i n g t h e e l a s t i c s c a t t e r i n g d a t a o f S c o t t e t a l . mentioned i n r e f . 26. A t 200 and 334 MeV t h e p a r a m e t e r s were p r o v i d e d by P. Schwandt and a t 800 MeV we used para- meter s e t 11 o f r e f . 28.

Two d i f f e r e n t t r a n s i t i o n d e n s i t i e s were used. One i s from a n P A c a l c u l a t i o n by t h e J i i l i c h group. The o t h e r i s a phenomenological d e n s i t y i n which t h e p r o t o n p a r t was o b t a i n e d d i r e c t l y from ( e , e l ) measurements 1251 a f t e r u n f o l d i n g t h e c h a r g e d i s t r i b u t i o n s of t h e p r o t o n and n e u t r o n ; t h e n e u t r o n p o i n t d e n s i t y was assumed t o s a t i s f y

The c o n s t a n t u was a d j u s t e d 1281 u n t i l t h e ( p , p l ) c r o s s s e c t i o n a t 800 MeV was de- s c r i b e d i n t h e d i s t o r t e d wave i m p u l s e approximation (DWIA). The v a l u e o f u i s 1.1 r e f l e c t i n g a n e a r l y p u r e c o l l e c t i v e e x c i t a t i o n i n t h e hydrodynamical s e n s e . Calcu- l a t i o n s u s i n g t h e s e two d e n s i t i e s a t 800 MeV a r e shown i n F i g . 6 . The u n a d j u s t e d J S l i c h t r a n s i t i o n d e n s i t y l e a d s t o c r o s s s e c t i o n s (denoted by B i n F i g . 6) which f a l l o f f somewhat t o o r a p i d l y w i t h a n g l e and which a r e s l i g h t l y s h i f t e d forward r e l a t i v e t o t h e d a t a . The phenomenological t r a n s i t i o n d e n s i t y ( A ) y i e l d s a n e x c e l - l e n t d e s c r i p t i o n of t h e s h a p e and magnitude of t h e e x p e r i m e n t a l d a t a . T h i s s h a p e d i f f e r e n c e a r i s e s from t h e s h a p e a n d magnitude d i f f e r e n c e s between t h e n e u t r o n t r a n s i t i o n d e n s i t i e s i n t h e J i i l i c h and phenomenological d e n s i t i e s . The J i i l i c h n e u t r o n t r a n s i t i o n d e n s i t y peaks a t a l a r g e r r a d i u s t h a n t h e p r o t o n t r a n s i t i o n den- s i t y ; i n t e r m s o f t h e r3y3 moments, a i n Eq. (21) i s 1.02 f o r t h e J i i l i c h t r a n s i t i o n d e n s i t y . The phenomenological t r a n s i t i o n d e n s i t y was used below 800 MeV.

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C4-242 JOURNAL DE PHYSIQUE

At 135 MeV n e i t h e r t h e G-matrix n o r t - m a t r i x c a l c u l a t i o n p r o v i d e s a good de- s c r i p t i o n o f t h e d a t a when t h e phenomenological (Woods-Saxon) o p t i 5 8 1 p o t e n t i a l i s used. (The s q u a r e s a r e a c t u a l l y d a t a I261 f o r t h e L=3 d o u b l e t i n 7 ~ b . ) From F i g . 6 each o f t h e s e c a l c u l a t e d c r o s s e c t i o n s r e q u i r e s a r e n o r m a l i z a t i o n o f 0.7 i n rough agreement w i t h t h e r e s u l t s f o r 'zC a t 120 MeV. The c o n s i s t e n t G-matrix c a l c u l a t i o n p r o v i d e s a n e x c e l l e n t d e s c r i p t i o n o f t h e c r o s s s e c t i o n between 20" and 40" w i t h o u t any r e n o r m a l i z a t i o n . T h i s is i n s h a r p c o n t r a s t t o t h e s i t u a t i o n f o r t h e 2 1 s t a t e i n 1 2 C where t h e agreement i n s h a p e between t h e c a l c u l a t e d and e x p e r i m e n t a l c r o s s sec- t i o n s d e t e r i o r a t e d when t h e c o n s i s t e n t o p t i c a l p o t e n t i a l was used. The c a l c u l a t e d e l a s t i c s c a t t e r i n g c r o s s s e c t i o n ( n o t shown) i s i n o n l y modest agreement w i t h t h e e x p e r i m e n t a l d a t a when t h e G-matrix f o l d e d o p t i c a l p o t e n t i a l i s used and t h e n e u t r o n ground s t a t e d e n s i t y i s t a k e n t o b e N/Z t i m e s t h e p r o t o n d e n s i t y . It i s u n f o r t u n a t e t h a t d a t a f o r t h e f i r s t maximum a r e u n a v a i l a b l e a t t h i s energy s i n c e t h a t i s t h e most r e l e v a n t peak f o r g i a n t r e s o n a n c e s t u d i e s and b e c a u s e t h e d i f f e r e n t m i c r o s c o p i c models p r e d i c t s i g n i f i c a n t l y d i f f e r e n t c r o s s s e c t i o n s a t t h e f i r s t peak. The s t a n - d a r d c o l l e c t i v e model, which p r o v i d e s a n e x c e l l e n t d e s c r i p t i o n o f t h e s h a p e of t h e c r o s s s e c t i o n f o r t h i s t r a n s i t i o n a t each energy c o n s i d e r e d , y i e l d s a peak c r o s s s e c t i o n o f -27 mb/sr n e a r 12.5' when s c a l e d t o match t h e a v a i l a b l e d a t a f o r

eCm 2 15'. Somewhat o l d e r d a t a /29/ a t 1 5 5 MeV s u g g e s t a peak c r o s s s e c t i o n n e a r 35 m b l s r .

A t 200 MeV and 334 MeV d a t a i s a v a i l a b l e o n l y f o r t h e f i r s t maximum. A t 200 (334) MeV b o t h t - m a t r i x and G-matrix c a l c u l a t i o n s r e q u i r e a r e n o r m a l i z a t i o n f a c t o r of 0 . 5 (-0.38) when phenomenological o p t i c a l model p a r a m e t e r s a r e used. The c o n s i s - t e n t G-matrix c a l c u l a t i o n s (GVFC) r e q u i r e r e n o r m a l i z a t i o n f a c t o r s o f -0.8 and 0.5 a t 290 and a t 334 MeV r e s p e c t i v e l y . The a v a i l a b l e d a t a do n o t d i f f e r e n t i a t e between t h e s h a p e s d e r i v e d from d i f f e r e n t c a l c u l a t i o n s ; l a r g e r a n g l e d a t a would b e most welcome.

A summary of r e n o r m a l i z a t i o n f a c t o r s d e f i n e d by ~ ( e x p t ) = Nxtr(theory) n e a r t h e f i r s t peak f o r t h e d i f f e r e n t t y p e s o f c a l c u l a t i o n s i s shown i n T a b l e 1 f o r b o t h

12c and 2 0 8 ~ b . The r e l a t i v e l y l a r g e s p r e a d i n t h e r e n o r m a l i z a t i o n f a c t o r s f o r t h e 3 i t r a n s i t i o n i n 2 0 8 ~ b i s confusing. The e x p e r i m e n t a l s i t u a t i o n i s e q u a l l y con- f u s i n g , e s p e c i a l l y between 100 and 200 MeV where t h e peak e x p e r i m e n t a l c r o s s s e c t i o n f o r t h e 3 i t r a n s i t i o n v a r i e s e r r a t i c a l l y ; t h e l a r g e s t J291 and s m a l l e s t 1281 peak c r o s s s e c t i o n s d i f f e r by a f a c t o r o f -2.5 i n t h i s energy i n t e r v a l ! Although t h e

102 L -

- 208~b(p,p'), 0%3-, Ex =2.61 MeV -

- -

-

- -

10-1 - - - -

- -

10-2 I I I I I I , I I I I I I

0 I0 20 30 40 50 6 O X - 0 5 10 I5

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F i g . 6. Cross s e c t i o n s f o r t h e 2 0 8 ~ b ( p , p F ) r e a c t i o n t o t h e 2 . 6 1 MeV 3- s t a t e a t p r o t o n e n e r g i e s of 135, 200, 334 and 800 MeV. The 200 MeV d a t a i s t h a t of t h e ORNL- Oregon c o l l a b o r a t i o n . The l e g e n d i s e x p l a i n e d i n t h e t e x t ( s e e s e c t , 4 . 1 . ) .

d a t a s e t s c o n t a i n i n g t h e s e e x t r e m e v a l u e s o f t h e c r o s s s e c t i o n were n o t used i n t h e p r e s e n t a n a l y s i s , t h e s e c o n s i d e r a t i o n s h e l p p l a c e t h e r e n o r m a l i z a t i o n f a c t o r s i n T a b l e 1 i n p e r s p e c t i v e .

T a b l e 1. R e n o r m a l i z a t i o n F a c t o r s f o r SzO, TTO T r a n s i t i o n s

Nucleus E (MeV) N(1AF'I-I) a) N ( G V P H ) ~ ) N(GVFC)

P

a ) t - m a t r i x i n t e r a c t i o n , phenomenological o p t i c a l p o t e n t i a l .

"G-matrix i n t e r a c t i o n , phenomenological o p t i c a l p o t e n t i a l .

matrix i n t e r a c t i o n , f o l d e d G-matrix o p t i c a l p o t e n t i a l .

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c4-244 JOURNAL DE PHYSIQUE

The d a t a f o r t h e 3 i t r a n s i t i o n i n 2 0 8 ~ b may a l s o b e d e s c r i b e d by t h e c o l l e c t i v e model which y i e l d s d e f o r m a t i o n l e n g t h s 6 5 BR of 0.73, 0 . 6 7 , 0 . 6 1 and 0.82 fm a t 135, 200, 334 and 800 MeV r e s p e c t i v e l y , w i t h a n a v e r a g e v a l u e of 0.71. ( T h i s a v e r a g e v a l u e i s unchanged when d a t a a t 6 1 1261 and 95 1301 MeV a r e i n c l u d e d . ) The v a l u e of 6R a t 334 (800) MeV i s l o w e r ( h i g h e r ) t h a n t h i s a v e r a g e by 1 4 (17)%.

Values of 6R a t t h e o t h e r e n e r g i e s a r e w i t h i n -5% o f t h e a v e r a g e .

Assuming t h a t t h e d a t a c o n s i d e r e d a r e p r o p e r l y n o r m a l i z e d , t h e 2 0 8 ~ b r e s u l t s i n T a b l e 1 imply t h a t n e i t h e r e f f e c t i v e i n t e r a c t i o n p r o v i d e s a s a t i s f a c t o r y d e s c r i p t i o n of t h e e x p e r i m e n t a l d a t a a s a f u n c t i o n o f e n e r g y and t h i s l i m i t s t h e i r p r e d i c t i v e power f o r g i a n t resonance s t u d i e s . There does a p p e a r t o b e a p r e f e r e n c e f o r a f o l d e d model t y p e of o p t i c a l p o t e n t i a l n e a r 135 MeV f o r 2 0 8 ~ b . For t h e 3: e x c i t a - t i o n i n '08pb t h e c h o i c e of o p t i c a l p o t e n t i a l between 100 and 200 MeV i n t r o d u c e s un- c e r t a i n t i e s a s l a r g e o r l a r g e r t h a n t h o s e a s s o c i a t e d w i t h t h e c h o i c e o f e f f e c t i v e i n t e r a c t i o n . The two o p t i c a l p o t e n t i a l s c o n s i d e r e d a r e n o t , however, e q u i v a l e n t f o r e l a s t i c s c a t t e r i n g ; t h e r e s o l u t i o n of t h i s o p t i c a l p o t e n t i a l problem c l e a r l y

w a r r a n t s h i g h p r i o r i t y .

5. ISOVECTOR EXCITATIONS: NUCLEON CHARGE EXCHANGE REACTIONS

The n u c l e o n c h a r g e exchange r e a c t i o n s ( p , n ) and ( n , p ) a r e e s p e c i a l l y s e l e c t i v e b e c a u s e o n l y i s o v e c t o r n u c l e a r e x c i t a t i o n s a r e allowed. T h i s r e n d e r s t h e S=O, T=O p a r t o f t h e N-N i n t e r a c t i o n , which dominates t h e ( p Y p 1 ) spectrum a t small momentm t r a n s f e r s i n o p e r a t i v e ( s e e F i g . 1 ) . Moreover, s i n c e most of t h e ( p y n ) d a t a /31/ a t i n t e r m e d i a t e e n e r g i e s have been t a k e n a t forward a n g l e s where t h e momentum t r a n s f e r i s s m a l l and t h e r e f o r e t h e n o n - c e n t r a l p a r t s o f vWE a r e r e l a ~ i v e l y u n i m p o r t a n t , t h e r a t i o o f 1 t o T / t T 12 shown i n F i g . 1 s h o u l d , a p a r t from n u c l e a r m a t r i x e l e m e n t s , b e a XP r e a s o n a b l e measure o f t h e r e l a t i v e c r o s s s e c t i o n s f o r S=l and S=O e x c i t a t i o n s and t h i s i s b o r n e o u t by experiment 1321.

From Eqs. (11-13) t h e n u c l e o n c o u p l e s t o t h e n u c l e u s through t h e t e n s o r s

f o r i s o v e c t o r e x c i t a t i o n s i n t h e c o n v e n t i o n Tz = (N-Z)/2; T~ i s t h e u t h component o f t h e P a u l i i s o s p i n o p e r a t o r . At s m a l l ( l o c a l ) momentum t r a n s f e r q t h e s e t e n s o r s may b e r e l a t e d t o t h e u s u a l i s o v e c t o r m u l t i p o l e o p e r a t o r s 132,331:

-+ _q^{L L}_ri

+ -

-

^(GT) ^-z q (IMSR) ¹ ²