Nuclear Physics A401 (1983) 157-174

0 North-Holland Publishing Company

THE HEAVY-ION POTENTIAL AND ITS INCREASING TRANSPARENCY AT INTERMEDIATE ENERGIES+

AMAND FAESSLER. L. RIKUS and S. B. KHADKIKAR’+

LfniwrsitBt Tilhingen. Institut ,ftir Theoretische Physik, D-7400 Tiihinyrn, W. Grrman>

Received 27 September 1982

Abstract: Starting from the Reid soft-core potential and the collision of two nuclear matters we solve the Bethe-Goldstone equation for the complex reaction matrix. The local density approximation

including a finite range correction enables us to calculate the volume part of the optical potential between two heavy ions up to 100 MeV per nucleon. The surface contributions are calculated from the second-order Feshbach expression including low- and high-lying collective states. The data for r60-“0 arc nicely reproduced including the reduction of the reaction cross section above

20 to 30 MeV per nucleon. The maximum of the reaction cross section at IO-20 MeV per nucleon is essentially due to the surface contributions. The description of the “C-“C data is not so satisfactory due to the assumption of weak coupling and the neglect of some rotational excitations

1. Introduction

The variation of the nuclear reaction cross section a,(E) for heavy ions has

recently come under both theoretical ’ - 3, and experimental scrutiny 4*5).

DeVries and Peng ‘) have suggested that the rapid decrease seen in the

experimental nucleon-nucleon scattering total cross section ggN up to 200 MeV

(lab) should be reflected in heavy-ion reactions provided that bulk effects (e.g.

collective excitations, hydrodynamic effects, etc.) do not play a dominant role.

Using the optical limit of Glauber theory they were able to quantitatively

reproduce the experimental data for light projectiles such as protons, a-particles,

‘He and deuterons in various target nuclei. However, the only heavier projectile for

which data exists over a wide range of energies is “C [refs. h.4)]. Recent results 5,

indicate that for 12C + 12C the reaction cross section oR reaches a peak at about 16

MeV/A (lab).

DeVries and Peng parametrized their results for a target of charge Z, and a

projectile of charge Z, and reduced wavelength R, at centre-of-mass energy E, as

a,(E) = 7L(R,ff+2)2 l- z&G

(R,,,+iZ)E (1-T)’ 1 ’ Work supported by the “Bundesminister fur Forschung und Technologie”.

” Now at Physical Research Laboratory. Ahmedabad 380009. India.

157

158 A. Faessler e’t ul. )I) The hrrrry-ion

where Refr is fitted to low-energy data and T, the transparency, represents the

difference between geometric result and the smaller values required to lit the data

at higher energies. Thus they attributed the decrease in oR at intermediate energies

as a surface transparency effect where the projectile has a longer mean free path at

higher energies as a consequence of the drop in (T&.

Another possible explanation suggested by Nishioka and Johnson 3, is that the

correct treatment of the internal motion of the nucleons in the projectile should

lead to a Relr which decreases at intermediate energies and thereby removes the

need for a large (1 - T) factor in eq. (1).

Brink and Sattiler 3, have also pointed out that the optical limit of Glauber

theory ignores Pauli blocking as well as the Fermi motion of the nucleons and

argued that the real part of the optical potential should have the effect of

increasing oR at low energies. Using a phenomenological optical potential fitted to

“C+ “C data about E,,, = 100 MeV they showed that the real part of the

potential does indeed play an important enhancing role at lower energies

particularly in the region of the peak but concluded that this behaviour is not

sufficient to fully explain the surface transparency effect. In addition they pointed

out that an energy-independent imaginary potential implies a mean free path that

increases with energy. Their results using a constant imaginary potential suggest

that the strength of this potential should increase with energy to some degree to lit

the data at 0.87 GeV (at least for “C + “C).

In this paper we seek the description of the decrease in oK with energy in terms

of a microscopically based complex optical potential. Our potential consists of a

volume part calculated in nuclear matter and surface part which takes into account

surface vibrational excitations. Thus it is a good tool for the investigation of the

role of collective effects in the energy variation of the heavy-ion excitation

function.

2. Basic theory

The rationale behind the nuclear matter approach to the volume part of the optical

potential is reviewed only briefly here since the details can be found

elsewhere ’ “).

Starting from the Reid soft-core potential (including J 5 2) the Bethe-Goldstone

equation,

VQG G=V+-

e ’ (2)

is solved for the Fermi sea corresponding to two infinite Fermi fluids flowing

through each other. This consists of two Fermi spheres in momentum space of

radii k,, and k, 1 with centres separated by K, (in fig. 1). Unlike the standard

A. Faessler et al. / The heavy-ion 159

F; ” F2

Fig. 1. The Fermi sea distribution in momentum space for a typical G-matrix calculation involving

spheres F, and F,. K,, the separation of the two spheres. is identified with the average relative

momentum per nucleon. The radii of the spheres are related to the density of the individual

nuclei at each point in configuration space and p,. pz, h, and h, represent the momenta of an

intermediate Zp-2h excitation which conserves energy and thus contributes to the imaginary part of G.

spherical Fermi sea this geometry allows the conservation of energy in the

intermediate states of eq. (2) causing simple poles and thereby producing a fully

complex G-matrix. There are a number of methods which use the G-matrix to

produce an optical potential. A complex effective nucleon-nucleon potential can be

calculated from G in the colliding nuclear matter system lo) and this, in turn, can

be used in some sort of folding procedure ’ ’ - 13). Alternatively the complex

potential energy density for a given nuclear matter configuration (kFIr kF2;Kr) can

be evaluated as

This can then be combined with a prescription for the kinetic energy density r to

evaluate the binding energy density

x=4 % G,k,lGlklk,)

elk,,, k,, ; K,) = T + n, (4)

a value of which is associated with each point in the r-space of two colliding heavy

ions. The optical potential for a separation d between the ions is then given by

&,,(d) = E(J; kF,, k,,; K,) - E(‘x ; k,,, k,,; K,), (5)

where

The nuclear matter

E(D; kF,, kF2 ; K,) = s

d3re(kF,, k,,; K,). (6)

results can be linked to the finite nucleus system in a variety

160 A. I;crcdrr BI ul. I) The huuq-ion

of ways. All start by associating K,, the separation of the centres of the two Fermi

spheres in nuclear matter, with the initial momentum per particle in the laboratory

system of the heavy ions. Then, given a model description of the two colliding

nuclei in r-space the results of a nuclear matter calculation of a specific

configuration can be associated with each point by the specification of the

remaining two parameters required to fix the geometry.

The simplest way to choose these two parameters at each point r is to directly

relate the two radii likl and /irL to the local densities i,,(r) and kj2(r) of the

individual nuclei.

This frozen density approximation (FDA) ignores the antisymmetrization

between the nucleons in different nuclei in the sense that it assumed that each

nucleon in a given volume element can be identified as belonging to a particular

nucleus. In addition the FDA does not allow any relaxation in the density

functions as the two heavy ions overlap. Both these assumptions are expected to

be valid when the relative velocity between the centres of the two ions is large in

comparison with the Fermi motion of the individual nucleons. i.e. at large incident

energies, or when the two Fermi spheres are small in comparison with their

separation in momentum space (for low ‘incident energies this means low densities).

These restrictions are the same as those required for the double density folding

technique r3.“) but are not expected to be important even at low energies since

the strong absorption present in heavy-ion potentials restricts the reaction to the

surface region, where the densitities are low.

As it is a form of local density approximation the FDA ignores the finite range of

the effective interaction but this can be included in an approximate way by folding

into the so derived optical potential a gaussian with the range of the nucleon-

nucleon interaction l’).

Since momentum conservation forbids Ip-1 h excitations in nuclear matter the

optical potential obtained in this way excludes the effects of Ip-1 h excitations of

either nucleus (with the other in its ground state) as well as particle transfer terms.

The most important of these is expected to be the coherent excitation of a single

nucleus and these can be taken account of in the collective surface vibration

model ‘). Denoting the volume potential obtained from nuclear matter as

Uy = V, +i WV, the total optical potential can be written as,

where

(10)

is the standard surface vibrational model form factor involving the collective

variables c$ and the equivalent sharp surface radius R, of the excited nucleus

(chosen to tit the experimental rms radius).

‘4. Fwsslrr PI 01. / Thhr h~ul~.~-i~n 161

Evaluating the propagator in a local plane wave approximation results in an expression for the surface potential term which is non-local:

wherein ,j,, trl are the regular. irregular spherical Bessel functions. respectively. The form factor is given by

(12)

We made the local wavenumber approximation, i.e.

tr2ki(r) = 2~1(E-h~0,-- V,(r)). 03)

The energies hto, and deformation parameters pi, defined by

/I: = ~l(Ola~ll>12, (14) P

were taken from systematics 15) and are given in table 1 along with the fractional exhaustion vi.) of the classical energy-weighted sum rule.

The potential (11) is localized using a modified Perey-Saxon 16) prescription in which the Fourier transform of the non-local potentiai is expanded about some constant k.’ :

j(k', R) = s U,N(R, ISl)eik’s P(k2, R)

dS = #(I?, R)+ (k’-r?) ‘-k’ , (15) k=K

where S = Y--Y’ and R = $ (r+r’). Ignoring, for the moment, the second term in the expression (15) the equivalent local potential is found to be (after angle averaging etc.)

u?L’o’(r) = - F c B:k, 2 (21+ 1)(21, + 1) I 1. lo

where

V IlolcrJr) = xA(r~.hoW) s

oixn(r’)j,,(kr’hi,(k,r, h(kAr, )rt2dry 0

162 A. Foes&r et al. I The heavy-ion

TABLE I

Table of parameters for the isoscalar surface excitations included in the calculations

Nucleus Excitation i.” Excitation energy

hi>, ( MeV) Fractional sum Square of deforma-

rule exhaustion fj tion parameter /jf

1% 2+ 2+ 3- 3-

3- 4+ 4’

‘T 2+ 7+

3. 3- 4+

6.92 0.06 0.037

23.0 0.94 0.181

6.10 0.04 0.06 1 9.92 0.26 0.262

42.46 0.70 0.157

24.60 0.5 1 0.330

60.32 0.42 0.111

4.44 0.04 0.108

25.33 0.96 0.454

9.64 0.28 0.73 I 46.74 0.72 0.388

66.39 0.49 0.3 19

These parameters are taken from systematics Is) with the inclusion of some

states.

In principle K can be chosen so that the second term

disappears, i.e.

h2k2(r) = 2M(E - U,(r) - r/FL(r)),

low-lying experimental

in the expansion

(17)

but this defines a self-consistency problem. Since all the steps in the derivation of

eq. (16) are valid for complex K (and continuous potentials), and to avoid the

recursive calculation implied by eq. ( 17). the following choice was made:

f?K2(r) = 2M(E- U,(r)). (18)

With this choice the second term in the Fourier transform (15) expansion produces

an effective-mass-type correction term to give

U,EL(r) = 1+ U~L’u’(r). (19)

Subsequent calculation showed that this additional correction was virtually

negligible in the all important surface region of the potential and hence made no

substantial difference to the calculated cross sections. In the interior, however, the

effective-mass term becomes large indicating that the evansion (15) and, hence,

the localization procedure, breaks down there.

In the case of intrinsically deformed nuclei the formalism described above has to

be modified. In addtion to surface vibrations we also need to account for

A. Faessler et al. / The heay-ion 163

rotational excitation of either or both nuclei. Using the standard definitions for

rotational states 25) the form factor for mutual rotational excitation can be

evaluated as

xi$&) = .f;,,,(r)y,T~,(P)YI,M,(P)~ (20)

where the radial form factor is defined in terms of integral over Legendre

polynomials and the volume potential in the intrinsic frame:

h,,,(r) = 7~ s sinBdOP,,(cos 0) s

sin ~d~P,z(cos$)V~“‘(r, 0, 4, fl”‘. /Y2’). (21)

in which PC’) and fi(” are symbolic parameters describing the intrinsic deformation

of nucleus 1 and nucleus 2 respectively. The angles B and C$ are defined as the

angles of the symmetry axes relative to the vector Y connecting the two heavy ions.

For the rotational excitation of a single nucleus (assuming the other remains in its

I, = 0 ground state) the form factor is easily found from the eq. (20) and (21) to

be 24)

with

X;,(r) = f;(~)CW)~ (22)

f,(r) = 6 s

sin BdBP,(cos O)V,,!“‘(r, 8, /I”‘), (23)

where Vi”’ is now angle-averaged over the deformation of the second nucleus. The

form factor for a surface vibration of multipole (i, p) built on the ground state is

given by 24)

x;“(r)= -R,(y~w+l)T(-)‘(; :, :,)c “,, ;)[gm] Y;(P).

(24)

Substituting all these form factors back into eq. (9) and summing over intrinsic

projection quantum number yields expressions which can be treated in almost

exactly the same fashion as those found in the undeformed surface vibration case

leading to eq. (16), provided that a reasonable prescription for I/vi”‘(r, 9, 4, p(l),

bC2’) can be found. Of course it is computationally impossible to fully account for

all rotational excitations of the two heavy ions and this should be reflected in

the prediction for the optical potential, particularly in the lower energy region

where these are especially important.

To evaluate the intrinsic potential I/;l”‘(r, 8, 4, /I(‘), /Y2’I) one should in principle

use deformed density distributions and do a series of FDA calculations on a grid

of 8, 4 values but such a procedure would be relatively expensive in terms of

164 ‘4. Faessler et al. / The hrat?y-ion

computing time. Thus the standard surface expansion approximation was used. i.e.

(the second-order term shown only contributes to the mutual rotational excitation

form factor), where V,(V) was identified as the volume potential calculated from

nuclear matter by the FDA.

The standard technique to account for strong deformation in the volume

potential is to use the strong coupling approximation (SCA) 26) in a coupled

channel calculation. Since we were mainly interested in the role of the optical

potential at fairly high energies the volume potential was assumed to be angle-

averaged and the coupling to low-energy rotational states approximated as

additional terms in the surface potential. This procedure is computationally

simpler than the SCA and is a good approximation at higher energies 26).

3. Discussion and results

The basic empirical inputs for the FDA are the matter density distributions of

the two nuclei involved. For protons these can be obtained by unfolding the finite

charge distribution of the proton from the charge density distribution of the

nucleus deduced from electron scattering. In the case of light nuclei with N = Z

one can then assume that the neutron density function is basically the same as the

proton point density. In this work we took the density functions for neutrons and

protons as those found in fits to intermediate-energy proton scattering from 160

[ref. 18)]. For 12C the situation is complicated by the fact that the ground state is

strongly deformed. Projected Hartree-Fock 19) calculations suggest in the intrinsic

system a radial deformation of the form

R = R,(l +P2YzoW)+P~Y4o(Q))r (26)

with B2 = -0.42 and /I4 = +0.12 in agreement with coupled channels fits to

(e, e’), (p, p’) and (a, ~1’) reactions. Unfortunately the PHF calculations also predict

an rms radius that is far too big. On the other hand, the spherical density

functions deduced from electron and intermediate proton scattering 18) will

compensate for the deformation in also predicting too large an rms radius. Thus,

we took the spherical density functions of ref. 18) but reduced the radius parameter

to reproduce the rms radius expected for an undeformed sphere which would give

the rms radius deduced from electron scattering when deformed as in eq. (26).

A. Faessier et al. / The heacy-ion 165

The real and imaginary potentials calculated in the FDA for I60 (with a

gaussian of range 1 fm folded in to correct for finite-range effects) are shown in

figs. 2 and 3. The real volume potential shows the effects of the weakening of the

NN interaction with increasing energy in that the potential becomes less attractive

(in fact repulsive in the surface region) at the highest energy considered here

(EC,,, = 93.4 MeVIA). This repulsion in the surface will tend to decrease the

reaction cross section by “pushing” some incident trajectories away from the

strong absorption radius. However, the effect of the real potential seems to

become less important at higher energies. [See for example fig. 1 of ref. “)I.

The imaginary volume potential shows an almost uniform increase with energy,

reflecting the weakening of the Pauli principle as the two Fermi spheres move

further apart in momentum space. The important feature which is not obvious

from the figure is that at the surface radius (z 7.8 fm) the potential increases with

energy at a rate slightly less than E*, the energy variation which should remove the

surface transparency 3).

Figs. 4 and 5 show the total potential (volume plus surface term). The real

Separation II Lfml

Fig. 2. The real part of the volume contribution to the optical potential for lb0 + “0, calculated in

the FDA and including the folding in of a gaussian form factor. The curves are for centre-of-mass energy. (a) 2.6 MeVIA, (b) 10.4 MeVIA, (c) 23.3 MeVIA, (d) 41.5 MeVIA, (e) 93.4 MeVIA.

166 A. Faesskr et al. i The heag+m

t I I I I I I I / i

0 2 4 6 a

Separation D ifml

Fig. 3. The imaginary volume potential for I60 + IhO. See lig. 2 caption for details

potential is not changed substantially but the imaginary potentials are drastically

modified. In particular the dominating role of the surface term at low energies can

be seen and the steady increase with energy in the surface region is no longer

present. The overall effect of the surface terms can be gauged by the comparison

(see fig. 6) of the reaction excitation function calculations for the total potential

and for the volume potential only. [All reaction calculations were carried out

using a modified form of MAGALI 21) and the higher energy results were later

checked against calculations with ATHREE “).I From this it can be seen that the

collective surface effects are crucial in explaining the drop off of crR with energy.

The contribution of the surface terms seems to reach a peak at about E,.,,/A = 10

MeV. From E,,,,/A = 20 MeV up the cross section due to the surface terms gets

smaller. If the real volume potential remained constant with energy the surface

part would produce a relatively constant correction to the total potential resulting

in a contribution to the reaction cross section which decreases as E-+. The

flatterning out of the real potential around E,,,,/A = 100 MeV in fact decreases

the surface correction term faster than this since the form factor XL(r) of eq. (12)

becomes small.

A. Faessler et al. 1 The heavy-ion 167

1

Fig. 6. The energy variation of total reaction cross section for ‘“O+ “‘0. The crorses represent the results of using just the volume potentink. while the calculations using the full potentials are shown as rectangles. The lines are drawn merely to indicate the trends in the results. The error

bar at E, m. ‘A = 2.5 MeV is an experinlcnt~~l vaiue “‘).

It should be remembered that the surface terms only add coherent 1 p-lh

excitations of the two-ion system to the volume term. The volume potentials

contain all possible 2p-2h excitations of the system as a whole (i.e. 2p-2h excitation

of one nucleus whilst the other remains in its ground state, mutual Ip-lh

excitations, two-particle transfer terms etc). Thus, at most, 2p-2h excitations of

either nucleus are taken into account and so the optical potential derived here

describes only very simple excitation processes of the type encountered in

peripheral interaction where the nuclei do not overlap appreciably. Vary and Dover ’ ‘) estimate that under these circumstances the optical potential cannot be expected

to give physically meaningful results when the elastic cross section G drops below

about 0.01 cMo,,. the corresponding Coulomb scattering cross section.

Fig. 7 shows the differential cross sections for “O+ “0 at Ellh = 80 MeV

calculated with the optical potentials for E,,,,/A = 2.6 MeV and compared with

experiment 20). The dashed curve shows a calculation using just the volume

potential and the full curve is the result for the full potential. Obviously a better fit

could have been obtained by varying the effective range correction and the

strengths of the surface excitations but as the potentials include only peripheral

processes such a variational fit may not be physical. Nevertheless, considering the

number of approximations made in the theory to facilitate the calculation the fit to

the data is quite impressive. Figs. 8 to 11 show angular distributions for 160-“‘0

scattering.

For ‘“C results are not expected to be so good. The problem is that “C has a

169

.001 1 I I I 1 I I

0 5 10 15 20 25 30 35 4

CENTRE OF MASS RNGLE (DEGREES)

Fig. 7. Calculations of differential cross sections divided by Mott scattering compared with experimentzO) (triangles), The dashed curve is the result for just the volume optical potential while

the full curve is the result using the full optical potential.

strongly deformed ground state. As discussed above the volume optical potential

does not contain all excitations, and, in particular, it does not contain rotational

excitations. Tanimura 13) has shown. within the scope of a pure rotational model,

that induced rotation of either or both “C nuclei plays an extremely important

role in grazing collisions, at least up to centre-of-mass energies of 40 MeV

(i.e. E,,,./A 2 3 MeV). In fact experimental results6) show that the single and

CENTE OF hviss ANGLE (DEGREES)

Fig. 8. Comparison of differential cross section divided by Mott scattering for the volume potential (upper curve) and the total potential (lower curve) for r60+ I60 at E,,,,/A = 10.4 MeV.

170

El A= 23.3MeV

0 5 10 15 20 25 30 35

CENTRE OF MASS ANGLE (DEGREES)

Fig. 9. Same as fig. 8 but for EC.,,/.4 = 23.3 MeV.

mutual excitation channels to the first 2’(4.44 MeV) excited state are still as important as the elastic channel at energies higher than this. (The data extends to

E c.m. = 63 MeV, i.e. E ,,,,/A z 5 MeV.) The data at intermediate energy4)

(LX = 516 MeV), however, indicate that the mutual excitation channel can be ignored and that the single excitation channel is no longer so important. Thus we expect the weak coupling formalism used to calculate the reaction cross section to break down at lower energies but should do reasonably well at the higher energies.

CENTRE OF MASS ANGLE (DEGREES f

Fig. 10. As for tig. 8 but for energy E,,,./A =. 41.5 MeV.

A. Faessler et al. / The heavy-ion 171

10

160+‘60 E/A=93.4 MeV c

CENTRE OF MASS ANGLE (DEGREES)

Fig. 1 I. As for fig. 8 but for energy E,,,,/A = 93.4 MeV.

Fig. 12 shows the differential cross section divided by Mott scattering for EC,,, =

508 MeV (i.e. E,,,,/A 5 42 MeV) calculated with the volume potential only (with

finite range correction folded in the full curve) in comparison with the experimental

results. The fit is excellent but as expected indicates that slightly more absorption is

required. However, when the surface terms are added (using the approximations

described above to include pure vibrational states as well as single and mutual

I0

1 ” ’ ” ” ’ ” ’ ’ ’ ! 12Ct12C El ab=l@lE;MeV

.001 I I I I I I I I I 1 1 1 I

I 2 3 4 5 6 7 8 9 I0 II 12 I3 14 15

CENTRE OF MASS FiNGLE (DEGREES)

Fig. 12. The differential cross section divided by Mott scattering for 12C+‘ZC at E,,, = 1016 MeV calculated with the volume optical potential (full line) and the full optical potential (dashed iine)

and compared with experiment 4, (triangles).

172

0.00

Fig. 13. The calculations of the reaction cross section for “C+ “C for only the volume potential

(crosses) and the total potential (rectangles) compared with various experimental parts”) (with error bars). The full lines are drawn merely to show the systematic trend of the calculations.

The dashed curve is the rough position of the calculation of DeVries and Peng ’ ).

rotational excitations of intrinsically deformed nuclei) the potential becomes too strongly absorbing as can be seen from the dashed curve in fig. 12. This is again not too surprising as the deformation of the matter density distribution of the ground state of “C has been totally ignored, in the calculation of the volume potential. In addition the assumption of the systematic strengths for the giant resonances in ‘*C is at best a poor approximation.

However, keeping these crude approximations we can still expect to be able to get the trend of the energy dependence of the reaction cross section (at least at higher energies where the weak coupling approximation is valid). This is compared with experiment and the rough trend of the calculation of DeVries and Peng in fig. 13. Again it can be seen how important the collective surface excitations seem to be in producing the drop in the excitation function. The poor descriptiotl at Iow energies reflects the need to include all rotational excitations as well as indicating that the weak coupling approximation breaks down.

4. Summary

The optical potentials with both real and imaginary parts have been calculated for the ‘60-‘60 and *‘C-r2C scattering starting from a realistic interaction, the Reid soft-core potential. The first step is the solution of the Bethe-Goldstone equation in two nuclear Fermi liquids which are flowing through each other with a

A. Faessler et al. : The heaqGon 173

velocity defined by the bombarding energy. Since the corresponding Fermi sea is

not spherical the propagator in the Bethe-Goldstone equation has poles and

produces a complex reaction matrix. This allows the calculation of a complex

energy density. With the help of a local density approximation and a finite range

correction this yields the volume part of the complex optical potential between two

heavy ions. The surface contribution due to the excitation of a collective state in

one of the two heavy ions is calculated using phenomenological surface vibrational

models. The angular distribution and the total reaction cross section for 160-‘60

is nicely reproduced as far as data are available. The maximum of the reaction

cross section found in this calculation between 10 and 30 MeV per nucleon is

essentially due to the surface contribution. Above 100 MeV per nucleon the

surface part practically does not contribute to the total reaction cross section. The

increasing transparency at higher energy is due to a drastic decrease of the real

part of the volume potential. At the surface it even becomes slightly repulsive and

diverts the relative wave function away from the imaginary part which is

increasing with energy.

The ‘2C-‘2C scattering is qualitatively reproduced but the agreement is not as

satisfactory as for r60-‘60. This may be due to the intrinsic deformation of ‘“C

which allows strong coupling to rotational excitations which should be treated in a

coupled channels calculation at least at lower projectile energies.

We wish to thank S. Krewald for supplying us with a preliminary version of the

FDA computer program used in these calculations. Two of us (L.R. and S.B.K.)

thank Prof. Amand Faessler for his kind hospitality during our time at Tubingen.

References

I) R. M. DeVries and J. C. Peng. Phys. Rev. C22 (1980) 1055 2) H. Nishioka and R. C. Johnson, Phys. Rev. C22 (1981) 2457

3) D. M. Brink and G. R. Satchler, J. of Phys. C7 (1981) 43

4) M. Buenerd,J. Pinston,J. Cole, C. Guet. D. Lebrun, J. M. Loiseaux. P. Martin, E. Monnand, J. Mougey, H. Niefenecker, R. Ost, P. Perrin,Ch. Ristori. P. desaintignon. F. Schussler, L. Carlen, H. A. Gustafsson, B.Jakobss0n.T. Johannson.G.Jiinsson,J.Krumlinde,J.Otterlund.H. Ryde.B.Schroder.G.Tibell,J, B. Bondorf and 0. B. Nielsen, Phys. Lett. 102B (1981) 242);

M. Buenerd, private communication 5) A. J. Cole, W. D. M. Rae, M. E. Brandon, A. Dacal, B. G. Harvey, R. Legrain. M. J. Murphy and R. G.

Stokstad, Phys. Rev. Lett. 47 (1981) 1705

6) R. M. Wieland, R. G. Stokstad, G. R. Satchler and L. D. Rickerton, Phys. Rev. Lett. 22 (1976) 1458; R. G. Stokstad, R. M. Wieland,G. R. Satch1er.C. B. Fulmer, D. C. Hensley, S. Raman, L. D. Rickerton. A. H. Snell and P. H. Stelson, Phys. Rev. C20 (1979) 655

7) T. Izumoto, S. Krewald and A. Faessler, Nucl. Phys. A341 (1980) 319; A357 (1981) 471 8) S. B. Khadkikar, L. Rikus, A. Faessler and R. Sartor. Nucl. Phys. A369 (1981) 495 9) F. Beck, K. H. Mtiller and H. S. Kohler, Phys. Rev. Lett. 40 (1978) 837

10) A. Faessler, T. Izumoto, S. Krewald and R. Sartor, Nucl. Phys. A359 (1981) 509 11) R. Sartor, A. Faessler, S. B. Khadkikar and S. Krewald. Nucl. Phys. A359 (1981) 467 12) R. Sartor and A. Faessler, Nucl. Phys. A376 (1982) 263;

A. Faessler, L. Rikus and R. Sartor Comp. Phys. Comm., to be published

13) G. R. Satchler and W. C. Love. Phys, Rep. 55 (1979) 183 14) J. P. Jeukenne. A. Lejeune and C. Mahaux. Phys. Rev. Cl6 (1977) 80 IS) C. C. Chang. in Topics in nuclear physics II, ed. T. T. S. Kuo and S. S. M. Wong, (Springer, Berlin, 1981)

16) F. C. Perey and D. S. Saxon, Phys. Lett. 10 ( 1964) 107 17) C. B. Vary and J. P. Dover, in Lecture Notes in Physics 33 (Springer. Berlin, 1975) 18) L.Ray.G. W.Hoffmann,G.S.Blankpied, W.R.CokerandR.P.Liljestrand.Phys. Rev.ClS(1978) 1756;

L. Ray, Phys. Rev. C20 (1979) 1857 19) Y. Abgrall. B. Morand and E. Carier, Nucl. Phys. A192 (1972) 372 20) 8. Fernandez, C. Gaarde, J. S. Larsen, S. Pontoppidan and F. Videbaek, Nucl. Phys. A306 ( 1978) 259 21) J. Raynal. in Computing as a language of physics (IAEA, Vienna 1972)

22) E. II. Auerhach, Comp. Phys. Comm. 15 (1978) 165 23) 0. Tanimura, Nucl<Phys. A309 (1978) 133

24) R. A. Broglia, C. H. Dasso, G. Pollarolo and A. Winter, Phys. Reports 48 (1978) 35 1; R. A. Broglia and A. Winter. Heavy ion reactions, (Benjamin. Reading, Massachusetts, 1981) vol. 1

25) D. M. Brink and G. R. Satchler. Angular momentum (Oxford University Press. Cambridge. 1968) 26) D. Jackson. Nuclear reactions (Chapman and Hall, London. 1975)