The heavy-ion potential and its increasing transparency at intermediate energies

Nuclear 0

Physics

A401 (1983) 157-174

North-Holland

Publishing

THE HEAVY-ION

POTENTIAL AND ITS INCREASING AT INTERMEDIATE ENERGIES+

AMAND LfniwrsitBt

Company

FAESSLER.

Tilhingen. Institut

L. RIKUS

TRANSPARENCY

and S. B. KHADKIKAR’+

,ftir Theoretische Physik, D-7400 Tiihinyrn, Received

27 September

W. Grrman>

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 a,(E) =

’ Work supported ” Now at Physical

wavelength

7L(R,ff+2)2 l-

by the “Bundesminister Research Laboratory.

R, at centre-of-mass

z&G (R,,,+iZ)E

1

(1-T)’

fur Forschung und Technologie”. Ahmedabad 380009. India. 157

energy

E, as

158

where

A. Faessler e’t ul. )I)The hrrrry-ion

Refr is fitted

difference between at higher energies. as a surface

to low-energy

data

and

T, the transparency,

represents

the

geometric result and the smaller values required to lit the data Thus they attributed the decrease in oR at intermediate energies

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, G=V+-

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

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 of the two spheres. is identified with the average relative spheres F, and F,. K,, the separation momentum per nucleon. The radii of the spheres are related to the density of the individual the momenta of an nuclei at each point in configuration space and p,. pz, h, and h, represent 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 x=4

%

G,k,lGlklk,)

This can then be combined

with a prescription

evaluate

density

the binding

energy

elk,,,

for the kinetic

energy

k,, ; K,) = T + n,

density

r to

(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

E(D; kF,,kF2; K,) = The nuclear

matter

results can be linked

s

d3re(kF,, k,,;

K,).

to the finite nucleus

(6) system in a variety

160

A. I;crcdrrBI ul.I)The huuq-ion

of ways. All start by associating K,, the separation of the centres spheres in nuclear matter, with the initial momentum per particle system

of the heavy

ions.

Then,

given

a model

description

of the two Fermi in the laboratory

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 even at low energies since technique r3.“) but are not expected to be important 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 nucleonnucleon 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 expression for the surface potential term which is non-local:

results in an

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)). The energies hto, and deformation

03)

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 dS = #(I?,

R)+ (k’-r?)

P(k2,R)

‘-k’

, k=K

(15)

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 VIlolcrJr) = xA(r~.hoW)

oixn(r’)j,,(kr’hi,(k,r, s0

h(kAr,

)rt2dry

A. Foes&r

162

et al. I The heavy-ion TABLE I

Table of parameters

Nucleus

Excitation

1%

for the isoscalar

i.”

principle disappears, i.e.

K can

In

in the calculations

6.92 23.0 6.10 9.92 42.46 24.60 60.32 4.44 25.33 9.64 46.74 66.39

0.06 0.94 0.04 0.26 0.70 0.5 1 0.42 0.04 0.96 0.28 0.72 0.49

3. 34+ are taken

included

Fractional sum rule exhaustion fj

2+ 3334+ 4’ 2+ 7+

These parameters states.

excitations

Excitation energy hi>, ( MeV)

2+

‘T

surface

from systematics

be chosen

the

0.037 0.181 0.06 1 0.262 0.157 0.330 0.111 0.108 0.454 0.73 I 0.388 0.3 19

of some low-lying

Is) with the inclusion

so that

Square of deformation parameter /jf

second

term

in

the

experimental

expansion

h2k2(r) = 2M(E - U,(r) - r/FL(r)), but this defines a self-consistency eq. (16) are valid for complex recursive

calculation

implied

K

(17)

problem. Since all the steps in the derivation (and continuous potentials), and to avoid

by eq. ( 17). the following f?K2(r)

=

2M(E-

With this choice the second term in the Fourier an effective-mass-type correction term to give

U,EL(r) = 1+

of the

choice was made:

(18)

U,(r)). transform

(15) expansion

U~L’u’(r).

produces

(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.

rotational

excitation

rotational evaluated

states 25) the as

of either

or both

form

factor

/ The heay-ion

nuclei. for

Using

mutual

163

the standard rotational

definitions

excitation

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

the

polynomials

radial

form

factor

and the volume

h,,,(r) =

is defined

potential

s

7~ sinBdOP,,(cos

in terms

in the intrinsic

0)

of integral

for

can

be

(20) over

Legendre

frame:

sin ~d~P,z(cos$)V~“‘(r,

0, 4, fl”‘. /Y2’).

(21)

s

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

(22)

with f,(r)

sin BdBP,(cos O)V,,!“‘(r, 8, /I”‘),

= 6 s

(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

computing

‘4. Faessler et al. / The hrat?y-ion

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 angleaveraged 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 NN interaction with increasing

potential shows the effects of the weakening of the 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.

A. Faesskr

166

t 0

et al.

i

i I

I 2

I

I 4

Separation

Fig. 3. The imaginary

volume

The heag+m

potential

I

I 6

D

I

/

a

ifml

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 1p-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

0

1

I

I

5

10

15

CENTRE

OF

I

1

I

I

20

25

30

35

MASS

RNGLE

4

(DEGREES)

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

strongly

deformed

ground

state. As discussed

above

the volume

optical

with while

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, “C nuclei plays an extremely important that induced rotation of either or both at least up to centre-of-mass energies of 40 MeV role in grazing collisions, results6) show that the single and (i.e. E,,,./A 2 3 MeV). In fact experimental

CENTE

OF hviss

ANGLE (DEGREES)

Fig. 8. Comparison of differential cross section divided by Mott potential (upper curve) and the total potential (lower curve) for r60+

scattering for the volume 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.

171

/ The heavy-ion

10 160+‘60

c

E/A=93.4

MeV

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

for EC,,, = 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”

.001 I

’ 12Ct12C ” ”

’ ’ ’ !

’ ”

El ab=l@lE;MeV

I

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

I0

CENTRE

OF

MASS

FiNGLE

1 II

1

1

12

I3

I 14

15

(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.

velocity

defined

by the bombarding

not spherical the propagator produces a complex reaction energy

density.

173

: The heaqGon

energy.

Since the corresponding

in the Bethe-Goldstone equation matrix. This allows the calculation

With the help of a local density

approximation

Fermi

sea is

has poles and of a complex

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) 2) 3) 4)

5) 6)

7) 8) 9) 10) 11) 12)

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