Microscopic analysis of nucleus-nucleus elastic scattering at intermediate energies

HAL Id: jpa-00210578

https://hal.archives-ouvertes.fr/jpa-00210578

Submitted on 1 Jan 1987

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Microscopic analysis of nucleus-nucleus elastic scattering at intermediate energies

B. Bonin

To cite this version:

B. Bonin. Microscopic analysis of nucleus-nucleus elastic scattering at intermediate energies. Journal

de Physique, 1987, 48 (9), pp.1479-1491. �10.1051/jphys:019870048090147900�. �jpa-00210578�

Microscopic analysis ^of nucleus-nucleus elastic

scattering ^at intermediate energies

B. Bonin

Service de Physique Nucléaire à Moyenne Energie ^CEN Saclay, 91191 Gif-sur-Yvette Cedex, ^France

(Requ le 19 aoat 1986, révisé le 8 avril 1987, accepté ^{le 5 mai} 1987)

Résumé.

L’article décrit d’abord une amplitude ^de diffusion nucléon-nucléon dans le noyau. Les effets du milieu nucléaire liés au principe ^{de Pauli et au mouvement de Fermi sont} explicitement pris ^en compte ^{dans la} construction de l’amplitude, calculée par ^une méthode de Monte-Carlo. Cette amplitude, ^{locale et} dépendante

de la densité, ^est ensuite utilisée via l’approximation ^de densité locale pour calculer ^un potentiel optique microscopique décrivant la diffusion élastique Noyau-Noyau ^aux énergies intermédiaires. Le couplage ^aux

états collectifs de basse énergie ^a été inclus dans la résolution de l’équation ^de Schrödinger. ^Cette analyse ^sans paramètre ^libre reproduit ^assez bien les données expérimentales pour des énergies incidentes comprises ^entre

30 et 200 MeV par nucléon.

Abstract.

An effective nucleon-nucleon amplitude for bound nucleons taking ^{into account} medium effects such as Fermi motion and Pauli blocking has been derived from the Bethe-Goldstone equation, using ^{a Monte-}

Carlo simulation of a nucleon-nucleon collision in the nuclear medium. The obtained amplitude is local and

density dependent. ^This amplitude has been used in a local density approximation ^to analyse microscopically

nucleus-nucleus elastic scattering ^at intermediate energies. ^The coupling ^to low-lying collective states has been

explicitly included. This parameter ^free analysis reproduces reasonably ^{well the} experimental ^{data down to an}

energy of 30- and up ^to 200 MeV/nucleon.

Classification

Physics ^Abstracts

24.10C - 24.10H - 25.70C

1. Introduction.

The 50 to 500 MeV/nucleon incident energy range is

a transition region ^for nucleus-nucleus (A-A) ^scatte- ring. ^At higher energies, ^A-A scattering ^{can be}

described in terms of independent, free nucleon- nucleon (N-N) collisions. At lower energies, ^the

mean-field effects dominate, ^{and a} microscopic description of the process becomes very difficult. In the transition region, medium effects such as Pauli

blocking and Fermi motion [1-21], ^{essential at low} energies, ^{still must} be taken into account.

The Fermi motion causes a large spreading ^{of the}

relative momentum distribution for the colliding

nucleons. In the low and medium energy region, ^the

N-N total cross section varies very rapidly ^{with the}

relative momentum (Fig. 1) ; ^{it is thus} important ^to

average properly ^{the N-N cross section over the}

Fermi momentum distribution. Another important

medium effect is connected with the Pauli principle,

which reduces the available phase space for ^a N-N collision in nuclei, ^as compared ^{to a free N-N}

collision. Both of these medium effects can legitima- tely ^be neglected ^at high energy, and microscopic

Fig. ^1.

^An example ^{of the} importance of Fermi motion in A-A collisions. When two slabs of nuclear matter collide at the average ^{c.m. momentum} ko = 1.24fm-l,

9 % of the N-N collisions occur at a c.m. momentum shown in the left grey bin, ^{with a N-N cross} section very different from the central bin value.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090147900

calculations of N-A and A-A elastic scattering using

the free N-N t-matrix have indeed proved ^{to be}

successful above 800 MeV/nucleon (see ^for example

refs. [22-28]). ^{This is no} longer ^{the case at} energies

lower than 500 MeV/nucleon. In the intermediate energy region, however, there is reasonable hope

that the mean field effects are still sufficiently weak,

and that a microscopic description of the A-A elastic

scattering ^is possible using ^an effective N-N interac- tion for bound nucleons. In the first part ^{of the} paper, ^an effective N-N amplitude ^{will be construc-} ted, taking ^{into account} medium effects such as

Fermi motion and Pauli blocking within the frame of

a simple model. The second part will deal with the

use of this effective amplitude ^{in a local} density approximation [20, 29-33], ^{to describe} microscopi- cally the A-A elastic scattering ^{for a} sample ^of

systems ^{and incident} energies. The method used is a momentum space folding procedure [2], ^{with the}

nice feature that it provides ^{a unified} description ^of

the real and imaginary parts ^{of the} optical potential.

In the third part, the results will be discussed and

compared ^with experiment, ^with special emphasis ^on

the recent 12C + 12C and 12C + 2°8Pb data recently

available from the Saturne National Laboratory [34].

2. The effective interaction.

In the present model, ^{the effective} amplitude ^for

bound nucleons has been taken to be proportional ^to

the probability Q (k ) ^{that a} N-N collision occurring

between bound nucleons with relative momentum k be permitted by ^{the Pauli} principle :

Equation (1) ^{does not take into account} any effect related to the Fermi motion of the nucleons in the nuclei. This can be cured conveniently by averaging equation (1) ^{over the momentum k :}

where P (ko, k) ^{is the} probability that the N-N collision occurs with relative momentum k when the A-A collision takes place ^with the average relative momentum per nucleon ko.

Equation (2) ^makes good physical sense, since it takes into account medium effects such as Pauli

blocking and Fermi motion. It can be justified ^on

account of the following argument :

Let us consider the (exact) version of the Bethe- Goldstone equation [35, 36] ^{for the} scattering ampli-

tudes

where teff, eeff,tfree ^{and efree are N-N} amplitudes ^and propagators in the nuclear medium and in the

vacuum respectively, ^{and Q the Pauli} operator. ^A frequently ^used approximation ^of equation (3) ^{is :}

The validity ^{of the} approximations (1 le’ - 1 le ,

and replacement ^of teff by t free in the second member of Eq. (3)) contained in equation (4) ^have already

been discussed [2, ^24, 35].

Application ^{of the} optical ^{theorem on} equation (4)

leads to :

where k and k’ are the initial and final centre of mass momentum of the two colliding nucleons and q ⁼ k’ - k is the momentum transfer. In equa- tion (5), ^{Q =} 0 if the collision k - k’ is forbidden,

and Q = 1 if it is allowed by ^{the Pauli} principle.

Im (teff(q =0)) ^{is not} easily ^evaluated through equation (5), except ^if Q (k -+ k’ ) ^is replaced by ^an angle-averaged Q(k) given by [11, 20]

Q (k) ^{is the} probability ^{that a N-N collision} occurring

with centre of mass momentum k be permitted by

the Pauli principle. Equation (5) ^can then be writ- ten :

Under the pion production threshold, the free NN elastic cross section Uelee equals the free NN total

cross section, ^{so that :}

Above the pion production threshold, ^{medium ef-}

fects practically disappear, Q(k) goes ^to 1 and

(eff becomes equal ^{to (free.} Equation (8) ^{can thus be}

considered to be valid also at high energy.

Equation (1) ^{has been} justified ^so far for the forward imaginary part of the effective amplitude.

The real part ^{can be} evaluated via the following, ^well

known dispersion ^relation [37] applied ^to the forward effective amplitude :

where the usual k-dependence ^of t (k, q ) ^and Q(k) has been converted into an energy depen- dence, and where the symbol S stands for principal

value.

Equation (9) ^can also be written :

where equation (8) ^{has been used to evaluate} Im teff.

The function Q (E ) is limited between 0 and 1 and has a smooth behaviour as a function of energy. A

dispersion relation similar to (9) ^can be written for the function Q , (free:

where 2: Ri is the contribution from the poles ^of

i

Q(E) t f" (E) in the upper half of the complex ^E plane (despite its smooth behaviour on the real axis,

the analytic continuation of Q (E) might ^have poles).

Comparison ^of equations (10) ^and (11) ^{then shows}

that :

The term ¿ Ri ^is independent of energy and ^must be

i

equal ^to zero, since medium effects vanish ^at high

energy. We are thus left with the relation :

The present argument ^{for the} justification ^of equation (12) ^is given only ^{as a hint, because it has}

not been proved thoroughly ^{that the effective} ampli-

tude should obey exactly ^a dispersion relation like

equation (9), and that the analytic continuation of

6(E) ^{in the} complex plane ^has only poles ^{and no}

branch cuts. Those topics certainly ^{deserve a more} complete discussion. They ^are beyond the scope of

this article, ^which mainly reports ^{on the success of}

equation (1) ^to reproduce experimental ^data.

The equation :

has now been established. Equation (1) ^is regained ^if equation (13) ^{is extended to} the q :A 0 domaine.

This extension, although unjustified, ^{is not crucial}

for the model. As is well known, ^the high energy nucleus-nucleus scattering ^is governed by ^{the for-}

ward N-N scattering amplitude ; besides, ^{the short}

range of nuclear forces implies ^a weak q dependence

of the scattering amplitude. Passing ^from equation (13) ^to equation (1) simply ^{assumes that}

the range of the effective interaction is equal ^{to the}

range of the free interaction, ^a rather conservative

assumption.

The computation of the effective amplitude through equation (2), requires ^the knowledge ^{of the}

free N-N amplitude t fee (k, q ), ^{and of} P (ko, k) ^and Q (k). ^As explained ^{in the} following section, ^{the two}

last quantities ^{have been} calculated by ^{means of a}

Monte Carlo simulation of a N-N collision within two colliding ^nuclei.

2.1 THE MONTE CARLO SIMULATION.

We use a

geometrical model similar to the one already ^used by

di Giacomo et al. [1-4]. ^The nuclei, ^both target ^and projectile, ^{are described as} free Fermi gases ; with

respective ^{Fermi momentum} kFr, kFp. ^{The corre-} sponding ^{two Fermi surfaces are assumed to be} spherical and non-deformable (Fig. 2). ^The projec-

Fig. ^2.

^A picture of the N-N collisions in ^momentum space. The projectile ^and target nuclei have Fermi spheres kFp, kFr, separated by the average ^{momentum 2} ko.

kP ^{and kt are} the initial momenta of the two colliding

nucleons ; kp ^{and kt are} the nucleon momenta after the

collision. The momentum q has been transferred.

tile nucleus is propelled against ^{the target nucleus}

with an average ^momentum per nucleon 2 ko. ^The

initial momenta kp ^{and kt of the two colliding}

nucleons are generated ^randomly within the Fermi

spheres ^{of the} projectile ^{and target nuclei} respect- ively. ^{The c.m. momentum} of the N-N collision is therefore The probability density P (ko, k) ^{that the collision occurs with} the relative momentum k is shown in figure 3 for various values of the Fermi sphere ^radii kFr, kFp. ^The spreading ^of

P (ko, k ) ^around ko simply ^reflects the Fermi motion of the two colliding nucleons. The two nucleons are

then made to scatter with a momentum transfer _q chosen randomly. ^{Below the} ’1T-production threshold, elastic N-N scattering ^{is the} only process which can occur, ^so that conservation of momentum and energy constrains the two colliding ^{nucleons to} stay ^{on the} sphere of diameter 2 k (Fig. 2). ^{After the} collision, ^their respective ^{momenta are :}

and

Fig. ^3.

^The probability density ^P (ko, k) ^{versus the c.m.}

momentum k of two colliding nucleons for various Fermi

sphere radii in the target ^and projectile nuclei. The average ^{c.m. momentum} is ko ^{= 1.24 fm - 1.}

The collision is allowed by ^{the Pauli} principle ^if

both kp ^and kt ^{fall outside the momentum space} region occupied by ^{the two Fermi} spheres. (This ^is

the case in Fig. 2.) ^{Then, the Monte} Carlo simulation

yields ^{the Pauli} blocking ^factor Q (k). ^This quantity

is shown in figure ^{4 for} various values of kFP, kFr.

2.2 FINITE SIZE EFFECTS.

Most of the authors

using geometrical ^{models similar to the one de-}

scribed above consider that the N-N collision is forbidden by ^{the Pauli} principle ^{if, after the collision,}

one of the nucleons falls inside the Fermi spheres kFP, kFr. ^In fact, ^{this is not} exactly ^{true since the}

Fig. ^4.

^{The Pauli} blocking ^factor Q(k) ^{as a} function of the c.m. momentum of the N-N collision for various target -Fermi sphere ^radii.

collision is Pauli allowed if both nucleons fall back

on their original nuclear orbits. The spatial ^wave

function of a nucleon is confined within a radius R, ^if

R is the size of the nucleus. The width of this nucleon

wave function in momentum space is thus of order

I1/R, implying that the nucleon can accept ^{a momen-}

tum transfer q 11/ R ^and yet keep a non zero

probability ^of staying ^{in its} original phase space cell.

This effect amounts to a reduction of the Pauli

blocking ⁱⁿ finite nuclei. The reduction is more

effective for small nuclei and vanishes for two infinite slabs of nuclear matter flowing through ^each

other. This finite size effect has been quantitatively

evaluated by ^{means of a} simple model. The projectile (resp. target) ^{nucleon is assumed to} occupy within the Fermi sphere kFP (resp. kFf) ^{a cubic} quantum ^cell of size 7Thl2 Rp ^(resp. 7rhl2 Rt) ^{in momentum}

space. After the collision, ^the wavepacket represent- ing the nucleon in momentum space is shifted by ^an

amount q with respect ^to the initial one. The

probability S(R, q ) that the nucleon stays in its cell after the collision is 1 if q ^: 7rhl2 R (in ^{this case the}

collision may be Pauli allowed, ^even though ^the

nucleon stays within the Fermi sphere), ^{and 0 if}

q, iT hl2 R.

As shown in figure ^5, many situations ^are possible

for the nucleons after the collision according ^{to their} respective ^{location inside or outside the two Fermi} spheres.

Situation (1) corresponds ^{to a} Pauli allowed collision. In situation (5), ^the target nucleon falls inside the target ^Fermi sphere ; ^{the collision is} permitted ^with probability S (Rt, q ). ^{In situation} (3),

the probability ^is S (Rt, q ) . S (Rp, q ) since both

nucleons must fall back on their original orbit is the

collision is to be allowed. All the eventualities

Fig. ^5.

The various possible configurations ^{of the two}

nucleons in momentum space after the collision. The dotted (full line) ^Fermi sphere corresponds ^{to the} projec-

tile (target) nucleus. The projectile nucleon is represented by ^{x ; the} target nucleon is represented by ^+.

pictured ⁱⁿ figure 5 have been included in the Monte Carlo simulation of the NN collision. The importance

of the finite size effect mentioned here can be seen in

figure ^6, where the effective N-N cross section

u eft is shown for various system ^sizes. Figure ^{6 shows}

that the importance of the Pauli blocking ^{is overes-} timated, ^{if one uses the infinite nuclear matter case}

to evaluate it. At an incident energy of 120 MeV/u,

and for realistic nuclear sizes, ^there may be varia- tions of 20 % in u eft according ^to the size of the system. ^{In the} limiting ^{case of a} pointlike system, ^the Pauli blocking ^vanishes and u eft becomes equal ^to 0" free.

Finally, ^{the Monte Carlo} simulation yields ^{the two} quantities ^P (ko, k) (Fig. 3) ^and Q (k) (Fig. 4). ^Those

two ingredients ^are sufficient to calculate the effec- tive N-N amplitude ^via equation (2).

2.3 THE FREE NUCLEON-NUCLEON AMPLITUDE. - As is usual in high energy physics, the free N-N

amplitude ^{has been} parametrized ^for neutron-neu- tron and neutron-proton scattering separately by :

where the parameters ufree, ^, a free, 13 le’ ^{have been}

taken from the Lehar et al. phase ^shift analysis [38].

afree represents ^the spin averaged ^ratio Re t f free at

1m tree

zero momentum transfer. The slope i3free ^{of the} amplitude ^has been fitted by ^{a least} square ^fit

procedure ^{over the momentum} transfer range 0-1.0 fm- l.

Fig. ^6.

Finite size effect. The effective proton-proton

cross section as a function of the target ^Fermi sphere

radius for various projectile ^and target box sizes : i) ^For

two pointlike ^nuclei (0-0), o,’ff = ol Ir,-e. ii) Rp ^{= 2.4 fm,}

Rt ^{= 2.4 fm} (12C _ 12C). iii) RP ^{= 2.4 fm, Rt = 6.5 fm} (12C - 20’Pb). iv) Rp ^{= oo,} Rt ^{= oo} (two infinite slabs of nuclear matters). The black dot is a calculation from di Giacomo et al. [4]) ^for KFp ^{= 0.3} fm-l, which compares

quite ^{well with our} (oo oo) calculation.

Note that the Gaussian q dependence ^{of the} amplitude ^{is a} rather poor parametrization ^{in the}

100 MeV/u energy range. This results in large ^uncer-

tainties in the determination of P. Altogether, 13 ^is certainly ^an important parameter, since it governs the range of the nucleon-nucleon interaction ; ^this

range influences in ^turn the diffuseness of the

microscopic nucleus-nucleus optical potential ^built

from the N-N amplitude.

The parameter -y ^{is the} phase of the N-N ampli-

tude. It is almost inaccessible to direct N-N scattering experiments ^{and is} usually ^{taken to be} equal ^{to zero.}

However, ^{it has been shown} [24, 39] ^{that the A-A} amplitude calculated in the frame of Glauber theory

is very sensitive to the value of ’Y. In the rest of this

study, ^{we shall} keep ^{y =} 0 and examine in a

separate section the effect of q on the A-A optical potential.

2.4 THE EFFECTIVE AMPLITUDE FOR BOUND NU- CLEONS.

The effective amplitude ^obtained

exhibits a complicated dependence ^{on various} quan- tities :

i) kFP, kFr dependence.

For small values of kFp, kF-r, ^{the effective} ampli-

tude is roughly equal ^to the free N-N amplitude.

For higher ^{values of} kFP, kFr, ^{teff is} depressed ^when compared ^to t free . This kFp, A;pr dependence ^{will be}

transformed into a density dependence ^{via a local} density approximation.

ii) ko dependence.

At high energy, ko ^becomes large ⁱⁿ comparison

with kFP, kFr. ^The medium effects then become

negligible ^{and teff = tfree. This} property built in the model automatically ^{ensures the success of} teff at

high energy, since the ^use of tfree is known to

reproduce satisfactorily ^the experimental ^{data for}

E -- 500 MeV/u (see ^for example ^refs. [22-28] ^and

references therein).

iii) q dependence.

At low energy, ( T ⁵⁰ MeVlu ), ^{the N-N scatter-} ing ^is isotropic, ^{and the} slope parameter 0 ^{free can}

thus be taken equal ^{to zero for} neutron-neutron and neutron-proton collisions. However, ^the amplitude

tfree(q ) ^{does not} retain its constant value for very

large ^values of q. The collision is assumed ^to take

place ^{on shell, and} therefore, ^the momentum trans- fer cannot be greater than the relative momentum,

so that t free (k, q ) = 0 if q ^{:::. 2} k. This truncation is

unimportant ^at high energy, since it ^{occurs at} very

high ^momentum transfer. At smaller energy, it amounts to giving a non zero range ^{k -} 11/ k ^{to the}

N-N interaction, ^even though j6 ^{free is} equal ^{to zero.}

Many previous microscopic ^{A-A studies} [4, 40-42]

have assumed /3 ⁼ 0 and have neglected ^{this trun-} cation, ^thus implicity taking ^{a zero} range N-N interaction. As we shall _see, the associated effect on

the A-A optical potential ^{is far from} negligible.

iv) Rp, Rt dependence.

teff also depends ^{on the radii} Rp, ^{Rt of the} projectile ^and target nuclei In the limit Rp, ^{Rt -+ 0,}

teff becomes ab constructio equal ^{to tfree since the}

Pauli blocking factor goes ^to 1. In the limit

Rp, ^{Rt -+ oo, teff becomes equal to tmatter, i.e. the}

scattering amplitude ^{for two} colliding slabs of nuclear matter. Our calculation of the effective N-N ^cross

section . Rp eff = oc), Rt = c)o ⁱ

(Fig. 6) compares quite well with the value from di

Giacomo et al. [4], which has been calculated

neglecting finite size effects.

In spite ^{of these} complicated dependences, ^the

effective amplitude obtained is still very crude for the following ^{reasons :}

i) Exchange ^{effects are not} completely ^included

since only ^{one nucleon} exchange is taken into account via the use of an antisymmetrized amplitude.

ii) Off-shell effects are neglected.

iii) ^The spin degrees ^of freedom, ^are partly neg- lected since t eff has only ^{a central} part : ^the

(T free, a free used to calculate teff take care of the spin only ^{in an} average way.

iv) The effective amplitude ^is local, ^{with a} simple

momentum transfer dependence. ^{As a} counterpart, the present ^effective amplitude ^is easy ^to compute

(1) ^{and easy to use. Besides, it turns out to be rather}

sucessful in the calculation of a nucleus-nucleus

optical potential.

3. Use of the effective amplitude ^{in a} microscopic

model of nucleus nucleus scattering.

3.1 THE OPTICAL POTENTIAL.

The effective am-

plitude for bound nucleons described above has been used for the analysis of A-A elastic scattering

at medium energy through ^{a momentum} space

folding procedure (the ^{so called « tp »} potential),

which can be summarized through ^{the formula :}

where pp ^{and p are the} projectile ^and target ^nuclear densities, ^m is the nucleon _mass, and t is the N-N

amplitude. ^The -sign stands for a 3 dimensional Fourier transformation. The observables are then calculated by solving ^the Schrodinger equation ^with

the potential Uopt. Besides its simplicity, ^{the tp} potential has many nice features : t being complex, ^it provides ^{a unified treatment} for the real and imagi-

nary part ^{of the} optical potential ; if it is taken to be

local, ^the optical potential obtained is also local.

Equation (16) is also attractive because of its clear connections with well known theories : it appears ^as the first order of KMT theory [22] ^{and as the} optical

limit of Glauber theory [40, ^43, 44]. ^If t (q) ^{is taken}

to be a constant, equation (16) ^can be written :

where the ^* sign ^{means a} 3-dimensional folding. ^This

r space folding ^{has been} widely ^used [40-42] ⁱⁿ previous ^A-A analyses. ^{A drawback of} equation (17)

lies in the fact that the range of the N-N interaction

(1) Numerical results are available from the author.

is neglected. ^{This error can be} compensated by ^the

choice of a phenomenological t [42] ^{or cancelled} partly by ^{the use of the} impulse approximation ^and by ^the neglect of medium effects [40, 41]. ^{We shall}

retain the full equation (16) ^{for the rest of this} study.

The isospin degrees ^{of freedom can be} explicitly

included in equation (16) by separating ^{out the roles} played by protons ^{and neutrons :}

where p8, p;, ^{PP,pn are the proton and neutron}

densities respectively.

Equation (16) ^{is used} successfully ^at high energy

by making ^the impulse approximation [22-28] :

Because of the medium effects described above, ^the impulse approximation ^is certainly faulty ^at energies

lower than 500 MeV/nucleon, ^{at least} in its crude form (19). ^{It is thus} tempting ^to try ^{to extend} downwards the domain of validity ^of equation (16) by using ^the prescription :

This attempt ^is expected ^{to be} correct at high

energy, since teff tfe, ^(tfee reproduces ^{the A-A}

data at high energy). ^{Below 500} MeV/nucleon, ^the prescription (20) ^should improve ^the quality ^{of the}

fit by taking ^{into account medium effects.}

3.2 THE LOCAL DENSITY APPROXIMATION

(LDA). - ^The kFP, kFr dependence ^of teff has been transformed into a density dependence by ^{means of}

a Local Density Approximation (LDA) [2, 4, 20, ^29- 33] :

assuming that the N-N amplitude ^is locally ^{the same}

as the one for two colliding Fermi gases of respective density _pp,pt.

The amplitude teff ^then depends upon the local densities pp, pt, and this density dependence ^{can be}

taken into account in equation (16) :

However, this 6 fold integral is tedious and expensive

to calculate. In order to simplify ^the calculation, ^we

have assumed the nuclear densities ^to be spherical.

The Fourier transform of p p, ^t then reads :

One can compute ^the quantity :

F is then readily ^Fourier transformed to yield ^the optical potential :

An usual _{way of} using ^{the LDA is to} consider that the N-N collision takes place ^{at the} density _{p p} ⁺

p t with ^{an effective} amplitude depending only ^{on the}

total density [2]. ^This prescription, however, ^{is not} unique since the interaction has a finite range and

projectile and target nucleon coordinates refer to different density points in the medium. Besides, ^this procedure ^{is not} well suited at medium energies

since the two nuclei occupy ^a very different location in phase space. In this paper, ^we have preferred ^to

treat the projectile ^and target ^densities separately

and this is why t eff depends upon p p ^{and p t.}

The LDA has met with much success in the past, possibly because the density dependence ^of Jeff ^is

weak [30, 33]. ^In spite ^{of its} simplicity, ^{the LDA} permits ^{one to take into account structure effects : it}

includes the coupling of the elastic channel to

inelastic channels in an average way. However, ^the

effective amplitude teff ^has been calculated assuming

that the projectile ^and target ^Fermi spheres keep

their original spherical shape during the collision.

This amounts to a neglect ^{of the} coupling ^{of the}

elastic channel to collective oscillations (correspond- ing ^{to a} deformation of the Fermi sphere) ^{or to}

nucleon transfer reactions (corresponding ^{to a modi-}

fication of the Fermi sphere radii). ^The neglect ^of

the coupling ^to collective states has been shown to

yield poor results which do ^not match the data, especially ^for energies lower than - 86 MeV/nucleon

[45-47]. ^{We shall cure this} by explicity taking ^the

main ones into account in a coupled ^{channel treat-}

ment of the elastic scattering.

3.3 NUCLEAR DENSITIES.

The nuclear density p p, t is ^an ingredient ^{needed to calculate the} optical potential. Experimental densities have been taken from the literature (from ^ref. [48] ^for proton ^den- sities, and from ref. [24] ^{for neutron densities,}

except ^for 12C, ^{where we} have taken P f2c ⁼ P f2c.) ^Of

course, point ^{densities must be used in} equations (16-24). ^These experimental ^densities

have thus been corrected for the finite size of the nucleon. No centre of mass correction has been made in equation (16), since this is automatically

included in the experimental ^densities.

Schrodinger equation ^{has been solved} for the A-A elastic scattering ^{with the} optical potential Uopt,

using ^the coupled channel code ECIS [49]. ^The coupling of the elastic channel to the low lying

collective levels has been explicitly taken into ac- count. The levels included in ECIS for each nucleus

are given ^{in table I,} together with the deformation parameters used. The channels taken into account were single excitation of the 2+ or 3- states but no

mutual excitation. The standard first order collective model was used for simplicity.

Table I.

List of ^{the low} lying ^{levels included in the} coupled ^{channel treatment} of the elastic scattering.

4. Results and discussion.

4.1 ENERGY DEPENDENCE OF THE OPTICAL POTEN- TIAL.

The microscopic optical potential ^calculated

at various energies ^{are shown in} figure ^{7 for the} 12C + i2C _system. The energy dependence ^{of the} microscopic optical potential is the result of a

complex interplay among four factors :

i) The energy dependence of the nucleon-nucleon forward amplitude ^{tends to} reduce the depth ^{of the} imaginary potential ^as the energy increases.

ii) The energy dependence of the Pauli blocking

has the opposite ^effect.

iii) ^The slope parameter of the nucleon-nucleon

amplitude ^varies sharply with energy ; this influences the diffuseness of the resulting potentials.

iv) The ratio of the real to imaginary part ^{of the}

forward nucleon-nucleon amplitude ^{has a} compli-

cated _energy dependence, very different _{for pp and} np amplitudes ; ^this influences the ratio of the real to the imaginary part ^{of the} potential.

Fig. ^7.

The energy dependence ^{of the 12C + 12C} microscopic optical potential.

The net results are a quasi constancy ^{of the} depth

of the real part ^{of the} potential ^as the energy

increases, ^{while the} imaginary part increases. (Note

that these remarks are limited to the 80- 200 MeV/nucleon energy domain.)

These findings ^{are in} qualitative agreement ^with previous theoretical [11, 33, 45, 50-54] ^and phenom- enological [34, 55] observations in the hundred MeV/nucleon energy ^domain.

4.2 THE PHASE OF THE N-N AMPLITUDE. - In this section we have taken the phase ^{y of the N-N} amplitude ^{as a free} parameter, ^{with the} assumption

Fig. ^8.

Influence of the phase of the N-N amplitude ^on

the microscopic optical potential. (Calculated ^{here with-}

out medium effects.)

Fig. ^9.

Influence of the phase of the N-N amplitude ^on

the calculated elastic cross section. (Without ^medium

effects.)

that _y is the same for proton-proton ^{and neutron-} proton scattering ^{and does not} depend ^on the energy of the N-N collision. The 120 MeV/nucleon 12C +

12C microscopic optical potential has been calculated for three values of y. It can be seen in figures ^{8 and 9}

that the potentials ^{and cross section are sensitive to}

the value of _y. The values of _y used are those from reference [39]. V. Franco obtained a good ^{fit to} high

energy ^aa scattering ^data using ^{y = 6.2} fm2. This value could thus be considered as realistic. It should be noted, however, that in the one hundred MeV energy region, ^the phase ^shift analysis ^{of N-N} scattering by Lehar et al. yields ^a y roughly equal ^to

zero [38]. ^Since y is essentially ^unknown experimen- tally, ^{the realistic} microscopic potential ^{could be}

considered to lie somewhere within the two extreme

curves of the figure 8. The value _y ⁼ 0 has been

kept ^{for the rest of this} study.

4.3 COMPARISON WITH EXPERIMENTAL CROSS SEC- TION AND POTENTIALS.

4.3.1 Cross sections.

The results of the optical

model calculation described above are shown in

figure ^{10 for the} 12C + 12C _system ^at

120 MeV/nucleon. (These very ^recent data are from

Fig. ^10.

The elastic differential cross section for the system 12C + 12C at 120 MeV/u.

microscopic potential

with all medium effects taken into account ;

microscopic potential without Pauli blocking [Q (k ) = 1].... ^micro- scopic potential without Fermi motion. [P (ko, k) = 8 (k - ko)]. ^-..-.. microscopic potential without finite size effects (RP ^{= Rt = oo ). -.-.} microscopic potential ^without

any medium effect, [teft ⁼ tfreel,

the Saturne National Laboratory [34].) The salient features are as follows :

i) ^The microscopic potential with all medium effects taken into account fits the experimental ^data reasonably ^well (it ^should be remembered that the model contains no adjustable parameters).

ii) It is essential to take into account medium

effects, ^{since the} calculation done with the free N-N

amplitude tfee(ko) (dash-dotted line) definitely ^does

not fit the experimental ^data.

iii) All the medium effects (Pauli blocking, ^Fermi motion, ^finite size) should be included since the

neglect ^{of one} of them results in a poor fit.

The remarks i, ii, ^{and iii are} valid for the two

systems ^{studied in the} energy range 30- 200 MeV/nucleon.

Figures 11-15 show the fit obtained with the

microscopic model for the 12C + 12C _system at 30, ⁸⁶

and 200 MeV/nucleon, and for the 12C + 208 Pb _system

at 120 and 200 MeV/nucleon.

The fit is reasonably good ^{at 86 and}

120 MeV/nucleon. At 120 MeV/u, the inelastic scat-

tering ^{to the 2 + level at} 4.4 MeV has also been measured [34]. ^Here again, the fit obtained from the

coupled ^{channel treatment with the} microscopic potential ^is quite good, ^{with a} deformation par- ameter value commonly used in the literature

(/3 2 + = - 0. 62) ) (Fig. 16). ^{The influence of the} coupling ^{of the} elastic channel to the low lying

collective states has been investigated ^{and found to}

be rather small for energies ^{of 120} MeV/nucleon and

higher (Fig. 17). ^{The model} might ^be expected ^to

Fig. ^11.

^{Fit to} the elastic differential cross section for the 12C + 12C _system at 30 MeV/nucleon. The experimental

data are from [63].

Fig. ^12.

^{Same as} figure ^{11 but at} 86 MeV/nucleon (data

are from [64].)

Fig. ^13.

^{Same as} figure ^{11 but at} 200 MeV/nucleon

(data ^{are from} [34].)

fail for an energy ^as low as 30 MeV/u, ^{and should}

work best at the highest studied energy, i.e.

200 MeV/u. The opposite ^indeed happens : ^{the fit}

for 12C + 12 C at 30 MeV/u is still acceptable, ^while

Fig. ^14.

^{Fit to} the elastic differential cross section for the 12C + 2°8Pb _system at 120 MeV/nucleon.

Fig. ^15.

^{Same as} figure ^{14 but at} 200 MeV/nucleon.

the fit at 200 MeV/u is less satisfactory. ^{As will be}

shown in the next section, ^a possible explanation ^lies

in the radial shape ^{of the} potentials.

4.3.2 Potentials.

A comparison ^{of the} microscopic

potential ^with phenomenological optical potentials

Fig. ^16.

^{Inelastic cross section to the 2+ level at}

4.4 MeV for the 12C + 12C _system at 120 MeV/nucleon.

Fig. ^17.

Influence of the coupling ^{to the low} lying

collective levels for the 12C + 12C _system at 120 MeV/nucleon.

No coupling.

Coupling ^{to the}

2+ ^{state at} E* ⁼ 4.4 MeV.

may be useful, ^{but is somewhat} delicate, ^{since the} phenomenological potentials fitting ^the heavy ^ion

elastic scattering ^{data are} generally ^not unique ^and

are only determined in a « sensitive region ^{» located}

near the surface [56, 57]. ^{Moreover, the} phenomeno- logical potentials usually describe the elastic channel

alone, while the microscopic potential ^{must be} explicitly coupled ^to collective states. The compari-

son is thus relevant only ^{when the} coupling ^{can be}

Fig. ^18.

Comparison ^of optical potentials ^{for the} 12C + 12C _system at 120 MeV/nucleon. Same convention as

in figure 10.. The shaded area represents ^{the zone where} the phenomenological potential ^is sensitively ^deter-

mined [34].

neglected, ^{i.e. above 120} MeV/nucleon. It can be

seen from figure ^{18 that the} microscopic potential ^is

in good agreement ^{with the} phenomenological ^one

in the sensitive region. However, ^{it is too} deep ^{in the}

interior. This feature is general and is found for both systems ^{studied at} 120 and 200 MeV/nucleon. Be-

cause of the onset of the so called « transparency effect ^» [34, ^{51, 55,} 58] ^{at the} highest energy studied,

the potential ^is probed ^{in a} very large ^{radial zone.}

This contrasts with the low energy situation, ^where

the potential ^is probed ⁱⁿ a narrow zone located around the strong absorption radius. It is thus more

difficult to obtain a good agreement ^{between the}

phenomenological and theoretical potentials ^at high

energy. A wrong diffuseness could therefore explain

the lesser quality of the fit to the elastic scattering

data at 200 MeV/u. Note that no attempt ^{has been} made to adjust the diffuseness of the potentials by varying the diffuseness of the nuclear densities and the range parameter f3 of the nucleon-nucleon ampli- tude, although ^{none of them are} very accurately

determined. The role of the various ingredients ^of

the model can be seen further in figure ^{18 :} i) ^{The Pauli} blocking ^{tends to} reduce the depth ^of

the potential in the interior and leaves the surface

region unaffected.

ii) Taking ^{into account} the Fermi motion has the

opposite ^{effect, for the} imaginary part ^{of the} poten-

tial. On the other hand the inclusion of Fermi motion reduces the depth of the real part, ^{at least in}

the 30-200 MeV/nucleon region.

iii) The finite size effects also tend ^to moderate the importance of the Pauli blocking.

4.4 VALIDITY DOMAIN OF THE MODEL. - The

model presented ^{above assumes that the} projectile

and target ^Fermi spheres ^are spherical ^{and non-}

deformable in momentum space. However, ^the

Pauli principle ^{forbids the} overlapping ^{of the two} spheres during the collision. The model is thus

expected ^to break down when 2 ko becomes of the order of, ^or smaller than kFP ⁺ kFr. ^{If the two Fermi} sphere ^{radii are taken} equal ^{to the} ordinary ^nuclear

matter Fermi sphere radii, ^{i.e. 1.36} fm-1, ^{this limit}

corresponds ^{to an incident} energy of 140 MeV/nucleon. In spite ^{of this} stringent limitation, ^{we have seen} that the model is still rather successful for energies ^{as low as 30} MeV/nucleon.

This is probably ^{due to} the fact that the A-A elastic

scattering mainly probes the outside part ^{of the} optical potential. Therefore, the relevant nuclear densities are much less than the usual nuclear matter

density, ^{and a} realistic lower limit for the validity

domain of the model is given by :

where (typical ⁼ (3 IT 2P typical )113 is much smaller than the nuclear matter value of 1.36 fm - 1 .

5. Conclusion.

The model of A-A interaction presented ^{here seems}

to reproduce satisfactorily ^the experimental ^elastic scattering ^{data for 12C + 12C and 12C + 2°8Pb from}

30 MeV/nucleon upwards. This has been achieved

by carefully including ^the physics ^{known to be} important ^at low energy, i.e. medium effects such ^as Pauli blocking and Fermi motion.

The present ^{model is a crude} simplification ^{of the}

G matrix approaches ^which reproduce ^{the data at}

low energy and up ^to 100 MeV/u [11, 29-33, 45-47, 50-54]. ^{It is} interesting ^{to note that this} simplification

proves ^to be sufficient, ^{even for} energies ^{as low as}

80 MeV/nucleon. The present model then represents

a bridge ^between the G matrix approach ^{which is} especially ^{used at} energies smaller than 100 MeV/u and the KMT or Glauber type theories successful above 500 MeV/u using ^{a standard} impulse approxi-

mation.

The effective N-N amplitude for bound nucleons described in this paper takes into ^account medium effects and has an intricate density, ^{momentum and}

box size dependence, ^{but it remains} rather easy ^to handle. It is hoped ^{that this effective N-N} amplitude

will be used as an input for other problems ⁱⁿ

Nuclear Physics, ^for example pion production ⁱⁿ heavy ^ion collisions [59] ^or ablation-abrasion model of fragmentation [60-62]. ^In the future it will be

interesting ^to include in the above description ^a

more careful treatment of the spin degree ^of freedom, ^{and to} apply ^{a similar} approach ^{to the} problem of the nucleon-nucleus interaction.

This paper is dedicated ^to the memory of Madame Faraggi.

References

[1] CLEMENTEL, ^{E. and} VILLI, C., Nuovo Cimento 2

(1955) ^176.

[2] ^{SALONER, D.} A., TOEPFFER, C. and FINK, B., ^Nucl.

Phys. ^{A 283} (1977) 108 and 131.

[3] KIKUCHI and KAWAI, M., ^{Nuclear matter} and nuclear reactions (North Holland, Amsterdam) ^1968.

[4] ^DI GIACOMO, ^N. J., ^DE VRIES, R. M. and PENG, J. C., Phys. Rev. Lett. 45 (1980) 527 ; Phys. ^Lett.

101B (1981) ^383.

[5] BEHERA, B., PANDA, K. C. and SATPATHY, ^R. K., Phys. Rev. C 21 (1980) ^1883.

[6] JEUKENNE, J. P., LEJEUNE, ^{A. and} MAHAUX, C., Phys. Rev. C 16 (1977) ^80.

[7] ^{BRIEVA, F. A. and} ROOK, ^J. R., ^Nucl. Phys. ^{A 291} (1977) ^299.

[8] ^{BRIEVA, F.} A., ^VON GERAMB, ^{H. and} ROOK, ^J. R., Phys. ^{Lett. 79B} (1978) ^177.

[9] ^{VARY, J. and} DOVER, C., Proc. 2nd High energy

heavy ^{ion summer} study (Berkeley) 1974, LBL 3675.

[10] ^{MULLER, K. H.,} Phys. ^{Lett. 93B} (1980) ^247.

[11] ^FAESSLER, A., DICKHOFF, W. H., TEFZ, ^{M. and} RHOADES-BROWN, ^Nucl. Phys. ^{A 428} (1984)

271c.

[12] ^{PRUESS, K. and GREINER,} W., Phys. ^{Lett. 33B} (1970)

197.

[13] ^{BRINK, D. A. and} STANCU, F., ^Nucl. Phys. ^{A 243} (1975) ^175.

[14] ZINT, ^{P. G. and} MOSEL, U., Phys. Lett. 56B (1975)

424.

[15] GÖRITZ, G. F. and MOSEL, U., ^Z. Phys. ^{A 277} (1976) ^243.

[16] FLECKNER, ^{J. and} MOSEL, U., ^Nucl. Phys. ^{A 277} (1977) ^170.

[17] ZINT, ^P. G., ^Z. Phys. ^{A 281} (1977) ^373.

[18] MOSZKOWSKI, ^S. A., ^Nucl. Phys. ^{A 309} (197) ^273.

[19] HERNANDEZ, E. S. and MOSZKOWSKI, ^S. A., Phys.

Rev. C 21 (1980) ^929.

[20] RIKUS, L., NAKANO, K. and VON GERAMB, ^H. V., Nucl. Phys. ^{A 414} (1984) ^413.

[21] ^VON GERAMB, ^H. V., ⁱⁿ The interaction between medium energy nucleons in nuclei, ^Indiana 1982, ed. H. O. Meyer, ^{AIP Conf.} Proc., ^1983.

[22] ^{KERMAN, A.} K., MCMANUS, ^{H. and} THALER, R. M., ^Ann. Phys. ⁸ (1959) ^551.

[23] ^{ALKHAZOV, G. D. et} al., ^Nucl. Phys. ^{A 280} (1977)

365.

[24] CHAUMEAUX, A., LAYLY, V. , SCHAEFFER, R., ^Ann.

Phys. (NY) ¹¹⁶ (1978) ^247.

[25] LAYLY, ^{V. and} SCHAEFFER, R., Phys. ^{Rev. C 17} (1978) ^3.

[26] BAUER, ^{T. S. et} al., Phys. Rev. C 19 (1979) ^1438.

[27] WALLACE, ^{J. et} al., Adv. Nucl. Phys. ¹² (1981) ^135.

[28] SATO, ^{H. and} OKUHARA, Y., Phys. Lett. 162B (1985)

217.

[29] JEUKENNE, J. P., LEJEUNE, ^{A. and} MAHAUX, C., Phys. Rev. C 10 (1974) ^1391.

[30] ^MAHAUX, C., Proc. Summer Institute in Nuclear

theory (Banff, Canada Plenum Press)1978.

[31] ^{BRIEVA, F. A. and} ROOK, ^J. R., ^Nucl. Phys. ^{A 291} (1977) ^317.

[32] ^VON GERAMB, H. V., BRIEVA, F. A. and ROOK, J. R., in Microscopic optical potentials, ^ed.

H. V. von Geramb (Springer) ^1979.

[33] MÜLLER, K. H., ^Z. Phys. ^{A 295} (1980) 79, Phys.

Lett. 93B (1980) ^247.

[34] HOSTACHY, ^{J. Y.} et al., Communication to Harrogate

Conference and Preprint ^to appear in Phys. ^Lett.

(1987).

[35] GOLDBERGER, ^{M. L. and} WATSON, ^K. M., Collision

theory (Wiley, ^{N. Y} ;) ^1967.

[36] HÜFNER, ^{J. and} MAHAUX, C., ^Ann. Phys. ⁷³ (1972)

525.

[37] NEWTON, ^R. G., Scattering theory of ^{waves and} particles (McGraw Hill) ^1966.

[38] LEHAR, F., private communication.

[39] ^{FRANCO, V. and YIN, Y.,} Phys. Rev. Lett. 55 (1985)

1059 and a paper ^to be published ⁱⁿ Phys. ^{Rev. C} (1986).

[40] CHAUVIN, J., LEBRUN, D., LOUNIS, ^{A. and} BUENERD, M., Phys. Rev. C 28 (1983) ^1970.

[41] CHAUVIN, R., LEBRUN, D., DURAND, ^{F. and} BUENERD, M., J. Phys. ^{G 11} (1985) ^261.

[42] ^VARY, J. P. and DOVER, C. B., Phys. Rev. Lett. 31

(1973) ^1510.

[43] GLAUBER, ^R. J., Lectures in Theoretical Physics.

BOULDER, ^W. E., ^{BRITRIN and} DUNHAM, ^{L. G. eds.}

(Interscience ^N. Y.) ^1958.

[44] DAR, ^{A. and} KIRZON, Z., Phys. Lett. 37B (1971) 166 ; ^Nucl. Phys. ^{A 237} (1975) ^319.

[45] TREFZ, M., FAESSLER, A., DICKHOFF, W. H. and RHOADES-BROWN, M., Phys. Lett. 149 B (1984)

459.

[46] KHADKIKAR, S. B., RIKUS, L., FAESSLER, ^{A. and} SARTOR, R., ^Nucl. Phys. ^{A 369} (1983) ^495.

[47] FAESSLER, A., RIKUS, ^{L. and} KHADKIKAR, S. B., Nucl. Phys. ^{A 401} (1983) ^157.

[48] ^{DE JAEGER, C. W. et al., Atomic Data} Nucl. Tables 14 (1974) ^479.

[49] RAYNAL, J., ^Lecture given ^{at a Seminar course on} computing ^{as a} language ^of Physics (ICTP, Trieste, 1971) ^and Phys. ^{Rev. C 23} (1981) ^2571.

[50] ^BECK, F., MÜLLER, ^{K. H. and} KÖHLER, ^H. S., Phys.

Rev. Lett. 40 (1978) ^837.

[51] NGUYEN VAN SEN et al., Phys. Lett. 156B (1985) ^185.

[52] ISMAIL, M., FAESSLER, A., TREFZ, M. and DIC- KHOFF, W. H., J. Phys. ^{G 11} (1985) ^763.

[53] TREFZ, M., FAESSLER, ^{A. and} DICKHOFF, W. H., submitted to Nucl. Phys. ^A (1985).

[54] ^VINH MAU, N., Preprint ^{submitted to Nucl.} Phys. ^A (1986).

[55] BONIN, ^{B. et} al., ^Nucl. Phys. ^{A 445} (1985) ^381.

[56] SATCHLER, ^G. R., ^Int. Conf. ^on Reactions between

complex ^nuclei, Nashville, Tennesse, ^{USA North} Holland Vol. 2, 1974, p. 171.

[57] SATCHLER, ^G. R., Direct nuclear reactions (Oxford University Press) ^1974.

[58] ^{DE VRIES, R. M. and PENG, J.} C., Phys. ^{Rev. C 22} (1980) ^1055.

[59] ^{BERTSCH, G. F.,} Phys. ^{Rev. C 15} (1977) ^713.

[60] HÜFNER, ^{J. et} al., Phys. ^{Lett. 73 B} (1976) ^289.

[61] ^{VARY, J. P. et} al., Phys. Rev. C 20 (1979) ^715.

[62] BLESZYNSKI, ^{M. and} SANDER, C., ^Nucl. Phys. ^{A 326} (1979) ^525.

[63] LOUNIS, ^{A. et} al., Proc. Int. Conf. ^{on Nucl.} Phys.,

Firenze (1983) eds P. Blasi and R. A. Ricci, Vol.1, 1983, p. 523.

[64] BUENERD, ^{M. et} al., Phys. Rev. C 26 (1982) ^1299.