SEMICLASSICAL CALCULATIONS OF THE IMAGINARY PART OF THE NUCLEON-NUCLEUS OPTICAL POTENTIAL

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SEMICLASSICAL CALCULATIONS OF THE

IMAGINARY PART OF THE NUCLEON-NUCLEUS

OPTICAL POTENTIAL

R. Hasse, P. Schuck

To cite this version:

JOURNAL

DE

PHYSIQUE

Colloque C6, supplément a u n06, Tome 45, juin 1984 page C6-213

S E M I C L A S S I C A L CALCULATIONS OF THE I M A G I N A R Y PART OF THE NUCLEON-NUCLEUS O P T I C A L P O T E N T I A L

R.W. Hasse and P. schuck*

I n s t i t u t Laue-Langevin, 38042 Grenoble Cedex, France

* ~ n s t i t u t des Sciences Nucléaires, 38026 Grenoble Cedex, France

Résumé - On c a l c u l e pour des noyaux f i n i s , l a p a r t i e i m a g i n a i r e du p o t e n t i e l o p t i q u e noyau-nucléon s u r e t h o r s couche u t i l i s a n t l ' a p p r o x i m a t i o n du gaz de Fermi l o c a l e t une f o r c e d ' é c h a n g e à deux c o r p s a y a n t une p o r t é e f i n i e . On compare l e s r é s u l t a t s a v e c ceux o b t e n u s p a r d e s c a l c u l s p o u r l a m a t i e r e i n f i n i e e t ceux o b t e n u s s o i t pour l a d e n s i t é l o c a l e s o i t pour l ' a p p r o x i m a t i o n de G l a u b e r .

A b s t r a c t - We c a l c u l a t e f o r f i n i t e n u c l e i t h e i m a g i n a r y p a r t of t h e n u c l e u s - n u c l e o n o p t i c a l p o t e n t i a l on and o f f s h e l l by u s i n g t h e l o c a l Fermi gas a p p r o x i m a t i o n and a f i n i t e r a n g e two-body exchange f o r c e . R e s u l t s a r e compared w i t h t h o s e o b t a i n e d by i n f i n i t e n u c l e a r m a t t e r c a l c u l a t i o n s a s w e l l a s u s i n g t h e l o c a l d e n s i t y o r Glauber a p p r o x i m a t i o n . I

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INTRODUCTION The i m a g i n a r y p a r t of t h e n u c l e u s - n u c l e o n o p t i c a l p o t e n t i a l e n t e r s d i r e c t l y i n v a r i o u s q u a n t i t i e s of i n t e r e s t i n s t a t i c p r o p e r t i e s of n u c l e i o r i n n u c l e a r r e a c t i o n t h e o r i e s , e . g . i n t h e e f f e c t i v e mass 1 1 1 and t h e s e l f e n e r g y 12-51, t h e n u c l e o n mean f r e e p a t h 16-7 / a n d q u a s i - p a r t i c l e l i f e t i m e s 1 8 1 o r i n t h e o r i e s of i n e l a s t i c s c a t t e r i n g . Whereas t h e o p t i c a l p o t e n t i a l h a s been o f t e n c a l c u l a t e d f u l l y micro- s c o p i c a l l y , s e m i c l a s s i c a l c a l c u l a t i o n s e x i s t o n l y f o r i n f i n i t e n u c l e a r m a t t e r / 3 , 9 , 1 0 / o r f i n i t e n u c l e i w i t h 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 / 5 / . The l a t t e r a p p r o x i m a t i o n c o n s i s t s of r e p l a c i n g t h e Fermi energy

A

by i t s l o c a l e q u i v a l e n t E _F (R). A r e v i e w o v e r e x i s t i n g s e m i c l a s s i c a l c a l c u l a t i o n s i s g i v e n i n r e f . 1111. I n t h i s p a p e r we a r e concerned w i t h f i n i t e n u c l e i and t a k e t h e f i n i t e s i z e e f f e c t s i n t o a c c o u n t e x a c t l y w i t h h e l p of a n a v e r a g e n u c l e a r p o t e n t i a l V(R) which we need n o t s p e c i f y e x p l i c i t l y . F u r t h e r m o r e , we a l s o c a l c u l a t e t h e o f f s h e l l b e h a v i o u r of t h e o p t i c a l p o t e n t i a l which h a s o f t e n b e e n assumed t o b e weak 171. I n o r d e r t o b e r e a l i s t i c , we employ a f i n i t e r a n g e exchange p o t e n t i a l which f o r c e s t h e o p t i c a l p o t e n t i a l t o d e c r e a s e a t h i g h e n e r g i e s . R e s u l t s w i l l a l s o b e compared w i t h t h o s e o b t a i n e d by u s e o f t h e Glauber a p p r o x i m a t i o n which n e g l e c o ; t h e P a u l i p r i n c i p l e . P r e l i m i n a r y r e s u l t s have a l r e a d y been p r e s e n t e d i n r e f s . / 1 2 , 1 3 / , II - THE MODEL Beyond Hartree-Fock, t h e f i r s t c o r r e c t i o n s t o t h e . n u c l e a r s e l f e n e r g y a r e g i v e n by c o r e p o l a r i z a t i o n and c o r r e l a t i o n c o n t r i b u t i o n s a c c o r d i n g t o F i g . 1 . F i g . 1 - The f i r s t t h r e e g r a p h s T

*

-

of t h e n u c l e a r s e l f e n e r g y .

O

_Zr-lh

_,

_{L J Z R Ip - 2 h , W LA}

Hartree-Fock Polarization Correlation

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W r i t t e n i n terms of t h e mass o p e r a t o r ,

-

where n , n and E a r e t h e p a r t i c l e and hole d e n s i t i e s and e n e r g i e s , r e s p e c t i v e l y , and 7t i s t h e two-body i n t e r a c t i o n , one s e e s t h a t t h e p o l a r i z a t i o n graph c o n t r i b u t e s only i f t h e energy w i s above t h e Fermi energy

X

and t h e c o r r e l a t i o n graph only i f w i s s m a l l e r t h a n

A.

Rewriting t h e imaginary p a r t of ( 1 ) i n space r e p r e s e n t a t i o n , one o b t a i n s the nonlocal imaginary p a r t of t h e o p t i c a l p o t e n t i a l ,

Here

z,

p and H a r e t h e p a r t i c l e and h o l e d e n s i t y o p e r a t o r s and t h e one p a r t i c l e Hamiltonian, r e s p e c t i v e l y .

I n o r d e r t o o b t a i n t h e c o o r d i n a t e and momentum dependent o p t i c a l p o t e n t i a l f o r a c o n t a c t exchange f o r c e

we go over t o r e l a t i v e and center-of-mass c o o r d i n a t e s a c c a r d i n g t o

-f + + +

and F o u r i e r transform with r e s p e c t t o t h e c.m. c o o r d i n a t e s , s + P, s i + p i . Then

the m a t r i x elements i n e q . ( 2 ) can be w r i t t e n i n s e m i c l a s s i c a l a p p r o x i m a t ~ o n as

and, s i m i l a r l ~ , f o r exp(-iHt). This y i e l d s

where 8 - i n d i c a t e s t h e same s t e p f u n c t i o n i n t h e s q u a r e b r a c k e t b u t taken a t

where we have i n t r o d u c e d t h e l o c a l Permi energy and t h e l o c a l Fermi momentum according t o

I n eq. ( 7 ) f i n i t e s i z e e f f e c t s e n t e r d i r e c t l y i n t h e energy conserving ô-function by t h e a d d i t i o n a l l o c a l Fermi energy.

For s i m p l i c i t y , we now use a Gaussian f o r c e of range ro i n c o o r d i n a t e o r of range k i n momentum space

+

Then the i n t e g r a t i o n over x i n eq. ( 7 ) can be performed. Furthermore, t h e momentum conserving ô-function s u g g e s t s t o i n t r o d u c e t h e r e l a t i v e and c.m. momentum t r a n s f e r s according t o

*

9 , ' F

-q/z

% = P

+ 5 / 2

(10)

and t o use t h e angle convention

C6-216 JOURNAL DE PHYSIQUE

where R i s t h e t u r n i n g p o i n t a t which V ( R ) = A .

III - ZERO RANGE FORCE

Eq. ( 1 2 ) s i m p l i f i e s c o n s i d e r a b l y i f t h e r a n g e of t h e f o r c e t e n d s t o z e r o o r k -t m.

The f i r s t l i n e o f e q . ( 1 2 ) becomes c o n s t a n t and t h e second and t h i r d l i n e s r e 8 u c e t o

The dependence of W on R h e r e i s o n l y c o n t a i n e d i n t h e l o c a l Fermi energy ( 8 ) s o t h a t e q . ( 1 3 ) c a n be employed f o r a r b i t r a r y p o t e n t i a l s . Here and i n t h e f o l l o w i n g we n o r m a l i z e W t o a t t a i n t h e v a l u e 1 / 4 X a t w = P = R = O. Momentum t r a n s f e r s a r e l i m i t e d t o

zfF(R)+

~ Z ~ ~ X - L O + L F CF))

cJ4

A

0 4

f f :

p,

( t

2

f ~ n

(w-h

+cr

CU)

w l

A

j which p r o v i d e s a l s o a s m a l l e n e r g y c u t o f f .

R e s u l t s f o r R = O a r e shown i n F i g s . 1 and 2 . One o b s e r v e s t h e w e l l known q u a d r a t i c b e h a v i o u r around w = X which p e r s i s t s even f o r l a r g e P and which i s f a i r l y indepen- d e n t of P . However, c o n t r a r y t o t h e c l a i m s i n r e f s . / 2 , 8 / , c o n t r i b u t i o n s from t h e p o l a r i z a t i o n and from t h e c o r r e l a t i o n graphs a r e n o t s t r i c t l y symmetric. This e f f e c t i s s t r o n g l y pronounced a t s m a l l P . As a consequence o f i n f i n i t e q u a s i p a r t i c l e l i f e - time a t w =

A,

W(w =

X ,

P , R) must v a n i s h . F i n a l l y , one o b s e r v e s t h a t t h e n o n l o c a l - i t y of W i s r a t h e r i m p o r t a n t f o r a l 1 e n e r g i e s excewt w A , which c a l l s i n q u e s t i o n t h e o f t e n made "innocous assumption" / 7 / of a l o c a l i m a g i n a r y p a r t of t h e o n t i c a l p o t e n t i a l . 5 F i g . 1

-

Contour p l o t of W(w, P , R = 0 ) f o r a z e r o r a n g e f o r c e . Energy E and energy t r a n s f e r p 2 / 2 r n a r e measured i n u n i t s of t h e Fermi energy. p z 3

1 ;

1 O - 3 - 2 - 1 O 1 2 3 4 5 - E

For z e r o momentum t r a n s f e r , W(w, P = O , R = 0) can b e c a l c u l a t e d a n a l y t i c a l l y

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and t h e Glauber auproximation a r e shown i n Fig. 3. Here t h e Glauber r e s u l t has been o b t a i n e d by n e g l e c t i n g t h e P a u l i p r i n c i p l e , i . e . dropping t h e second and t h i r d 8-

f u n c t i o n s i n eq. ( 7 ) and n e g l e c t i n g t h e h o l e energy p12/2m i n t h e energy conserving

Local F G. 0.3

-

O1

-

- 4 -3 - 2 -1 O 1 F i g . 3

-

The imaginary p a r t of t h e o p t i c a l p o t e n t i a l a t R = O f o r v a r i o u s v a l u e s of P and i n t h e Glauber approximation.

ô-function. One s e e s t h a t t h e l o c a l e q u i v a l e n t and P = O v a l u e s a r e r a t h e r s i m i l a r f o r p o s i t i v e e n e r g i e s b u t t h a t t h e on s h e l l value d i f f e r s d r a s t i c a l l y f o r l a r g e e n e r g i e s . The Glauber approximation l i e s always above by v i r t u e of t h e a d d i t i o n a l phase space gained by n e g l e c t i n g t h e P a u l i p r i n c i p l e .

I n t u r n i n g t o f i n i t e s i z e e f f e c t s , W can be c a l c u l a t e d a n a l y t i c a l l y on s h e l l , w = p2/2m

+

v ( R ) ,

The dependence on P' h e r e i s e x a c t l y t h e same a s i n i n f i n i t e n u c l e a r m a t t e r /2,10/ b u t w i t h

X

r e p l a c e d by t h e l o c a l cF(R). However, eq. (16) i s not s t r i c t l y t h e same r e s u l t a s one would employ t h e l o c a l d e n s i t y approximation, i . e . r e ~ l a c i n g h by E ( R ) everywhere s i n c e t h e r e g i o n s of v a l i d i t ~ of t h e d i f f e r e n t branches do depend

Fig. 4 shows t h e dependence of W on E ~ ( R ) o r , i n o t h e r words, on t h e r a d i u s R. The upper curve then corresponds t o t h e n u c l e a r i n t e r i o r and t h e lower ones t o t h e s u r f a c e . I n assuming f o r i n s t a n c e a Woods-Saxon p o t e n t i a l , f o r f i x e d energy, W i s almost c o n s t a n t i n t h e i n t e r i o r and f a l l s o f f s h a r p l y a t t h e s u r f a c e . The l o c a l d e n s i t y approximation, on t h e c o n t r a r y , would y i e l d curves s h i f t e d t o t h e l e f t and, t h u s , v i o l a t i n g t h e b a s i c f e a t u r e of W(w = A, P , R) = 0.

I V

-

FINITE RANGE FORCE

Gaussian two-body e f f e c t i v e i n t e r a c t i o n s have t y p i c a l ranges of 2.25 fm 1141, i . e . k /kF = 0.625. We t h e r e f o r e solved eq. (12) numerically f o r v a r i o u s r a n g e s . The

aussia an

exp (-2q2/k02) c u t s o f f e f f e c t i v e l y t h e high momentum t r a n s f e r contribu-

t i o n s and, hence, f o r c e s W t o d e c r e a s e a t high energy a s shown i n W(w, O . S . , R = 0) of Fig. 5 . The s t r e n g t h of t h e f o r c e , h e r e a g a i n , has been a d j u s t e d t o a t t a i n

Finite range Local F. G. F i g . 5

-

Imaginary p a r t of t h e o p t i c a l p o t e n t i a l on s h e l l a t R = O f o r v a r i o u s ranges k /k of the two-body e f f e c t i v e O F Gaussian i n t e r a c t i o n .

W(w = O, P = O , R = 0) = 114 A . The l a r g e r the range, i . e . t h e s m a l l e r ko i s , t h e more c o n c e n t r a t e d i s W around small e n e r g i e s .

For f i n i t e r a d i i , W now depends a l s o on t h e n u c l e a r mass number v i a R i n t h e RI- i n t e g r a t i o n . Fig. 6 t h e r e f o r e shows W(w, O . S . , R) f o r a 4 0 ~ a h a r m o n i c O o s c i l l a t o r and ko/kF = 0.625 a t v a r i o u s E (R). A cornparison w i t h t h e z e r o range e q u i v a l e n t ,

F F~nite range Nonlocal E G. On shell k0/kF=0625

i/znw,/~

=0.154 F i g . 6

-

Imaginary p a r t of t h e on s h e l l o p t i c a l p o t e n t i a l f o r 4 0 ~ a and v a r i o u s l o c a l Fermi e n e r g i e s .

Fig. 4, shows a g a i n t h e d e c r e a s e of t h e o p t i c a l p o t e n t i a l i n t h e s u r f a c e and a c o n c e n t r a t i o n near small p o s i t i v e and n e g a t i v e e n e r g i e s .

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10 I I I F i g . 7

-

Imaginary p a r t of t h e on s h e l l

w ( ~ ) Local F.G. o p t i c a l ~ o t e n t i a l a t R = O and r a n g e On shell ,-')k.=l0 ko/kF = 0.625 compared w i t h t h e Glauber

-

Finite range - a p p r o x i m a t i o n .

0.6-

O 2 4 6

phase s p a c e g a i n e d by t h e n e g l e c t i o n of t h e P a u l i p r i n c i p l e enhances W r o u g h l y by a c o n s t a n t amount f o r a l 1 e n e r g i e s .

V . SUMMARY AND OUTLOOK

We have a p p l i e d s e m i c l a s s i c a l methods f o r t h e c a l c u l a t i o n o f t h e imaginary p a r t of t h e o p t i c a l p o t e n t i a l W(w, P, R). F i n i t e s i z e e f f e c t s have been i n c o r p o r a t e d by means of t h e n o n l o c a l Fermi g a s a p p r o x i m a t i o n and a f i n i t e range e f f e c t i v e i n t e r - a c t i o n . As r e s u l t s we found t h a t t h e f i n i t e r a n g e of t h e f o r c e is r e s v o n s i b l e f o r t h e f a 1 1 o f f of W a t h i g h e n e r g i e s ; t h a t t h e n o n l o c a l i t y of W i s s t r o n g l y pronounced e s p e c i a l l y f o r s m a l l P ; t h a t t h e Glauber a p p r o x i m a t i o n y i e l d s v a l u e s of W always t o o h i g h by a b o u t a c o n s t a n t amount and t h a t t h e s t r i c t 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 o n l y r o u g h l y s i m u l a t e s f i n i t e s i z e e f f e c t s . F u r t h e r s t u d i e s on t h i s s u b j e c t w i l l be d e v o t e d t o t h e c a l c u l a t i o n s of t h e r e a l D a r t of t h e o p t i c a l p o t e n t i a l v i a s u b t r a c t e d d i s p e r s i o n r e l a t i o n s , t h u s o b t a i n i n g e . g . t h e c o r r e c t i o n t o t h e Hartree-Fock v o t e n t i a l and r e a l i s t i c l e v e l d e n s i t i e s around t h e Fermi e n e r g y , e f f e c t i v e masses and momentum d i s t r i b u t i o n s .

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_-

/ 4 / BERNARDV. andMAHAUX C . , Phys. Rev.

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/ 8 / BROWN G. i n "Many Body Problems" (North-Holland/Elsevier, Amsterdam, 1972) ch. 29.

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1121 HASSE R.W. and SCHUCK P., P r o c . I n t . Conf. on Nuclear P h y s i c s , F l o r e n c e , 1983 ( T i p o g r a f i a Compositori, Bologna, 1983) p . 777.

1131 HASSE R.W. and SCHUCK P . , P r o c . I n t . Symp. on Highly E x c i t e d S t a t e s and N u c l e a r S t r u c t u r e , Orsay, 1983 (Orsay, 1983) p . 16.