PHYSICSBUDAPEST INSTITUTE FOR RESEARCH CENTRAL I

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KFKI-1980-123

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I W. B A U H O F F H.V. G E R A M B G. PALLA

STUDY OF NONLOCAL AND LOCAL

EQUIVALENT MICROSCOPIC OPTICAL POTENTIALS

H u n g a r i a n A c a d e m y o f S c i e n c e s CENTRAL

RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

KFKI-1980-123

STUDY OF NONLOCAL AND LOCAL

EQUIVALENT MICROSCOPIC OPTICAL POTENTIALS

W. Bauhoff*, H.V. Geramb*, G. Pállá Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

♦Theoretische Kernphysik, Univ. Hamburg

2000 Hamburg 50, Luruper Chaussee 149, W.-Germany

HU ISSN 0368 5330 ISBN 963 371 768 X

generated nonlocal nucleon nucleus optical model and new features are delin­

eated. A thourough discussion is presented in terms of exact phase equivalent local potentials that reveal a strong repulsive Я-dependent core which is beyond phenomenological potential models. The numerical part is limited to 40, 180, 200 MeV scattering from 12C and 30, 180 MeV scattering from 40ca.

Possible experiments to check the new features in angular distributions are discussed.

АННОТАЦИЯ

Упругое рассеяние нуклонов при средних энергиях исследовалось на основе микроскопического оптического потенциала, и было открыто новое специфическое явление. По анализу эквивалентного по фазе локального потенциала показано существование отталкивающего, зависящего от Я кора, не существующего в фено­

менологических моделях. Сделаны расчеты в случаях рассеяния протонов с энер­

гией 40, 180 и 200 МэВ на 12С и протонов с энергией 30 и 180 МэВ на 40са.

Обсуждается возможность экспериментального доказательства этого нового явле­

ния в измерении углового распределения протонов.

KIVONAT

Közepes energiájú rugalmas szórást vizsgáltunk mikroszkopikusan származ­

tatott nukleon-mag optikai potenciál alapján és uj sajátságos jelenséget der ritettünk fel. Ezen optikai potenciál diszkussziója során az exakt fázis equivalens lokális potenciál alapján kimutatunk egy taszitó, Я-függő törzs jelenlétét a potenciálban, amelyről a fenomenologikus modellek nem tudnak számot adni. A numerikus vizsgálatokat a l2C-en való 40, 180 és 200 MeV-es szórásra, továbbá a 40ca-on való 30 és 180 MeV-es szórásra korlátoztuk.

A feltárt uj jelenség szögeloszlásban való kísérleti igazolásának lehető­

ségét diszkutáljuk.

1

1. Introduction

The t h e oretical study of ela s t i c s c a t t e r i n g p r o c e s s e s in nuclear physics is a f u n damental step in u n d e r s t a n d i n g the nuclear many body problem. To its solution the last three decades have put forward various a p p r o a c h e s , l y ing b e t w e e n

purely p h e n o m e n o l o g i c a l and fully microscopic. N u c l e o n n ucleus s c a t t e r i n g r e p r esents the r e b y the forefront to s tudies w i t h more c o m p l e x proj e c t i l e s and we cons i d e r the d e r i v a t i o n of a c o m p l e x single particle potential, the optical m o del p o t e n t i a l

(OMP), f r o m the e l e m e n t a r y n u c l e o n - n u c l e o n i n t e r a c t i o n as 1

ultimate goal . The nuc l e a r m atter appr o a c h has itself e s t a b l i s h e d as a q u a l i t a t i v e a n d q u a n t i t a t i v e m e t h o d to r e ­ concile the success of the p h e n o m e n o l o g i c a l local o p t i c a l model p o t e n t i a l s with a purely m i c r o s c o p i c model 2

It lies in the nature of the m i c r o s c o p i c t h e o r y that various appr o x i m a t i o n s are used,of w h i c h some are m odel t r u n c a t i o n s and some are solely c o m p u t a t i o n a l conveniences. W i t h the latter appr o x i m a t i o n s reference is made to solution te c h n i q u e s of the B e t h e - G o l d s t o n e equa t i o n 2 3' , c o m p u t a t i o n s of the nonl o c a l folded O M P w i t h nuclear m a t t e r t-matrices , use of various forms for the d i a g o n a l a n d m i x e d s ingle part i c l e g r o u n d state

4

density and, w h a t is of o u r concern, the t r a n s i t i o n from n o n ­ local to local e q u i v a l e n t potentials.

In this paper w e resume to study prob l e m s and eff e c t s a r i s i n g from the m i c r o s c o p i c nonlo c a l OMP in d i f f e r e n t i a l cross section, and p o l a r i z a t i o n data. C l o s e l y c o n n e c t e d wit h this a i m is a study of local e q u i v a l e n t p o t e n t i a l s in order to d i s c l o s e i m ­ p ortant shortc o m i n g s of p h e n o m e n o l o g i c a l OMP's. W i t h this r emark we r e c a l l that for a nonlocal o p e r a t o r one c a n n o t tell if the i n t e r a c t i o n is p u r e l y a t t r a c t i v e or p u r e l y r e p u l s i v e s i mply by kn o w i n g the over-all sign.

One mus t cons i d e r the d e t a i l e d struc t u r e o f the n o n l o c a l i t y a n d its b e h a v i o u r as a function of energy and angular momentum.

Some potentials b e h a v e a t t r a c t i v e at low e n e r g y and r e p u l s i v e at h i g h energy. S i m i l a r l y the b e h a v i o u r m a y change for d i f f e r e n t a n g u l a r momenta a n d thus r esult in an e f f e c t i v e Я- d e p e n d e n t p o ­ tential, here OMP.

The i n v e s t i g a t i o n o f P e rey and Buck ^ for n eutron s c a t t e r i n g b e l o w 25 M e V wit h the F r ahn a n d Lemmer^ type nonl o c a l p o t e n t i a l is w e l l known. The i m p o r t a n t r e s u l t of this analysis w a s the

r e p r o d u c t i o n of the e n e r g y d e p e n d e n c e of local p o t e n t i a l s w i t h a n e n e r g y - i n d e p e n d e n t n o n l o c a l p o t e n t i a l a n d the r e p r o d u c t i o n of local p h e n o m e n o l o g i c a l OMP as e q u i v a l e n t potentials.

Furthermore, the t r a n s i t i o n fro m n o n l o c a l p o t e n t i a l s to e q u i v a ­ lent local p o t e n t i a l s was a c h i e v e d w i t h w e l l w o r k i n g a p p r o x i m a -

7 tive analytic e x p r e s s i o n s w h i c h w e r e l ittle a l t e r e d since

8 9

It has bee n p o i n t e d o u t by A u s t e r n , later by F i e d e l d e y a n d other authors that n o n l o c a l p o t e n t i a l s are b e s t e l u c i d a t e d by e x a c t e q u i v a l e n t local p o t e n t i a l s (LEQ) w h i c h are u n i q u e l y d e t e r m i n e d by a pair o f linear i n d e p e n d e n t solutions to the n o n ­ local problems. An e s s e n t i a l i n g r e d i e n t in this form u l a t i o n is the da m p i n g of the n o n l o c a l wa v e function as c o m p a r e d to the local wav e functions of the e q u i v a l e n t local potential. This

О -1Л

effect, k n o w n as P e rey e f f e c t ' , m a n i f e s t s an i m ­ p o r t a n t d i f ference b e t w e e n S c h r o e d i n g e r e q u a t i o n s w i t h a local and a nonlocal p o t e n t i a l operator. A p a i r of linear i n d e p e n d e n t s o l u t i o n s to local p r o b l e m s y i elds a W r o n s k i d e t e r m i n a n t w h i c h

i s , i n d e p e n d e n t of the radius, a constant. This radial i n d e p e n d e n c e of the W r o n s k i a n is g e n e r a l l y not true for nonl o c a l potentials.

M o s t o f our results are b u i l t on this difference, w h i c h is a rigorous m a t h e m a t i c a l property. In other words, it makes it i m ­ poss i b l e to e s t a b l i s h the full e q u i v a l e n c e in phase and m a g n i t u d e b e t w e e n local and n o n l o c a l p o t e n t i a l s o l u t i o n s w i t h o u t i n t r o ­ d u c i n g singular potentials.

3

In section II w e review the s a l i e n t features of the m i c r o s ­ copic n o n l o c a l o ptical p o t e n t i a l as it may be g e n e r a t e d from n u c l e a r m a t t e r t-matrices . This p o t e n t i a l is u n d e r s t o o d as e x e m p l a r i q m e a n of a m i c r o s c o p i c t h eory and it may be sub- s t i t u t e d s t r a i g h t forward by o t h e r a p p roaches ' .

F o l l o w i n g the theo r e t i c a l and m a t h e m a t i c a l b a c k g r o u n d the d e f i n i t i o n of e x a c t phase e q u i v a l e n t potentials in t e r m s of nonl o c a l Wro n s k i a n s and their d e r i v a t i v e s is given.

Sec t i o n III c o n t a i n s the f o r m u l a t i o n of the LEQ w h e n r e p l a c i n g the n o n l o c a l Wronskians and its d e r i v a t i v e s by pure pro p e r t i e s of the n o n l o c a l potential. The final result of this sec t i o n is a r apidly c o n v e r g i n g series e x p a n s i o n of LEQ. Se v e r a l n u m e rical examples are g i v e n to show q u a l i t a t i v e l y and q u a n t i t a t i v e l y the d i f ferences to other m ethods to d e f i n e e q u i v a l e n t l o c a l p o t e n ­ tials. A n a p p l i c a t i o n w i t h q u a n t i t a t i v e results for 4o, 18o,

2oo M e V C(p,p) and 16o,18o M e V °Ca(p,p) are g iven in section IV. With t h i s c a l c u l a t i o n we show the e s s e n t i a l d i f f e r e n c e s b e ­ tween p h e n o m e n o l o g i c a l o ptical m o d e l s and the m i c r o s c o p i c o p t i ­ cal p o t e n t i a l as they m a n i f e s t t h e m s e l v e s at low a n d h i g h energy.

The e s s e n t i a l s of this r e s u l t s h a l l be the o c c u rence o f a re­

pulsive core in the L E Q w h ose r a d i u s increases wi t h a n g u l a r momentum.

As c o m p a r e d w i t h p h e n o m e n o l o g i c a l p o t e n t i a l s this e s t a b l i s h e s the importance o f nonl o c a l p o t e n t i a l a n a l y s e s for s c a t t e r i n g above

1oo MeV. For lower energies we c o n f i r m that e nergy d e p e n d e n t a n d

£ - i n d e p e n d e n t p o t e ntials are s u f f i c i e n t to describe the global OMP. This is a n u m e rical result i r r e s p e c t i v e l y that the r e p u lsive core exists as e n e r g y i n d e p e n d e n t entity. T o g e t h e r w i t h a c o m ­ p a r i s o n of 2oo M e V proton s c a t t e r i n g data w e discuss ^-de p e n d e n t features in the an g u l a r d i s t r i b u t i o n w h i c h should be e x p e r i m e n t a l l y verified. The latter s t a t e m e n t is e q u i v a l e n t to p o s t u l a t e m e d i u m e n e r g y op t i c a l m o d e l analyses w i t h n o n l o c a l potentials.

2. Theoretical background

The study of i n t e r a c t i n g nucleons in i n f i nitely e x t e n d e d nu c l e a r m atter is w e l l e s t a b l i s h e d and a p p r o x i m a t e t r e a t m e n t s for finite nuclei s e e m justified. M e t h o d s d e v e l o p e d by

B r ü c k n e r and B e the (BB) have thereby b e e n w i d e l y a p p l i e d a n d the theory wit h c a l c u l a t i o n a l pr o c e d u r e s for the u n d e r s t a n d i n g of n u c l e o n - n u c l e u s e lastic s c a t t e r i n g s t a r t i n g from a r e a l i s t i c

1 2 3 NN force is on f r o m ground ' ' .

The a p p r o a c h p u r s u e d in our studies is b a s e d on the e v a l u a t i o n o f the effective i n t e r n u c l e o n t-ma t r i x f r o m the free NN inter-

3

a c t i o n . The real a n d i m a g inary o p t i c a l model for nucle o n s we calcu l a t e to f i r s t order in the e f f e c t i v e NN inte r a c t i o n

w i t h an improved v e r s i o n of the local d e n s i t y a p p r o x i m a t i o n (LDA) in a folding a p p r o a c h with s ingle p a r t i c l e target densities.

T h e m o del relies on the q uite general a p p r o a c h to gene r a t e in first a p p r o x i m a t i o n the OMP as a sum o f a direct t e r m and a n o n ­ local exchange t e r m

U(r,r';E) = 6 (r-r ' ) Е/ф* (x) t Q ( |r-x| ;E) <}>n (x) d 3x n

+ E <P*(r) tE ( |r-r'

I

n

The coordinates r a n d r' r e fer to p r o j e c t i l e coordinates, w i t h the summation of s i n g l e part i c l e wave functions w e r e p r e s e n t the bes t possible (Hartree-Fock) p a r t i c l e densities - d i a g o n a l a n d m i x e d densities - for p r o t o n s and neutrons. The basic i n g r e ­ dients of the LDA e n t e r s he r e in the c h o i c e of t_ and t_ w h i c h are

D ~

m i x t u r e s of d irect a n d e x c h a n g e e f f e c t i v e NN i n t eractions

In p r i n ciple it s h o u l d be c a l c u l a t e d in the finite s y s t e m w i t h its full s t r uctural details. The

h y p o t h e s i s is made that this e f f e ctive i n t e r a c t i o n c a n be a p p r o x i ­ m a t e d by the one c o r r e s p o n d i n g to the l o cal de n s i t y a n d e n e r g y d e p e n d e n t situation in nu c l e a r matter. Th i s e f f e c t i v e i n t e r a c t i o n is our version of LDA.

I

5

The inte r a c t i o n contains a u t o m a t i c a l l y real and i m a g i n a r y parts and the c o r r e c t features of the finite range of the interaction.

This is i m p o r t a n t since the ranges are d i f f e r e n t for real and imaginary p a r t s and the relative spi n and isospin channels.

A n y o t her a p p r o x i m a t i o n inherent in the nuclear m a t t e r approach r #

in c o m p u t i n g the effective i n teractions remains u n a l t e r e d to previous calculations.

The s t a t i o n a r y S c hroedinger e q u a t i o n

Дф(г,к) + (k2- v D ) ф(г,к) = /u(r,r';E) ф (r',k) dr' (2)

for the single particle OMP s c a t t e r i n g solutions is m o s t easily solved in the s t a n d a r d par t i a l w a v e decomposition, w h e r e the numerical p r o b l e m is r educed to an ordin a r y second o r d e r inte- g r o d i f f e r e n t i a l equation, viz.

2 oo

[§pr - — + k2 - vD (r)]uL J (r) = /wL (r,r') uL J (r')dr' (3) о

The diagonal p o t e n t i a l v ß conta i n s the s t a n d a r d h o m o g e n o u s c h a r g e d sphere C o u l o m b p o t e n t i a l a n d the spin o r b i t p o t e n t i a l

3 w h i c h we kept in a local form

The m u l t ipole d e c o m p o s i t i o n of the nonlocal O M P is f o r m a l l y o b ­ tained for a r o t a t i o n a l i n v a r i a n t symmetric p o t e n t i a l fr o m

u (r ,r ' ) = Uj^írJőír-r') + uE (r,r')

E 0)L (r,r')/ r r , (YL (f). YL ( f ) ) L

n (2L+1) > a 1 _ * л л .

£ > . ■ u>_ (r,r') PT (r- r ' ) , ,

T 4ir L L /rr1

(4)

The local d i r e c t p o t e n t i a l is s u b s u m m i z e d into the n o n l o c a l k n o c k o u t e x c h a n g e p o t e n t i a l w h i c h rep r e s e n t s the p r o p e r source of nonlocality. The e n e r g y d e p e n d e n c e in (1) results from the s m all e nergy d e p e n d e n c e of the e f f e c t i v e interaction. T h e m u l t i ­ pol e decomposition, eq. (4) is t e c h n i c a l l y straight f o r w a r d but is n u m e r i c a l l y quite i n v o l v e d due to r e q u i r e d e n e r g y a n d density i n t e r p o l a t i o n o f n u m e r i c a l l y s t o r e d e f f e c t i v e interactions.

T h e fo l d i n g integral for the d i r e c t p o t e n t i a l is s imple

and w a s g e n e r a t e d w i t h a G a u s s - L e g e n d r e i n t egration r o u t i n e w h e n p e r f o r m i n g the radial and angular integrations.

w i t h s p e c i f y i n g the o c c u p a t i o n n u m b e r in the s i n g l e part i c l e o r b i t (lj) for p r o t o n s / n e u t r o n s (т ). T h e radial wa v e functions Ф, . (r) are s o l u t i o n s o f a Frahn-Leimer type non local b o u n d state

J

T I

p o t e n t i a l w i t h p a r a m e t e r s Vq = - 72 MeV, R = 1,2 A 1/3 fm, a = o . 6 5 fm, a n d the r ange of n o n l o c a l i t y ß = о . 8 fm. C o u l o m b and spin o r b i t p o t e n t i a l s are kep t in the standard l o cal form w i t h v la = 7 MeV, R lg = 1.1 A u* fm, a = 0 . 6 5 fm.

The exch a n g e p o t e n t i a l is bes t d i r e c t l y g e n e r a t e d in its m u l t i ­ pol e d e c o m p o s i t i o n

oo 1

un (r) = 2tt/ / dr' dx Z S

0 - 1 lj,x

(/ r 2+ r '2+2rr'x)

(5) t ° (r ' ,kF (r),E(r) )

z ф (г) ф (r*) t ET (|r-r' I

ST, n n n E

(6)

7

In the limit of п о - s p i n / isospinflip this e x p r e s s i o n as s u m e s the for m

■ие*Ст,~) > E (т) ф jr') sj. < e v*

( - ) S "T ^ <Vx % n | S v ,f>1 < > * % x £ I T ( 7 )

^t/JLCx‘) t sr ( I Z- x'l ; -kp. , E )

T o g e t h e r w i t h a m u l t i p o l e e x p a n s i o n of the e f f e ctive i n t e r a c t i o n t|T (|r-r*|,kp ,E) I t^T (r, r 1) (Yx (r). Y x (r‘) )

X

(8) we o btain

wL (r,r‘) = l ф _(r )ф . . (r’)< l A o o I L o >2 1j /T 1 3 'T l 3 'T

X (9)

S lTj -- ^ {j (tX

3 2тг L‘ 4 Л

X P*T 16

♦ t? 1 - t00 - 3 t ]1) (1-6 )

X pt

The isospin (proton/neutron) of the p r o j e c t i l e enters through the index p (for projectile) on the K r o n e c k e r symbol 6^ т and accounts for like or u nlike p r o jectile a n d t arget nucleons. As a l r e a d y mentioned, the requ i r e d m u l t ipoles t, (r,r’) are com-ST p u t e d from t a b u l a t e d v a l u e s t ST ( s / k ^ E ) . To e l i m i n a t e possible

Г

errors in inte r p o l a t i o n s w e apply a double t r a n s f o r m a t i o n and o b t a i n

00 00

t xT (r,r') = 8 / kdk j x (kr) j x (kr1 ) f rdr sin(kr) t ST ( | r - r • ]) (1o)

о о

A n impression of the nonl o c a l e x c h a n g e k e r n e l is o b t a i n e d from j?

Fig. 1 and 2 w h e r e the radial d e p e n d e n c e w “'(r,r') is shown for

4o °

Ca, 6o M e V w i t h cuts across the diagonal r + r' = 6 fm and L = o(1)5. The m u l t i p o l e e x p a n s i o n of the local d i r e c t p o t e ntial is stra i g h t forward and yields an L i n d e p e n d e n t p u r e l y diago n a l one;

w£(r,r) = U p (r )6(r-r') (1 1)

W i t h the g e n e r a t i o n of the nonlocal kernel for the c entral r e a l / i m a g i n a r y p o t e n t i a l s and the spin orbit p o t e n t i a l s all ingredients of eq. (3) a r e available. It r e m a i n s to solve the r a d i a l i n t e g r o d i f f e r e n t i a l e q u a t i o n and e x t r a c t the S-ma t r i x e l e m e n t s in the usual m a t c h i n g procedure.

S p u r i o u s states or b o u n d states in the c o n t i n u u m are k n o w n to e x i s t for i n t e g r o d i f f e r e n t i a l equations 15. W e defer further d i s c u s s i o n s to this p r o b l e m since w e c h e c k e d o u r n u m e rical r e s u l t s c a r e f u l l y and d i d not find a n y such case.

W i t h this c o m m e n t w e m a y consider the p r o b l e m solved and the c o m p a r i s o n of t h e o r e t i c a l p r e d i c t i o n s w i t h e x p e r i m e n t a l data can be d o n e .

S i nce s t a n d a r d p h e n o m e n o l o g i c a l O M P analy s i s uses local p o ­ t e n t i a l s it a p p e a r s d e s i r e a b l e to fu r t h e r delineate properties in the language o f local potentials. W e t h e r e f o r e o u t l i n e the s a l i e n t features of the t r a n s f o r m a t i o n of a S c h r o e d i n g e r equation w i t h a nonlocal p o t e n t i a l eq. (3) to a S c h r o e d i n g e r e q u a t i o n wi t h a l o cal p o t e ntial

+ - vD (r)]U L J (r) = V e q (r)UL J (r) . (12) The local p o t e n t i a l V e^(r) is said to be e q u i v a l e n t to the n o n ­ local kernel о^(г , г ' ) if it can be c o m p l e t e l y s p e c i f i e d in terms of eq.(3) and its s o l u t i o n s and if it a n a l y t i c a l l y r e p roduces o b ­ s e r v a b l e features as a f u n c t i o n of energy.

The t r a n s f o r m a t i o n for the nonl o c a l e q u a t i o n requires two linear i n d e p e n d e n t s o l u t i o n s w i t h a s y m p t o t i c a l l y u nique b o u n d a r y con-

9 1o d i t i o n s for i n c o m i n g and o u t c o m i n g w a v e s

B o u n d a r y conditons in the i n t e r a c t i o n region m a y d e p e n d on the a s s u m p t i o n s a b o u t the i n t e r a c t i o n w h i c h are n o t subject to direct observation. T h e as y m p t o t i c p r o p e r t i e s are m a i n t a i n e d in the t r a n s ­ f o r m a t i o n and are the same for both eqs. (3) and (12).

9

These s p e c i f i c a t i o n s and the a n a l y t i c p r o p e r t i e s of any two linear i n d e p e n d e n t solutions d e f i n e the p r o b l e m uniquely. L e t f^ 2(k,r) be two solutions to the n o n l o c a l r adial e q u a t i o n and F^ 2(k,r) the e q u i v a l e n t solutions to the local equation.

Both s o l u t i o n s satisfy the a s y m p t o t i c b o u n d a r y c o n ­ ditions

lim (f. (k,r) - F . (k,r )) = о (13)

r->°° x 1

In the resi d u a l r a d i a l range we r e late the s o l u tions by a d a m p i n g function

f ± (к , r ) = A ( k , r ) F ± (k,r) (14)

w h i c h be h a v e s a s y m p t o t i c a l l y

lim A(k,r) = 1 (15)

r->°°

Q

This ansatz w a s first s u g g e s t e d by A u s t e r n and it is i n t r i ­ guing as it d e p i c t s so many r e l e v a n t p h y s i c a l q u a n t i t i e s a n d yields a s m ooth r e g u l a r e q u i v a l e n t local potential.

Any two linear i n d e p e n d e n t solut i o n s sa t i s f y the w r o n s k i a n r e l a t i o n

W ( f1,f2 ) = f1(r) f^(r) - f2(r) f'(r) ^ (16)

This W r o n s k i a n shows the e s s e n t i a l d i f f e r e n c e b e t w e e n local and nonlocal S c h r o e d i n g e r equations. Eq. (16) is g e n e r a l l y a func t i o n of the radius and stands as such c o n t r a r y to the p r o p e r t y of W ( F1,F2 ), the local Wronskian, w h i c h is r a d i a l l y i n d e p e n d e n t

for a regular potential. A n s a t z (14) t o g e t h e r w i t h (15) re l a t e s the Wronskian

W ( f r f 2 ) = A 2 W ( F r F 2 ) = W ( F r F 2 ) | r = oo (17)

b y the d a m p i n g functions. The d a m p i n g function is in other 8 11

literature o f ten i d e n t i f i e d as P erey e f f e c t '

A(k,r) = / W ( f1 (k,r) , f2(k,r))/W(f1 (k,») , f2 (k,°°)) (18) F o r s y m m e t r i c kernels шт (r,r’) = шт (r',r)

Li h

l i m A(r) = 1 and l i m W (r) = о

r - > o r->o

The e q u i v a l e n t local p o t e n t i a l is c o n s t r u c t e d in the same 9 1 о

m a n n e r ' . C o m b i n i n g e q s . (3) and (12) w e may e l i m i n a t e any r e f e r e n c e to a p a r t i c u l a r pai r of s o l u tions and find by pure a l g e b r a i c m a n i p u l a t i o n s that

V (2)

eq (r) 1 W " (f1'f 2 > . 3 r (f1.f2>, 2 W (f 1 » f 2) 4 lW ( f r f2 ) ;

+ w T f ^ T / 0)T (r,r‘) [f 1 (r') f'0 (r) f \ (r)f2(r')]dr'

A ' 2 2(5 “ )

CO

+ f w L (r,r') [f1 (r’) f'2 (r) о

(19)

- f2(r')f’ (г) I d r1

T h e e x p r e s s i o n for the d a m p i n g functions and e q u i v a l e n t local p o t e n t i a l s provides a unique and m a t h e m a t i c a l l y r i g o r o u s d e ­ fini t i o n of these ^ - d e p e n d e n t q u a n t i t i e s t h r o u g h o u t the d o m a i n of e x i s t e n c e of solutions to the n o n l o c a l equation. H e r e w i t h w e have a m e a n to c ompare p r o p e r t i e s of m i c r o s c o p i c ^-de p e n d e n t

p o t e n t i a l s w i t h p h e n o m e n o l o g i c a l p o t e n t i a l s working.

3. Analytic representation of the local l-dependent poten t i a l s . T h o u g h eq. (19) gives an e x a c t e x p r e s s i o n for the local equ i v a l e n t potential, it is q uite c u m b e r s o m e to be used in practice since t w o i n d ependent solutions are n e e d e d for the

calculation. Therefore, it seems d e s i r a b l e to d e v e l o p an a p p r o x i ­ mation scheme for the c a l c u l a t i o n of V e ^(r). The qu a l i t y of the a p p r o x imation c a n then be t e s t e d by c o m p a r i s o n w i t h the e x a c t

formula (19). Suc h a co m p a r i s o n was not poss i b l e for other approximations u s e d so far b e c a u s e they did not m a k e use o f an angular m o m e n t u m decomposition. The b a s i c idea of the a p p r o x i ­ mation scheme is to use an p e r t u r b a t i o n e x p a n s i o n w i t h r e s p e c t to the n o n l o c a l i t y range of the integral kernel. It is a s imple exercise to p r o v e the following e q u a l i t y for any s y m m etric i n t e ­ gral k e r n e l w(r,r')s

ш (r, r ' ) - E n=o

(-1) nl

n ik (r-r') , . ,r+r' . .

- e ° V n ) (r-r') vn <— !ko> (2o)

where k Q = k Q (— ^— ) is an a r b i t r a r y f u n c t i o n to be c hosen later.

The e x p a n s i o n c o e f f i c i e n t s v n (R;kQ ) are d e t e r m i n e d by the f o l l o w i n g equation:

v (R;k ) =

n о

2R d k s . .

f e S u ) ( R - - ^ s , R + 4s) ds

- 2 R г

(2 1)

So they are the n - t h - m o m e n t s of the inte g r a l k e r n e l ш(г,г')*

The a p p r o x i m a t i o n scheme to be d e s c r i b e d may be c a l l e d m o m e n t u m expansion. For the special case к (R) = о this type of e x p a n s i o n

о 16

has b e e n used in the framework of g e n e r a t o r c o o r d i n a t e t h e o r y Since in our case the arguments of oj(r,r') are r e s t r i c t e d to positive values the integration wit h r e s p e c t to s = r-r' is on a finite interval only.

It is n o w i m p o r t a n t to note that the m o m e n t s v n (R;kQ ) w i l l d e ­ crease r a t h e r r a p i d l y wi t h i n c r e a s i n g n due to the p e c u l i a r

form of the i n t e g r a l kernel. Since ш(г,г') is stro n g l y p e a k e d at s = r - r' = o, an appreciable c o n t r i b u t i o n to the integral (21) may come from the region of small s. T h e factor s11, however, is small there for n > o, so the value of the w h ole integral w i l l

d e c r e a s e w i t h i n c r e a s i n g n. F r o m this consideration, it is s u g gestive to consi d e r (2o) as a p e r t u r b a t i o n e x p a n s i o n of the i n t e g r a l kernel. To e m p h a s i z e this idea, a p a r a m e t e r X w i l l be i n t r o d u c e d to c h a r a c t e r i z e the magni t u d e o f the various o r d e r s so t h a t eq. (2o) can be written:

CO

Cú(r,r') = E An Vn (r,r';k ) (22a)

n=o w i t h

V n (r-r '!ko )

, .. n lk (r - r ') , . , , (-1) _ о r (n), 1% ,r+r' , *

e 6 (r-r' ) v (— =— ;k )

n 2 ' о П !

(22b)

If this e x p a n s i o n is i n t r o d u c e d into the nonl o c a l S c h r o e d i n g e r equation, a c o r r e s p o n d i n g e x p a n s i o n for the w a v e - f u n c t i o n is appropriate:

f (r ) = E АПфп (г) (23)

n

E q u a t i n g n o w equal p o w e r s of Л on both sides, the nonl o c a l S c h r o e d i n g e r equat i o n is t r a n s f o r m e d into a set of local equations. T h e lowest o r d e r e q u a t i o n s r e a d explicitly:

{h T - + k2 - v o ( r ) H o (r) = о

{H F r " — + k2 - vo (r)}l|»1(r) = / V1( r , r " ,k o )^o ( r ' ) d r I

(24) - ---'^21 ^ + k2 - v q (r ) } ф 2 (r ) = / V 1 (r,r' ;ко ) ф1 (r* ) dr '

+ / V2(r,r';ко )Фо (г') d r '

So the zeroth order e q u a t i o n is a usual local Schroe d i n g e r

e q u a t i o n w h e r e a s all h i g h e r e q u a t i o n s c o n t a i n an i n h o m o geneous term, w h i c h is d e t e r m i n e d by the solution of the lower equations.

13

The first t e r m on the r.h.s. vanishes d u e to eq. (26).

U p to o r d e r X we have:

W (r) = W1(r) W(r) W Q (“ )

Wit h the def i n i t i o n (25) we find for the derivative:

(28)

W'(r) = ф£(г)^(г) + ф ” (г)^о (г) - фо (г)у>''(г) - ф 1 (г ) (г) (29)

The s e c o n d deri v a t i v e s of the w a v e - f u n c t i o n s may n o w be e l i m i ­ nated w i t h he l p of the S c h r o e d i n g e r e q u a t i o n s (24). Many terms cancel, a n d it r emains

w (r) =

у^(г)

;kQ)Ф0 (г'

- Ф

(

)/V1

;kQ)

)

(3o)

Finally, w e use the d e f i n i t i o n (22b) for (r,r';kQ ) to o b t a i n

W!j(r) = v1 (r;kQ ) [yQ (r) ф^(г) - y>'o (r)4>o {r)] (31) The e x p r e s s i o n in square b r a c k e t s is W Q (r) = W q (°°). W q (<»)

cancels in eq. (28) and we get as c o n t r i b u t i o n to the p o t e n t i a l V (r) f r o m the t e r m (W TTir i ) up to o r d e r X 2:

eq w (r )

,W' (г) 42 2 2 , , ' ч ,-лч

{~ Ш ) ) = X v1(r;ko ) (32)

As c l a i m e d at the beginning, this e x p r e s s i o n does n o t co n t a i n the w a v e - f u n c t i o n anymore, b u t only the (first) m o m e n t of the integral kernel. Similarly, all other c o n t r i b u t i o n s to the p h a s e e q u i v a l e n t p o t e n t i a l c a n be calculated. T h e final r e s u l t up to order X2 reads:

(33)

Of course, A = 1 in the final e x p ression. The inclusion of higher o r der terms does not p r e s e n t p r i n c i p a l d i f f i c u l t i e s but the c a l c u l a t i o n s become r a t h e r lengthy.

If the w h o l e series is summed, the r e s u l t w i l l n o t d e pend

on the c h o i c e of k Q (r). It is, however, i m p o rtant to c hoose it in an a p p r o p r i a t e manner, if o n l y a few terms are t a ken into

account. By c o m p a r i n g the e x a c t p o t e n t i a l eq. (19) w i t h the partial s u m of the m o m e n t e x p a n s i o n it is p o s s i b l e to see w h e t h e r the p a r t i c u l a r c h o i c e of kQ (r) is a good o n e or not.

It turns o u t that the a g r e e m e n t b e t w e e n exact p o t e n t i a l a n d the zeroth o r der of the m o m e n t e x p a n s i o n is c l o s e s t whe n

kQ (r) is c h o s e n s e l f - c o n s i s t e n t l y ^ as the local m o m e n t u m of the p r o j ectile

A l l r esults p r e s e n t e d in the f o l l o w i n g are o b t a i n e d w i t h this special k Q (r). Also, only the zeroth o r d e r of the m o m e n t e x p a n ­ sion is c o n s i d e r e d because it will be s h own to be a c c u r a t e l y e n o u g h .

In o r der to c o m p a r e the v a r i o u s a p p r o a c h e s for a p h ase e q u i v a l e n t local p o t e n t i a l w e consider first a m o d e l of a s i m p l e anal y t i c form, the s o - c a l l e d F r a h n - L e m m e r ^ or P e r e y - B u c k potential, w h i c h is g iven by

(34)

vо (35)

15

As V (г,г';к ) c o n t a i n s deri v a t i v e s of the 6- function, the i nhomogeneties are in fact local e x p r e s s i o n s c o n t a i n i n g d e ­ rivatives of the w a v e functions. S econd and h i g h e r d e r ivatives of t h e m m a y be e l i m i n a t e d by using the lower o r d e r equations.

The two i n d e p e n d e n t solutions ip (r) , <f(r) are e x p a n d e d in the form (23), a c o r r e s p o n d i n g e x p a n s i o n w i l l result also for the Wronskian W(r) w h i c h reads exp l i c i t l y up to o r d e r X

W(r) = W Q (r) + X W1(r) + ...

t

= 0 o ( r ) ^ ( r ) - 4^(r)<fo (r)]

(25) + х[фо (г)^>’ (г) + ф., (r)«^(r) - ♦0<-Г)?1 (г)- Ф !j (r)y>o (r)]

+ . . .

F r o m the set of e q u a tions (24) it is, however, s e e n that W (r) is a W r o n s k i a n c o r r e s p o n d i n g to a usual local S c h r o e d i n g e r

equation. It is w e l l - k n o w n th a t in this case the Wrons k i a n do e s not d e p e n d on the radius,so we have the i m p o rtant result:

W Q (r) = W Q (») = const. (26)

A f t e r t hese pr e l i m i n a r i e s w e return to the q u e s t i o n of a p p r o x i ­ m a t i n g the phase e q u i v a l e n t p o t e ntial V e g (r), eq. (19). F r o m the e x p a n s i o n in p owers of X for the integral kernel a n d the W r o n s k i a n we w i l l get also an e x p a n s i o n for V ( r ) . It w i l l be shown t h a t

eq

the va r i o u s co n t r i b u t i o n s c a n be c a l c u l a t e d from.the k n o w l e d g e of the m o m e n t s of the integral k ernel and tha t any w a v e - f u n c t i o n

drops o u t o f the final formula. The full c a l c u l a t i o n is r a t h e r lengthy, so we w i l l c o n s i d e r o n l y a s i n g l e term t o d e m o n s t r a t e

• MI / v j ^

the method. As an example take the t e r m • F r o m (25) w e have

W' (r) = W^(r) + XW!^ (r) + ... (27)

Its var i o u s multipole c o m p o n e n t s can be c a l c u l a t e d a n a l y t i c a l l y . 4 о

T h e p a r a m e t e r s have b e e n c h o s e n for Ca i.e. R = 4.1 fm, a = 0.65 fm. The d epth was t a k e n as Vq = - 72 M e V a n d the n o n ­ loca l i t y r a n g e was у = о.8 fm. V arious forms of the e q u i v a l e n t

5

local p o t e n t i a l are shown in Fig. 3 for the s-wave at a p r o j e c t i l e e n e r g y of 3o MeV. F i r s t of all, the tri v i a l e q u i v a ­ lent p o t e n t i a l has so man y s i n g u l a r i t i e s t h a t a s m o o t h inter­

p o l a t i o n is rather arbitrary. T h e exact p h a s e - e q u i v a l e n t p o t e n ­ tial eq. (19) has a c o n v e n t i o n a l W o o d s - S a x o n shape at larger radii, b u t deviates f r o m this f o r m s t r o n g l y at s m all distances.

F o r r -> о it is even repulsive. The p o t e n t i a l r e s u l t i n g from the m o m e n t e x p a n s i o n d e v i a t e s o n l y s l i g h t l y from this form:

it does n o t show the d i p a r o u n d 1 fm, and it vanishes at the origin. F o r comparison, also the usual P e r e y - B u c k a p p r o x i m a t i o n is shown w h i c h does n o t d e p e n d o n the a n g u l a r momentum. It

c o i n c i d e s w i t h the m o m e n t e x p a n s i o n e v e r y w h e r e e x c e p t close to the o r i g i n w h e r e it a s s u m e s a finite value.

In Fig. 4, w e cons i d e r the e n e r g y d e p e n d e n c e of the local p o t e n ­ tial r e s u l t i n g from the m o m e n t expansion. The b e h a v i o u r at short d i s t a n c e s is found to be i n d e p e n d e n t of the energy w h e r e a s for l a rger d i s t a n c e s the p o t e n t i a l decreases a l m o s t l i n e a r l y w i t h the energy. T h e b e h a v i o u r for d i f f e r e n t a n g u l a r momenta at the same energy, s h o w n in Fig. 5, is c o m p l e m e n t a r y to this: F o r large distances, the p o t e n t i a l does n o t depend o n 1, w h e r e a s the short- dist a n c e is c o m p l e t e l y d i f f e r e n t for d i f f e r e n t a n g u l a r momenta:

T h e p o t e n t i a l can be s h own to s t a r t like r21+3

T h e c o r r e s p o n d i n g r e s u l t s for the real p a r t of the m i c r o s c o p i c o p t i c a l p o t e n t i a l are s h own in Fig. 6 to 8. In this case one has to a d d a l o cal (repulsive) p o t e n t i a l to the nonl o c a l a t t r active one. Therefore, the b e h a v i o u r c l o s e to the origin is domin a t e d by the d i r e c t term w h i c h leads to the e x i s t e n c e of a repulsive core. The various e q u i v a l e n t l o c a l p o t e n t i a l s are c o m p a r e d in Fig. 6. T h e trivial e q u i v a l e n t potential do e s not h a v e s i n ­ g u l a r i t i e s because the wav e func t i o n is n o w complex. Instead it s h ows finite jumps at the z eroes of the r e a l part of the wav e function. T h e exact .potential h a s various o s c i l l a t i o n s for small d i s t a n c e s w h i c h are i n t e r p o l a t e d by the m o m e n t expansion.

17

Such a b e h a v i o u r h a d to be expected s ince the zeroth o r d e r is e s s e n t i a l l y g i v e n by a W i g n e r t r a n s f o r m w h i c h cannot r e p r o d u c e all q u a n t u m fluctuations. The local m o m e n t u m a p p r o x i m a t i o n used

3

by B r i e v a and R o o k is seen to d e f e r s t r o n g l y for all radii, so this a p p r o x i m a t i o n seems to be q u e s t i o n a b l e .

The e n e r g y dep e n d e n c e of the local p o t e n t i a l Fig. 7 is s i m i l a r to t h a t obta i n e d for the F r a h n - L e m m e r potential. Much m o r e i n t e r e s t i n g is the angular m o m e n t u m d e p e n d e n c e shown in Fig. 8. One observes the e x i s t e n c e of a p r o n o u n c e d o s c i l l a t i o n b e t w e e n

1 a n d'2 fm w h i c h is a m p l i f i e d for l arger an g u l a r momenta.

For c o m p l e t e n e s s we add in Figs. 9a to 9c the numerical re s u l t s for the imaginary and spin orbit p o t e n t i a l s together w i t h the d a m p i n g function for L = o. In fact, all these qua n t i t i e s are A - d e p e n d e n t . We d e fer its r e p r e s e n t a t i o n due to its m i n o r im­

p o r t a n c e or l ittle physi c a l s i g n i f i c a n c e in i n t e r p r e t i n g results.

4. Applications

M i c r o s c o p i c op t i c a l pot e n t i a l s h a v e proven to r e p r o d u c e the global features of p h e n o m e n o l o g i c a l results. T o these features we c o u n t the e n e r g y dependence of volume integrals of real and

imaginary c e n t r a l potentials, the rms radii and in some cases with a d j u s t m e n t s of the strength by 1o -2o% the r e p r o d u c t i o n of d i f f e r e n t i a l cross sections and a n a l y z i n g p o w e r data. With the here p r e s e n t e d analysis we do not expect to a c h i e v e better fits b u t r a t h e r indicate c h a r a c t e r i s t i c w h i c h are i n h e r e n t in nonlocal p o t e n t i a l s .

The e n e r g y d e p e n d e n c e of (equivalent) local p o t e n t i a l s is we l l k n own as w e l l as the dam p i n g of n o n l o c a l wav e functions. The success of S a x o n - W o o d s pot e n t i a l s b e l o w 80 M e V on the o ther h a n d indicate that the inner r e g i o n of the t a r g e t nucleus does not e n t e r s e nsi­

tively in a n g u l a r distributions. I n c l u s i o n of the ^-dep e n d e n t core, as shown in p r e v i o u s sections, m u s t therefore leave t h e o r e t i c a l

an g u l a r d i s t r i b u t i o n unaltered. In Fig. 1o p r o t o n s c a t t e r i n g from 12C is d i s p l a y e d for 4o and 18o MeV. The angular d i s t r i b u t i o n s are ca l c u l a t e d w i t h the full p h ase e q u i v a l e n t m i c r o s c o p i c O M P (LEQ MOP) Fig. 11 and 12 w i t h / w i t h o u t spin o r b i t potentials (USq = o ) . The

s e c o n d g r oup o f curves are o b t a i n e d w h e n the p o t e n t i a l w a s r e p l a c e d by the m i c r o s c o p i c potential for l=o and its core e l i minated. (LEQMOP w i t h o u t c o r e ) . T h e results prove the importance of the cor e for

high energ i e s a n d being there c o m p a r a b l e with the spin o r b i t effects.

In Fig. 13 and 14 we compare the r e s u l t s wit h e x p e r i m e n t a l data.

Since LEQ MOP r e p r esents an ab i n i t i o calculation, parameterfree, its fit is e x c e l l e n t also w h e n c o n s i d e r i n g p h e n o m e n o l o g i c a l fits.

The 2oo M e V d a t a give reason to b e l i e v e that large a n g l e sca t t e r i n g is mo s t s t r o n g l y i n f l uenced by the core. The m i n i m u m in a(0) a round 95° is f o l l o w e d by a back rise. S i m i l a r results are o b t a i n e d for 16o,

18o M e V s c a t t e r i n g from ^°Ca /. Figs. 15 - 1C. A g a i n we e m p h asize the back r i s e a f t e r the m i n i m u m a r o u n d 98°. The forward r e g i o n is well r e p r o d u c e d w i t h a strongly d a m p e d diffr a c t i o n p a t t e r n between

3o and 7o degrees.

19

In p h e n o m e n o l o g i c a l analyses this d a m p i n g was e x p l a i n e d as a pure spin o r b i t effect. We i n t e r p r e t it rather as a m u t u a l

e ffect of the effective ^ - d e p e n d e n t core and spin o r b i t potential.

// Q

In view of the un d e r l y i n g 1o% t h e o r y it is not our i n t e n t i o n to o v e r e s t i m a t e the predictive p o w e r of micr o s c o p i c O M P a n d their

fits. On save ground is the b a c k a n g l e rise of the d i f f e r e n t i a l cross section. We interpret this as the e s s e ntial f e a t u r e of nonlocal p o t e n t i a l s w h i c h p r o d u c e s trong trapping e f f e c t s re­

s ulting in s c a t t e r i n g paths into the b a c k w a r d region. In the language of L E Q MOP, we see r a d i a l region of rapidly v a r y i n g attraction / repulsion. It r e m a i n s as an e x p e r i m e n t a l challenge to measure c o m p l e t e an g u l a r d i s t r i b u t i o n s w i t h 2oo M e V p r o j ectile b e y o u n d 9o d e g r e e s for a few nuclei. Furt h e r m o r e the analyses of future h i g h energy data is p r o p o s e d to be p e r f o r m e d w i t h n o n ­

local p o t e n t i a l s w hich are fully m i c r o s c o p i c or at l e a s t guided b y it. T e c h n i c a l l y the p h e n o m e n o l o g i c a l F r a h n - L e m m e r a n satz is c o n v e n i e n t to handle in its m u l t i p o l e d e c o m p o s i t i o n a n d it c o n ­ tains less free p a r a meters as s uperpositions of s e v e r a l S a x on-Woods p o t e n t i a l s .

Re ferences

1) M i c r o s c o p i c Opt i c a l Model Potential, ed. H.V. G e r a m b Lecture N o t e s in Physics, Vol. 89, Spri n g e r 1979 2) J.P. Jeukenne, A. Lejeune and C. M a h a u x

Phys. R e p o r t s 2 5 C , (1976) 83 3) F .A . B r i e v a and J.R. Rook

N u c l . Phys. A 2 9 1 , (1977) 299, Nucl. Phys. A 2 9 1 , (1977) 317 N u c l . Phys. A 2 9 7 , (1978) 2o6

4) J.W. N e g e l e and D. Vautherin, Phys. Rev. C5 (1972) 1472;

Phys. Rev. Cl 1, (1975) 1o31

X. Campi a n d A. Bouyssy, Phys. Lett. 7 3 B , (1978) 263 A. Tielens, Diplomarbeit, H a m b u r g 1979

5) F.G. P e rey a n d B. Buck Nucl. Phys. 32, (1962) 353 6) W.E. F r a h n a n d R.H. Lemmer

N u o v o Cim. 5, (1957) 523; N u o v o Cim. (5, (1957) 664 7) F.G. Perey and D.S. Saxon, Phys. Lett, ho, (1964) 1o7

W.E. Frahn, Nucl. Phys. £6, (1965) 358

R. Peierls and N. V i n h Mau, N u cl. Phys. A 3 4 3 , (198o) 1 H. Horiuchi, Progr. Theor. Phys. 6_4, (198o) 184

8) N. A u s t e r n

Phys. Rev. TJ7, (1965) B752 9) H. F i e d e l d e y

Nucl. Phys. 77, (1966) 149 Nucl. Phys. A96, (1967) 463 Nucl. Phys. A 1 15, (1968) 97

10) M. Coz, R.G. A r n o l d a n d A.D. M a c K e l l a r

Ann. Phys. 5j), (197o) 5o4, Ann. Phys. 59, (197o) 2 19 11) F.G. Perey

Direct I n t e r a c t i o n s and N u c l e a r Reaction M e c h a n i s m s

(E. Clement, C. Villi, eds.) G o r d o n and B r each (1958) p 125 12) N. Vinh M a u and A. B ouyssy

Nucl. Phys. A 2 5 7 , (1976), 189 13) M.K. W e i g e l and G. W egmann

F o r t s c h r i t t e d. Phys. J^9, (1971 ) 451 14) T. Ueberall

D i p l o m a r b e i t H a m b u r g (1979)1

21

15) L.G. A r n o l d and A.D. M a c K e l l a r Phys. Rev. C3, (1971) 1o95

Th. 0. Krause, M. M u l l i g a n Ann. Phys. 94j (1975) 31

B. Mulligan, L.G. Arnold, B. Badchi and T.O. K r a u s e Phys. Rev. СГЗ, (1976) 2131

16) B. G i r o u d and B. G r a m m a t i c o s Ann. Phys. 1 o 1 , (1976) 67o 17) B.W. Riedley a n d J.P. T u r n e r

Nucl. Phys. 58, (1964) 497

18) .H.O. Meyer, P. Schwandt, G.L. Moake and P.P. Singh IUCF (1980) 385

19) A. Nadasen, P. Schwandt, P.P. Singh, W.W. Jacobs,

A.D. Bacher, P.T. Debecev, M.D. K a i t c h u c k a n d J.T. Meek IUCF (1979) 118

Fig. 1 T r i a x i a l r e p r e s e n t a t i o n of the real n o n l o c a l e x c h a n g e kernel w^(r,r')- The u n d e r l y i n g geome t r y is ^°Ca for 6o MeV.

Fig. 2 Cuts of n o n l o c a l e x c h a n g e k e r n e l s ш (r,r') real - solid lines, i m a g i n a r y - d a s h e d lines for various a n g u l a r momenta.

Fig. 3 C o m p a r i s o n of e q u i v a l e n t local potentials, g e n e r a t e d w i t h i n d i f f e r e n t pres c r i p t i o n s . The W r o n s k i m ethod

(solid line) is the e x a c t r e s u l t for eq. (19) and

m oment e x p a n s i o n r efers to e q . (33). P e rey Buck a n d trivial e q u i v a l e n t p o t e ntial c o i n c i d e w i t h its d e f i n i t i o n in ref. 5.

Fig. 4 E n ergy d e p e n d e n c e of the local e q u i v a l e n t F r a h n - L e m m e r typ p o t e n t i a l g e n e r a t e d with m o m e n t expansion.

Fig. 5 Study of the «.-dependence L = o , 1 . . . 2 o of the e q u i v a l e n t F r a h n - L e m m e r p o t e n t i a l for 3o M e V nucleons.

Fig. 6 C o m p a r i s o n of v a r i o u s e q u i v a l e n t local p o t e ntial p r e s c r i p t i o n s w h e n a p p l i e d to the full m i c r o s c o p i c opt i c a l m o d e l kernel. W ronski r e fers to the exact solution of eq. (19) a n d m oment e x p a n s i o n to the a p p l i c a t i o n of eq. (33). . Local m o m e n t u m a p p r o x i m a t i o n refers to the p r e s c r i p t i o n u s e d in ref. 3. T r i v i a l e q u i ­ v alent is i d e n t i c a l w i t h ref. 5.

Fig. 7 Study of the e n e r g y d e p e n d e n c e o f LEQ w i t h i n the m oment expansion.

Fig. 8a «.-pedendence of the real part o f LEQ w i t h i n the m o m e n t expansion.

Fig. 8b «.-dependence of the i m a g inary p a r t of L E Q w ithin the m o m e n t e x p a n s i o n

Fig. 9a Study of the e n e r g y d e p e n d e n c e o f full imaginary central p o t e n t i a l .

Fig. 9b Local spin o rbit p o t e n t i a l s g e n e r a t e d w i t h expr e s s i o n s d e v e l o p e d in ref. 3.

23

Fig. 9c E n e r g y dependence of the r e a l part of the d a m p i n g function (Perey effect) for L = о b a s e d on the full n o n l o c a l kernel.

The d a m p i n g functions are Л- d e p e n d e n t a n d change visually like the potentials shown in fig. 5.

Fig. 1o E ffects of the repulsive Л- d e p e n d e n t co r e on differ e n t i a l cross sections at low and h i g h energy. The geometrical i n g r e d i e n t s are from 12C .

Fig. 11a 4o M e V Л- d e p e n d e n t real c e n t r a l pot e n t i a l s u n d e r l y i n g the c a l c u l a t i o n s in fig. 1o.

Fig. 1 1b 4o M e V Л- d e p e n d e n t i m a g inary central p o t e n t i a l s un d e r l y i n g the c a l c u l a t i o n s in fig. 1o

Fig. 12a 18o M e V Л-de p e n d e n t real c e n t r a l p o t e n t i a l s u n d e r l y i n g the c a l c u l a t i o n s in fig. 1o.

Fig. 12b 18o M e V Л-de p e n d e n t i m a g i n a r y central p o t e n t i a l s underlying the c a l c u l a t i o n s in fig. 1o.

Fig. 13 12C p r o t o n sc a t t e r i n g a n a l y s e s for two energies. T h e data 17 18

are f r o m literature ' . P h e n o m e n o l o g i c a l OMP 1 a n d 2 are the b e s t fit potentials o f ref. 18 w h i c h apply single and double Saxon - W o o d s form f actors in their fit procedures.

Fig. 14 P o l a r i s a t i o n s to the 2oo M e V analyses in fig. 13.

Fig. 15a C o m p a r i s o n of 16o M e V p r o t o n e x p e r i m e n t w i t h t h e oretical p r e d i c t i o n wit h the LEQ. D a t a from ref. 19.

Fig. 15b C o m p a r i s o n of 18o M e V p r o t o n e x p e r i m e n t w i t h theoretical p r e d i c t i o n with the LEQ. D a t a from ref. 19.

Fig. 16a 18o M e V Л-de p e n d e n t real c e n t r a l p o t e n t i a l u n d e r l y i n g the c a l c u l a t i o n s in fig. 15.1,.

Fig. 16b 18o M e V Л- d e p e n d e n t i m a g i n a r y central p o t e n t i a l underlying the c a l c u l a t i o n s in fig. 15 L .

I

Radius lfm 1

F I G, 4

P o te n ti a l S tr e n g th [M e V ]

F I G, 5

F I G. б

P O T E N T IA L S T R E N G T H [M e

F I G. 7

P o te n ti a l S tr e n g th [M e V ]

F I G. 8 A

RADIUS [fm]

FIG. 9 A

P O T E N T IA L S T R E N G T H [M e

FIG, 9 В

FIG. 9 ^С

'1

F I G. I o

{MillI

P O T E N T IA L S T R E N G T H [M e V

F I G ,

12

0 / 0 .

FIG, 13

FIG. 14

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Szegő Károly

Szakmai lektor: Doleschall Pál Nyelvi lektor: Kluge Gyula

Példányszám: 450 Törzsszám: 80-739 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1980. december hó