NUCLEUS-NUCLEUS POTENTIAL ; THEORETICAL ASPECTS IN THE 20-100 MeV/u RANGE

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ASPECTS IN THE 20-100 MeV/u RANGE

A. Faessler, R. Linden, N. Ohtsuka, F. Malik

To cite this version:

A. Faessler, R. Linden, N. Ohtsuka, F. Malik. NUCLEUS-NUCLEUS POTENTIAL ; THEORET-

ICAL ASPECTS IN THE 20-100 MeV/u RANGE. Journal de Physique Colloques, 1986, 47 (C4),

pp.C4-111-C4-123. �10.1051/jphyscol:1986415�. �jpa-00225781�

JOURNAL DE PHYSIQUE

Colloque C4, supplement au n o 8, Tome 47, aoiit 1986

NUCLEUS-NUCLEUS POTENTIAL ; THEORETICAL ASPECTS IN THE 20-100 MeV/u RANGE( 1 )

A. FAESSLER, R. LINDEN, N. OHTSUKA and F.B. MALIK +

Institut fiir Theoretische Physik der Universitat Tubingen, 0 - 7 4 0 0 Tubingen, F.R.G.

Abstract

-

The method developed by us previously f o r c a l c u l a t i n ~ the real a ~ d

-

imaginary part of the optical potential from the nucleon-nucleon (NN) inter- action i s extended and refined in several ways: ( i ) We request now ;elfcon- sistency so t h a t the same force (Reid s o f t core) determines the ground s t a t e properties of the two interacting nuclei including binding energies and mass d i s t r i b u t i o n s and also the optical potential. ( i i ) A Weizsacker l i k e surface term (0,)2 i s added, which cannot be determined in i n f i n i t e unclear matter.

( i i i ) We use f o r the density of the two i n t e r a c t i n g nuclei two limiting assumptions: In the sudden approximation the two d e n s i t i e s are added f o r each distance R of t h e two nuclei. In the a d i a b a t i c approach we do not allow t h a t the density gets l a r g e r than the saturation density. That means t h a t the total density adjusts optimally f o r each distance. The real and the imaginary p a r t of the energy per nucleon in two nuclear matters flowing throuqh each other i s shown a s a function of the density f o r d i f f e r e n t average r e l a t i v e k i n e t i c en- e r g i e s . The real an im inary parts o the sudden an adiabatic potentials

6 4

a r e given f o r ~ Z C + ~ C , fg0+160, 0°ca+4 Ca and 208pbt2 8 ~ b f o r differen bom- barding energies. E l a s t i c and i n e l a s t i c cross sections a r e given f o r JCt12C i n a coupled channel approach f o r ELab=300 and 1016 MeV.

I

-

INTRODUCTION

For describing the s c a t t e r i n g and the reactions between two nuclei the optical model i s always the s t a r t i n g point. Thus the heavy ion (HI) optical potential i s an essential quantity f o r HI nuclear physics. Therefore we have i n the past develooed a method how t o derive the real and imaginary part of t h e optical potential between h.ro nuclei from a r e a l i s t i c nucleon-nucleon (NN) interaction [I-81.

For the real part of the optical model already before our work a simple and trans- parent derivation had been achieved in the folding model [9]. How well such a folding model can reproduce the 12c+12c s c a t t e r i n g data with a phenomenolopical l y f i t t e d imaginary part has been shown f o r example by von Oertzen and coworkers [10,111.

In our approach [I-81 the s t a r t i n g point i s the c o l l i s i o n of two i n f i n i t e nuclear matters which flow through each other. F i r s t we solve the Bethe-Goldstone equation.

Since the sum of the Fermi spheres of the two interacting nuclear matters i s non- spherical the Bruckner reaction matrix gets complex. This reaction matrix allows t o calculate a complex energy density. With the help of a generalized local density approximation we are able t o calculate the real and imaginary part of the optical potential between two nuclei.

upport ported by the GSI-Darmstadt and the Deutsahe Forschungsgemeinschaft. F.R.C.

+permanent address : Dept. of Physics. University of Southern Illinois a t carbondale. IL 92902 Carbondale.

U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986415

In the present contribution we extend and r e f i n e our previous approach [I-81 i n four ways: ( i ) We c a l c u l a t e the ground s t a t e properties of nuclei across the whole mass t a b l e with the same Y N force a s the optical p o t e n t i a l . Thus t h e same NN force determikes the mass d i s t r i b u t i o n of each nucleus and the interaction between the HI'S.

( i i ) We add t o the enerSy density a WeizsZcker l i k e surface term ( ~ p ) ' . Only with t h i s term we a r e able t o describe the nuclear mass d i s t r i b u t i o n s and the radii i n arreement with the data. ( i i i ) To c a l c u l a t e the density d i s t r i b u t i o n of the two in- teracting HI'S we use two limiting approaches: In the sudden approximation we add the two d e n s i t i e s and i n t h e adiabatic approach we allow t h a t the d e n s i t i e s a r e appro- ximately optimized s t a t i c a l l y f o r each distance R. ( i v ) We distinguish f o r the elas- t i c and i n e l a s t i c s c a t t e r i n g of two HI'S between the nuclear radius RN (obtained by f i t t i n g t h e real part of our optical potential by a Saxon-Woods p o t e n t i a l ) and the Coulomb radius ~ ~. z . A ' / ~ - 1 fm. W i t h B l a i r ' s scaling rule

BC RC = BN R N

t h i s y i e l d s also d i f f e r e n t deformations f o r t h e t r a n s i t i o n s .

The optical potential obtained i n t h i s way describes q u i t e well the experimental data 110-121 f o r t h e 12C

+

e l a s t i c and i n e l a s t i c s c a t t e r i n g .

In chapter 2 we give a very s h o r t survey of the theoretical description, while the r e s u l t s a r e presented in Chapter 3. Chapter 4 summarizes the main conclusions.

I1

-

THEORETICAL DESCRIPTION

The s t a r t i n g point f o r the calculation of the real and imaginary p a r t of the potential between two heavy ions i s t h e c o l l i s i o n of two i n f i n i t e nuclear matters.

They a r e flowing through each other. The i n t e r a c t i o n between the d i f f e r e n t nucleons i s taken i n t o account using the Bethe-Goldstone equation.

The reaction matrix < k i , k ; lG(W=ekl + E ~ kFl ~ ,kF2,kr) ;

I

kl,k2> depends on the momenta k l , k 2 of the two i n i t i a l nucleons and on t h e momenta of the two nucleons k i , k; in t h e intermediate s t a t e s . In addition i t depends on the s t a r t i n g energy W = € k l + €k2 and on the Fermi momenta k F l , k ~ 2 of t h e two Fermi spheres. The two Fermi momenta a r e connected with the local densities i n p r o j e c t i l e and t a r g e t .

The solution of t h e Bethe-Goldstone equation ( 2 ) i s complex since the energy denomi- nator

can have a pole due t o the non-sphericity of t h e two Fermi spheres. The Pauli opera- t o r f o r the two Fermi spheres allows intermediate energies c k i

+

~ k i which have the same values a s the s t a r t i n g energy W = ckl

+

^{E k 2 .} I t i s obvious t h a t f o r only one i n f i n i t e nuclear matter corresponding t o a spherical Fermi sphere the intermediate

energy o f t h e two nucleons i s always l a r g e r than t h e s t a r t i n g energy W and thus one f i n d s t h e r e o n l y r e a l r e a c t i o n s m a t r i c e s G .

The t o t a l energy d e n s i t y f o r t h e c o l l i s i o n o f t h e two i n f i n i t e n u c l e a r matters nNM(;) = T ( ~ F ~ ( ; ) , k ~ ~ ( f ) . kr) + n(kF1(:). kF2(f). kr) ( 5 ) i s c a l c u l a t e d as t h e sum o f t h e t o t a l k i n e t i c energy d e n s i t y and t h e Hartree-Fock p o t e n t i a l energy.

4 + 2 3

T = I (k-k,) d k + + 4 ) / ( 2 m)

(2n) _F

<k,, k2/~(kFl(i). k F 2 i i ) , kr, WIkl. kp>

The k i n e t i c energy d e n s i t y ^T(;) and t h e p o t e n t i a l energy d e n s i t y n(;) a r e c a l c u l a t e d by i n t e g r a t i n g over t h e content o f t h e two u n i t e d Fermi spheres and by summing over protons and neutrons w i t h s p i n up and s p i n down ( t h i s i s t h e f a c t o r 4). These energy d e n s i t i e s depend i n o r b i t a l space on

?

due t o t h e f a c t t h a t t h e Fermi momenta depend on l o c a l d e n s i t i e s p(;) as i n d i c a t e d i n eq.(3). The o p t i c a l model p o t e n t i a l can then be d e f i n e d as t h e energy o f two heavy i o n s obtained by i n t e g r a t i n g the energy density

(F)

o f eq.(5) over t h e two heavy i o n s a t d i s t a n c e R andsubtracting t h e correspon-

NM

d i n g value a t i n f i n i t y .

This expression depends n o t o n l y on t h e d i s t a n c e between t h e two heavy i o n s i n o r b i - t a l space b u t a l s o on t h e r e l a t i v e k i n e t i c energy represented by t h e average r e l a t i v e momentum p e r nucleon kr.

Ile have here added t o t h e nuclear m a t t e r energy d e n s i t y nNM(?. R) a surface c o r r - e c t i o n which i s s i m i l a r t o t h e Weizsacker s u r f a c e term [14] ( b u t has a s l i g h t l y d i f f - e r e n t d e n s i t y dependence). The parameter a = 8.30 fm has been a d j u s t e d t o reproduce 3 t h e experimental r o o t mean square r a d i i o f t h e n u c l e i (see below).

I t can be shown [ 4 ] t h a t expression ( 7 ) can be obtained approximately u s i n g t h e Feshbach expression [ I 3 1 f o r t h e H I o p t i c a l p o t e n t i a l .

V I @ )

u(R) = (+oIvrI@o) + ( @ o I v p Q E

-

^{QHQ + in QP p (8)}

Here, t h e round brackets i n d i c a t e i n t e g r a t i o n over a1 1 v a r i a b l e s a p a r t o f t h e r e l a - t i v e d i s t a n c e R between t h e two heavy i o n s . @, i s t h e i n t r i n s i c w a v e f u n c t i o n o f t h e two heavy i o n s a t d i s t a n c e R w i t h o u t i n c l u s i o n o f t h e r e l a t i v e wavefunction. P pro- j e c t s on t h e two ground s t a t e s w h i l e Q includes a l l t h e o t h e r s t a t e s . Vr i s t h e r e - s i d u a l i n t e r a c t i o n between t h e nucleons i n heavy i o n one and i n heavy i o n two. I n using eq. (8) one must be aware t h a t i t i s n o t an exact expression since t h e space Q should n o t c o n t a i n break-up i n t o t h r e e fragments.

The numerical c a l c u l a t i o n proceeds now i n t h e way t h a t we choose a r e a l i s t i c nucleon-nucleon i n t e r a c t i o n ( R e i d - s o f t - c o r e - p o t e n t i a l ) and s o l v e f o r t h i s i n t e r a c t i o n V the Bethe-Goldstone eq.(2). The P a u l i o p e r a t o r i s d e f i n e d i n t h e angle averaged approximation. The complex r e a c t i o n m a t r i x

i s c a l c u l a t e d f o r t h e d e n s i t i e s p = 0.25 pO, 0.5 pO, 0.75 po, 1.00 p0...,2.5 p o and f o r t h e K = p / ( P +p ) Galues (pp,pT d e n s i t i e s i n the p r o j e c t i l e and t h e t a r g e t , res-

P P T

p e c t i v e l y ) K = 0, 1/8, 1/4, 3/8, 1/2 and t h e average r e l a t i v e momenta kr 0.5, 1.0, 1.5, 2.0, 2.5, 3 fm-l. For t h e s t a r t i n g energy we choose an averaged value

W

[2,3,4].

I n a d d i t i o n we t a b u l a t e t h e k i n e t i c energy d e n s i t y ^T and t h e p o t e n t i a l energy den- s i t i e s n ( 6 ) f o r d i f f e r e n t values o f p and kr. The average r e l a t i v e momentum kr i s determined by t h e bombarding energy o f t h e heavy i o n beam. The t o t a l d e n s i t y p i s taken i n each volume element u s i n g f o r b o t h heavy i o n s mass d e n s i t i e s determined from e l e c t r o n s c a t t e r i n g by s c a l i n g t h e charge d i s t r i b u t i o n A/Z. This d e n s i t y a l l o w s t o c a l c u l a t e t h e l o c a l k i n e t i c energy d e n s i t y 1151. With t h e h e l p o f p and ^T one can determine K . With t h e h e l p o f p and K one i n t e r p o l a t e s f o r a given r e l a t i v e average momentum kr t h e p o t e n t i a l energy d e n s i t y n(?). T and n determine t h e t o t a l energy d e n s i t y q(F). This energy d e n s i t y i s then i n t e g r a t e d f o r a given d i s t a n c e R between t h e two heavy ions t o o b t a i n t h e t o t a l energy E(kr,R) which determines according t o (7) t h e complex o p t i c a l p o t e n t i a l .

I n t h i s way one o b t a i n s t h e volume p a r t o f t h e H I o p t i c a l p o t e n t i a l b u t one can n o t i n c l u d e t h e surface v i b r a t i o n a l e x c i t a t i o n s . They can be handled i n two d i f f e r e n t ways: e i t h e r they a r e e x p l i c i t l y i n c l u d e d [ 4 ] i n t h e f i n i t e nucleus u s i n g t h e Fesh- bach expression ( 8 ) o r they a r e i n c l u d e d by e x p l i c i t l y t a k i n g t h e i n e l a s t i c s c a t t e r - i n g i n t o these s t a t e s i n t o account by the coupled channel approach.

I n t h e

+

s c a t t e r i n g i t t u r n e d o u t t o be enough t o i n c l u d e i n t h e coupled channel treatment the 2+(4.44 MeV) and 3-(9.64 MeV) s t a t e s . From B l a i r ' s s c a l i n g r u l e ( 1 ) and t h e reduced EX t r a n s i t i o n p r o b a b i l i t i e s we f i n d f o r t h e Coulomb and n u c l e a r deformation parameters :

~ ~

+

) (=

-

20.586 ~ ~ ( 2 ' ; 1016 MeV) =

-

^0.418

BC(3-.) = 0.942 BN(3-; 1016 MeV) = 0.672

The n u c l e a r deformation parameters B~ vary s l i g h t l y w i t h t h e bombarding energy ( f o r example: EL = 1016 MeV), since o u r r e a l p a r t o f t h e 12c

+

o p t i c a l p o t e n t i a l de- pends on t h e energy.

I 1 1

-

^RESULTS

From the energy d e n s i t y nNM i n n u c l e a r m a t t e r ( 5 ) one can e a s i l y c a l c u l a t e t h e energy p e r nucleon as a f u n c t i o n o f t h e d e n s i t y p f o r d i f f e r e n t average r e l a t i v e momenta o f a nucleon i n nuclear m a t t e r 1 and an average nucleon i n n u c l e a r m a t t e r 2.

The r e a l p a r t f o r d i f f e r e n t average r e l a t i v e momnta kr [ f m - l ] i s g i v e n i n F i g u r e 1.

For kr d i f f e r e n t f r o m zero one o b t a i n s a l s o an imaginary p a r t which i s shown i n FiGure 2

Fig. 1:

Real part of the energy per nucleon i n nuclear ma. t e r as a function of t h e density p in units of the satu- ration density p0 = 0.17 ~ I I I - ~ . The proton density i s equal t o the neu- tron density. The interaction i s the Reid-soft-core potential the energy per nucleon E / A i s obtained by dividing the energy density ( 5 ) by the density p . The d i f f e r e n t curves a r e f o r d i f f e r e n t average r e l a t i v e momenta hkr of a nucleon i n nuclear matter 1 r e l a t i v e t o a nucleon i n nuclear matter 2.

Fig. 2: Imaginary part of t h e t o t a l energy per nucleon as a function of t h e d e n s i t y p i n u n i t s o f t h e saturation density p O = O . 17 f ~ n - ~ . tlkris the ave- rage momentum of nucleon in nuclear matter 1 r e l a t i v e to a nucleon in nuclear matter 2.

The t o t a l energy density ( 5 ) i s calculated in nuclear matter and therefore does not contain any surface corrections. We therefore a r e adding ( 7 ) a Weizsacker surface correction term (vp12. The parameter a i s now adjusted by f i t t i n g in d i f f e r e n t nuclei the root mean square radius. We parametrize the nuclear mass d i s t r i b u t i o n by expres- sions used t o describe the charged d i s t r i b u t i o n in electron s c a t t e r i n g . [I71

p ( r ) = pO{1+ W

(-t

) 2 l / ( l + e x p [ ( r V - ~ & ) / a ~ l ) (10) R~~

To a l l o w a l s o f o r asymmetric n u c l e a r m a t t e r we add t o t h e energy d e n s i t y ( 5 ) a term

w i t h : cr = 8.30 fm 3 C = 90 MeV

We c a l c u l a t e now f o r d i f f e r e n t n u c l e i t h e t o t a l energy by i n t e g r a t i n g t h e eneroy d e n s i t y (11) w i t h t h e d e n s i t y (10) over the whole nucleus. The parameters a a n d C a r e then a d j u s t e d t o reproduce as good as p o s s i b l e the r o o t mean 'square r a d i i and t h e b i n d i n g energies of 12c, 160, 4 0 ~ a and *08pb. The r a d i u s parameter RWS i s v a r i e d t o minimize t h e energy f o r each nucleus. The r o o t mean square r a d i u s o b t a i n e d i n t h i s way agrees extremely w e l l w i t h t h e measured value (see Fig.3). TO perform t h i s c a l -

c u l a t i o n we assumed i n l ' ~ , 160 and 4 0 ~ a t h a t t h e p r o t o n and t h e neutron d e n s i t y a r e p r o p o r t i o n a l t o each o t h e r and a r e normalized t o Z and A - Z. Only f o r *08pb we scale t h e r a d i a l coordinate f o r the neutrons i n such a way t h a t r o o t mean square r a d i u s i s by 0.2 fm l a r g e r t h a n t h e r o o t mean square r a d i u s f o r t h e protons.

F i g . 3: Experimental b i n d i n g energy p e r nucleon and r o o t mean square r a - d i u s f o r 12c, 160,40~a and '08pb compared w i t h our t h e o r e t i c a l r e s u l t obtained by i n t e g r a t i n g energy den- s i t y (11) over t h e whole nucleus and m i n i m i z i n g t h i s energy as a f u n c t i o n o f t h e r a d i u s parameter RWSin eq. (10).

, O - - - - o ~ o

,--,--- - 6 - - - - - - _ _ _ _ ,

A Theory

0 Experiment

,

10 Lo 100 2M) MASS NUMBER A

To c a l c u l a t e the o p t i c a l p o t e n t i a l between two n u c l e i according t o eq. ( 7 ) we must know t h e d e n s i t y o f t h e two n u c l e i which are a p a r t by t h e d i s t a n c e R. We use here f o r the p r e s c r i p t i o n t o o b t a i n t h e d e n s i t y of the two i n t e r a c t i n g n u c l e i two I i m i t i n g recipes:

i n the sudden approach we j u s t add t h e d e n s i t y o f t h e f i r s t and t h e second nucleus.

I n t h e a d i a b a t i c approach we assume t h a t t h e two n u c l e i have a t each d i s t a n c e R

enough time t o r e a d j u s t t h e i r d e n s i t y d i s t r i b u t i o n . Thus we should c a l c u l a t e the den- s i t y d i s t r i b u t i o n o f t h e two i n t e r a c t i n g n u c l e i i n a s t a t i c two c e n t r e Hartree-Fock

approximation. To obtain t h i s adiabatic density d i s t r i b u t i o n of the t t I 1 s we use the followinq recipe: we assume t h a t the saturation density i s given by the maximum den- s i t y of the independent mass d i s t r i b u t i o n s of the two nuclei. As long a s the additive density of the two nuclei i s not l a r g e r than t h i s saturation density we j u s t add t h e two d e n s i t i e s . I f the additive density surpasses the saturation density we take as the limiting value the saturation density and s c a l e the s p a t i a l variables so t h a t the integral over the t o t a l density y i e l d s the sum of the mass numbers. I f one would take t h i s recipe l i t e r a l l y one would end up with a density which i s not smooth and has d i f f i c u l t i e s a t the distance R = 0. A smooth adiabatic t o t a l density i s obtained by the following prescription:

+ R

p p ( r + --; a I? ) f o r Z < -R/2

2 P P

+ R.

apRp) + ~ - ~ ( r

- 7 ,

aTRT);BI f o r -R/2 < Z ^c R/2 f o r Z > R/2

( 1 2 )

The scaling variables ap(R) f o r t h e p r o j e c t i l e and aT (R) f o r t h e t a r g e t a r e unity f o r distances l a r g e r than Ro f o r which the sum of the two densities p p

+

^{p T} i s l e s s than the saturation density defined by the maximum of the s e p a r a t e d e n s i t i e s . A t sepa- ration R = 0 t h e Saxon-Woods radius parameter f o r the p r o j e c t i l e Rp and the Saxon- Woods radius parameter f o r the t a r g e t RT should be identical

Ye know therefore the r a t i o of the scaling parameters of the p r o j e c t i l e and the t a r - oet f o r distance R=O and f o r t h e c r i t i c a l distance Ro above which no scaling i s nec- essary.

I f we do a l i n e a r interpolation f o r the r a t i o 5 of the scaling parameters f o r the p r o j e c t i l e and t h e t a r g e t between the two values given in eq. (14) we obtain f o r t h i s r a t i o as a function of R:

c(R) = (1

-

^{i(0))R/RO + ~ ( 0 ) f o r 0 s R ,< RO}

f o r R > RO ⁽¹⁵⁾ c(R) = 1

The absolute values of t h e scaling parameters a p ( R ) and aT(R) i s obtained from the normal ization.

Figure 4 and Figure 5 show the sudden and adiabatic mass d i s t r i b u t i o n f o r 12c +12c a t a distance R = 4 fm.

Fig. 4:

Sudden density d i s t r i b u t i o n of two 12c nuclei a t a distance R = 4 fm.

The l i n e s of equal density in the lower part a r e one tenth of the maximum density apart from each other.

Fig. 5:

Adiabatic density of 12c

+

^{a t a}

Z [ f m l

distance of R = 4 fm. The lower p a r t

-4 0 4

shows equal density l i n e s which a r e ^{1 1 1 ) , 1 1 1}

one tenth of the maximum density ^Adiabnt~c

-

apart of each other. The adiabatic _-

density i s defined in eqs.(lZ) t o

(16). ^R12

We have s t u d i e d t h e sudden and the a d i a b a t i c o p t i c a l p o t e n t i a l f o r d i f f e r e n t k i n e t i c energies and d i f f e r e n t p a i r s o f n u c l e i 12c

+

"c, 160

+

160, 4 0 ~ a

+

4 0 ~ a and

2 0 8 ~ b

+

2 0 8 ~ b . To save space we show here o n l y t h e r e a l and the imaginary p a r t o f t h e o p t i c a l p o t e n t i a l f o r 12c

+

^12c a t a l a b o r a t o r y energy EL = 300 MeV i n F i g u r e 6 and 7.

DISTANCE R l f m l

Fig. 6: o ^{2 4} _{6 8}

Real p a r t o f the o p t i c a l p o t e n t i a l o f 12c on "C w i t h a l a b o r a t o r y

energy E L = 300 MeV i n t h e sudden _{- 2 0}

-

^-

and a d i a b a t i c l i m i t i n g case.

-

-

sudden -

- - - - ad~abatic

F i g . 7:

Imaginary p a r t o f t h e o p t i c a l p o t e n t i a l o f 12c on 12c w i t h a l a b o r a t o r y energy EL = 300 MeV f o r t h e sudden and adiaba- t i c approach.

- 8 0

I,, ,I- I “ _I I _I

/

_/

^yv

-100

DlSTANCE R f f m )

0 2 4 6 8

-

^sudden

-

---- a d ~ a b a t ~ c

-20

I

- ^I

/ I

-

t

,,;

_I

,'

^\

- // -

- /

I 4 I I

I

-

I I I /

- -

I I I I

[ \ ,

Figures 8, 9 and 10 show e l a s t i c and i n e l a s t i c cross-sections f o r the 12c on

Y C

a t 1016 MeV and 360 MeV.

Fig. 8:

El a s t i c s c a t t e r i n g cross, section of on 12c a t EL = 1016 MeV The data a r e taken from ref- erence 12. The s o l i d l i n e i s the e l a s t i c cross section calculated in a coupled channel approach [16] including the ~'(4.44 MeV) and 3-(9.64 MeV) s t a t e s . A t small angles both recipes y i e l d the same cross-section.

I I I

c t? xjl=

0 -

\ -

v -

- -

ld:

^-

-

sudden

- ---- ad~abat~c

I I

5 10 15

1:

20

Fig. 9:

I n e l a s t i c excitation of the ~'(4.44 bkV) s t a t e i n the s c a t t e r i n g of 12c on12c with a laboratory energy of EL = 1016 Mev.

The data a r e taken from reference [12].

The difference between the sudden (solid l i n e ) and the adiabatic (dashed l i n e ) approach i s small.

"C ⁺ "C , 360 MeV

-

^sudden

t ^-

^adiabatic

^j

Fig. 10: E l a s i j c sca ering cross-section i n units of the Rutherford cross-section f o r C on " C a t a laboratory energy of EL = 360 MeV. Within the angle range given the sudden ( s o l i d l i n e ) and the adiabatic (dashed l i n e ) approach y i e l d s p r a c t i c a l l y the same cross-section. The agreement a t these lower

energies i s not a s good as a t the higher energies.

Judging the agreement between theory and experiment in Figures 8, 9 and 10 one has t o take i n t o account t h a t the cross-sections have been calculated parameter f r e e s t a r t i n g from a real i s t i c nucleon-nucleon i n t e r a c t i o n . Even the root mean square radius i s reproduced i n agreement with the experimental data with the same NN inter- action using expression (10) and varying the radius parameter RWS t o minimize the t o t a l energy which one obtains from the energy density (11) by integrating over the nucleus. The only ingredient taken from experiment i s the w parameter and the diffuse- ness a of equation (10) and the t r a n s i t i o n p r o b a b i l i t i e s i n t o the 2' and 3- s t a t e s i n 12c. One should not mix the type of calculation presented here with a calculation of the cross-section where one uses f o r t h e real p a r t a folding potential and one f i t s the imaginary p a r t t o the data. [10,11]. I t i s obvious t h a t i f one f i t s a large nu, ber of parameters f o r the imaginary p a r t d i r e c t l y t o the cross-sections f o r each bombarding energy separately one has t o obtain a much b e t t e r agreement. In the pre- s e n t calculation no parameter i s adjusted t o the cross-section data. In t h i s sense the present approach y i e l d s a parameter f r e e theoretical cross-section with t h e essential irnput of the r e a l i s t i c N N interaction.

I V

-

CONCLUSIONS

We s t a r t e d from a r e a l i s t i c nucleon nucleon i n t e r a c t i o n (Rei d - s o f t - c o r e - p o t e n t i a l ) and c a l c u l a t e d t h e r e a c t i o n m a t r i x f o r t h e c o l l i s i o n o f two i n f i n i t e nuclear matters f o r d i f f e r e n t r e l a t i v e average momenta kr and w i t h d i f f e r e n t d e n s i t i e s p T and p p f o r t h e t a r g e t and t h e p r o j e c t i l e y r e s p e c t i v e l y . Using a aeneral i z e d l o c a l d e n s i t y a p p r e x i m a t i o n we were a b l e t o c a l c u l a t e t h e volume c o n t r i b u t i o n t o the r e a l and imaginary p a r t o f t h e heavy i o n p o t e n t i a l . Compared t o former c a l c u l a t i o n s [ l - 8 1 we requested now t h a t t h e same r e a l i s t i c NN i n t e r a c t i o n (Reid-soft-core) should a l s o reproduce the ground s t a t e p r o p e r t i e s o f t h e n u c l e i i n v o l v e d and n o t o n l y t h e r e a l and t h e imagin- a r y p a r t o f the o p t i c a l model. I n a d d i t i o n we added a Weizs'dcker l i k e s u r f a c e term which cannot be c a l c u l a t e d i n n u c l e a r matter. The s t r e n g t h parameter i s a d j u s t e d t o reproduce as w e l l as p o s s i b l e the r o o t mean square r a d i i o f several n u c l e i across the p e r i o d i c t a b l e . F o r the d e n s i t y o f the two i n t e r a c t i n g n u c l e i we used two l i m i t i n g assumption: i n t h e sudden approximation t h e two d e n s i t i e s a r e added f o r each d i s - tance R o f t h e two n u c l e i . I n t h e a d i a b a t i c approach we do n o t a l l o w t h a t t h e density gets l a r g e r than t h e s a t u r a t i o n d e n s i t y d e f i n e d by t h e h i g h e s t d e n s i t y o f t h e two i n - d i v i d u a l n u c l e i . Although the sudden and a d i a b a t i c o p t i c a l model p o t e n t i a l s a r e q u i t e d i f f e r e n t f o r small distances R between t h e two n u c l e i t h e c r o s s - s e c t i o n s a r e r o u g h l y the same s i n c e t h e imaginary p a r t prevents t h a t t h e n u c l e i see t h e d i f f e r e n c e of t h e p o t e n t i a l s a t s h o r t distances.

The agreement i s v e r y good f o r h i g h energies (1016 MeV f o r on "C s c a t t e r i n g ) . A t s m a l l e r energies t h e t h e o r e t i c a l cross-section overestimates t h e experimental values a t l a r g e r angles. This i s probably due t o t h e f a c t t h a t t h e imaginary p a r t o f the o p t i c a l model p o t e n t i a l i s underestimated due t o t h e f a c t t h a t t r a n s f e r r e a c t i o m a r e n o t e x p l i c i t l y included.

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