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

NUCLEUS-NUCLEUS POTENTIAL ; THEORETICAL 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 ; THEORETICAL 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> HAL Id: jpa-00225781 https://hal.archives-ouvertes.fr/jpa-00225781 Submitted on 1 Jan 1986 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. 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, 0-7400 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) interaction i s extended and refined in several ways: ( i ) We request now ;elfconsistency 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 distributions 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 interacting nuclei two limiting assumptions: In the sudden approximation the two densities are added f o r each distance R of t h e two nuclei. In the adiabatic approach we do not allow t h a t the density gets larger 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 part of the energy per nucleon in two nuclear matters flowing throuqh each other i s shown as a function of the density for different average r e l a t i v e kinetic energies. The real an im inary parts o the sudden an adiabatic potentials are given f o r ~ Z C + ~ Cfg0+160, , 0°ca+4 Ca and 208pbt2 8 ~ bf o r differen bombarding energies. Elastic and i n e l a s t i c cross sections a r e given f o r JCt12C in a coupled channel approach f o r ELab=300and 1016 MeV. 6 4 - INTRODUCTION For describing the scattering 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 in the past develooed a method how t o derive the real and imaginary part of the 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 transparent derivation had been achieved in the folding model [9]. How well such a folding model can reproduce the 12c+12c scattering data with a phenomenolopical ly 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 collision 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 nonspherical the Bruckner reaction matrix gets complex. This reaction matrix allows to 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. I 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 C4-112 JOURNAL DE PHYSIQUE In the present contribution we extend and refine our previous approach [I-81 in four ways: ( i ) We calculate the ground s t a t e properties of nuclei across the whole mass table with the same Y N force as the optical potential. Thus the same NN force determikes the mass distribution 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 are able t o describe the nuclear mass distributions and the radii i n arreement with the data. ( i i i ) To calculate the density distribution of the two interacting HI'S we use two limiting approaches: In the sudden approximation we add the two densities and i n the adiabatic approach we allow t h a t the densities a r e approximately 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 elast i c and i n e l a s t i c scattering of two HI'S between the nuclear radius RN (obtained by f i t t i n g the real part of our optical potential by a Saxon-Woods potential) 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 yields also different deformations f o r the transitions. The optical potential obtained in t h i s way describes q u i t e well the experimental e l a s t i c and i n e l a s t i c scattering. data 110-121 for the 12C + In chapter 2 we give a very short survey of the theoretical description, while the r e s u l t s are presented in Chapter 3. Chapter 4 summarizes the main conclusions. - THEORETICAL DESCRIPTION The s t a r t i n g point f o r the calculation of the real and imaginary part of the potential between two heavy ions i s the collision of two i n f i n i t e nuclear matters. They a r e flowing through each other. The interaction between the d i f f e r e n t nucleons i s taken into account using the Bethe-Goldstone equation. I1 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 the two Fermi spheres. The two Fermi momenta are connected with the local densities in p r o j e c t i l e and target. The solution of the Bethe-Goldstone equation ( 2 ) i s complex since the energy denominator can have a pole due t o the non-sphericity of the two Fermi spheres. The Pauli operat o r f o r the two Fermi spheres allows intermediate energies cki + ~ k which i 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 to 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 finds there only real reactions matrices 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 I (2n) F = T <k,, The k i n e t i c energy d e n s i t y + (k-k,) k2/~(kFl(i). T(;) 2 3 d k + + 4 ) / ( 2 m) k F 2 i i ) , kr, WIkl. kp> and t h e p o t e n t i a l energy d e n s i t y n(;) are calculated 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 on l o c a l d e n s i t i e s p(;) ? due t o t h e f a c t t h a t t h e Fermi momenta depend 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 NM (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 correspond 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 3 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 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 . u(R) = (+oIvrI@o) + ( @ o I v p QE - QHQ + VQPI @p ) in (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 the i n t r i n s i c wavefunctionof the 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 proj 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. C4-114 JOURNAL DE PHYSIQUE 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 ) V the Bethe-Goldstone eq.(2). and s o l v e f o r t h i s i n t e r a c t i o n 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 calculated f o r the densities f o r the K pectively) 1.5, 2.0, = 0.25 pO, 0.5 pO, 0.75 po, p 1.00 p0...,2.5 and po = 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 , P K 2.5, P T = 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, 3 fm-l. For t h e s t a r t i n g energy we choose an averaged value 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 s i t i e s n ( 6 ) f o r d i f f e r e n t values o f p T W res- 1.0, [2,3,4]. and t h e p o t e n t i a l energy den- 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 is 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 determine K. With t h e h e l p o f p and K momentum kr t h e p o t e n t i a l energy d e n s i t y n(?). d e n s i t y q(F). p and T one can 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 T and n determine t h e t o t a l energy 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 Feshbach 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. + 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 I n the 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 BC(3-.) = 0.942 ~ ~ ( 2 ' ;1016 MeV) = BN(3-; 1016 MeV) = - 0.418 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. I11 - 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 in nuclear ma. t e r as a function of the density p in units of the saturation density p0 = 0.17 ~ I I I - ~The . proton density i s equal to the neutron 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 different curves are f o r different average r e l a t i v e momenta hkr of a nucleon in nuclear matter 1 r e l a t i v e t o a nucleon in nuclear matter 2. Fig. 2: Imaginary part of the total energy per nucleon as a function of thedensityp i n u n i t s o f t h e saturation tlkris . the avedensity pO=O. 17 f ~ n - ~ rage momentum of nucleon in nuclear matter 1 r e l a t i v e to a nucleon in nuclear matter 2. The total energy density ( 5 ) i s calculated in nuclear matter and therefore does not contain any surface corrections. We therefore are adding (7) a Weizsacker surface correction term (vp12. The parameter a i s now adjusted by f i t t i n g in different nuclei the root mean square radius. We parametrize the nuclear mass distribution by expressions used t o describe the charged distribution in electron scattering. [I71 2 (10) p ( r ) = pO{1+W (-t) l / ( l + e x p [ ( r V - ~ & ) / a ~ l ) R~~ C4-116 JOURNAL DE PHYSIQUE 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 with: 3 cr = 8.30 fm 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 ~ aand *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 ~ at 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 dens 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 RWSineq. (10). , O - - - A Theory 0 Experiment - o -6----- ~ o , 10 Lo 100 MASS NUMBER A 2M) 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 dens 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 distribution of the ttI1s we use the followinq recipe: we assume that the saturation density i s given by the maximum dens i t y of the independent mass distributions of the two nuclei. As long as the additive density of the two nuclei i s not larger than t h i s saturation density we j u s t add the two densities. I f the additive density surpasses the saturation density we take as the limiting value the saturation density and scale the spatial variables so t h a t the integral over the total density yields 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 total density i s obtained by the following prescription: R a I? ) f o r Z < -R/2 p p (+r + --; 2 PP R.r aTRT);BI apRp) + ~ - ~- (7, + f o r -R/2 for Z > < Z c R/2 R/2 (12) 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 the t a r g e t are unity f o r distances larger 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 separatedensities. A t separation 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 SaxonWoods 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 the c r i t i c a l distance Ro above which no scaling i s necessary. If we do a linear 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 the 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 c(R) = 1 - i(0))R/RO + ~ ( 0 ) for 0 s R for R > RO ,< RO (15) The absolute values of the scaling parameters ap(R) and aT(R) i s obtained from the normal ization. JOURNAL DE PHYSIQUE Figure 4 and Figure 5 show the sudden and adiabatic mass distribution f o r 12c +12c a t a distance R = 4 fm. Fig. 4: Sudden density distribution of two 12c nuclei a t a distance R = 4 fm. The lines 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 + at a distance of R = 4 fm. The lower part shows equal density l i n e s which are one tenth of the maximum density apart of each other. The adiabatic density i s defined in eqs.(lZ) t o (16). Z -4 1 1 [fml 4 0 1 ) , 1 Adiabnt~c 1 1 - 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 ~ aand 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 o Fig. 6: 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 laboratory energy EL = 300 MeV i n t h e sudden -20 - - and a d i a b a t i c l i m i t i n g case. - - sudden ---- - ad~abatic I -80 I I,, I“ ,I- I I I / / yv -100 t - I / ,,;,' - // -/ \ [\, I I 4 I I F i g . 7: DlSTANCE R f f m ) 2 4 6 0 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 8 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 adiabat i c approach. - - sudden ---- ad~abat~c I I I I / - -20 I I I I C4-120 JOURNAL DE PHYSIQUE YC 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 a t 1016 MeV and 360 MeV. Fig. 8: El a s t i c scattering cross, section on 12c a t EL = 1016 MeV of The data are taken from reference 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 yield the same cross-section. I I I c t? 0 \ v xjl= - - ld: - - - sudden 1: ---- ad~abat~c I I 5 10 15 20 Fig. 9: I n e l a s t i c excitation of the ~'(4.44 bkV) s t a t e in the scattering 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 t + "C , 360 MeV -sudden - adiabatic j Fig. 10: Elasijc sca ering cross-section in 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 practically the same cross-section. The agreement a t these lower energies i s not as 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 into account that 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 interaction. Even the root mean square radius i s reproduced in agreement with the experimental data with the same NN interaction using expression (10) and varying the radius parameter RWS t o minimize the total 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 diffuseness a of equation (10) and the t r a n s i t i o n probabilities into 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 the real part a folding potential and one f i t s the imaginary p a r t to 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 directly to 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 pres e n t calculation no parameter i s adjusted t o the cross-section data. In t h i s sense the present approach yields a parameter f r e e theoretical cross-section with the essential irnput of the r e a l i s t i c NN interaction. C4-122 IV - JOURNAL DE PHYSIQUE 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 pT and pp for 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 imagina 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 scattering). 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. REFERENCES [I] T. Izumoto, S. Krewald, A. Faessler, Nucl-Phys.= 121 (1980) 319 T. Izumoto, S. Krewald, A. Faessler, Nucl . P h y s . K (1981) 471 A. Faessler, T. Izumoto, S. Krewald, R. S a r t o r , Nucl . P h y s . e (1981) 509 S.B. Khadkikar, L. Rikus, A. Faessler, R. Sartor, N u c 1 . P h y s . E A. Faessler, W.H. D i c k h o f f , M. T r e f z , Yucl .Phys.= M, T r e f z , A. Faessler, W.H. D i c k h o f f , Nucl .Phys M. I s m a i l , A. Faessler, M. Trefz, W.H. G.R. S a t c h l e r , W.G. Love, Phys.Rep. (1984) 271c .A443 (1985) D i c k h o f f , J.Phys. M. I s m a i l , M. Rashdan, A. Faessler, M. Trefz, H.M.M. 55c (1981) 495 G 499 2 (1985) 763 Mansour, t o be published (1979) 183 W. von Oertzen, i n " F r o n t i e r s i n Nuclear Dynamics" (1985) 241; eds. R.A. g l i a , C.H. Dasso; Plenum P u b l i s h i n g Corporation. Bro- Ell] H.G. Bohlen, X,S, Chen, J.G. Cramer, P. Frolerich, B. Gebauer, H. Lettau, M. Miczaika, W . von Oertzen, R. Ulrich, T. Wilpert, Z.Phys.= (1985) 341 - (1982) 1299 and N u c 1 . P h y s . Z (1984) 313 [12J M. Buenerd a t a l . , Phys.Rev.CP6 19 (1962) 287 1131 H. Feshbach, Am.Phys.5 (1958) 357; 1141 I. Reichstein, F.B, Ma1 i k , preprint 1986 [15] R. Sartor, A. Faessler, Nucl .Phys.= (1982) 263 El61 M. Rhoades-Brown, M.H. MacFarlane, S.C. Pieper, Phys ,Rev.C21(1980) 2417 and 2436 [I71 C.W. De Jager, H. de Vries, and C. de Vries, At.Nucl.Data Tables 14 (1974) 479