Unified description of statistical excitations, deformations and charge transfer in a dynamical theory of deep-inelastic heavy-ion collisions

Nuclear 0

Physics

North-Holland

A406 (1983) 557-573 Publishing

Company

UNIFIED DESCRIPTION OF STATISTICAL EXCITATIONS, DEFORMATIONS AND CHARGE TRANSFER IN A DYNAMICAL THEORY OF DEEP-INELASTIC HEAVY-ION COLLISIONS

P. FROBRICH’ Hahn-Mritner-Institut

and B. STRACK”

.ftir Kertzforschung, Berlin West, Germane

M. DURAND Institut des Sciences NucEaires, 53, Avenue des Martyrs, 38026 Grenoble-Cedex, France Received

5 January

1983

.\hstract: In order to describe dissipative heavy-ion collisions a multi-dimensional Fokker-Planck equation including relative motion, quadrupole deformations and charge transfer is solved within the quasilinear approximation (moment expansion up to second order). We aim at a unified description of Coulomb-like. focusing and orbiting systems by using a universal parametrization. Calculations for 136Xe+209Bi at E,,, = 940. II30 and 1422 MeV are compared with the experimental data.

1. Introduction The increasing amount of experimental data on deep-inelastic heavy-ion collisions calls for a systematic unified theoretical description. In order to explain the measured quantities in a satisfactory way it is desirable to study the dynamical coupling of all relevant degrees of freedom. If one wants to know the fluctuations around the -mean values which are usually described by classical trajectory calculations, one must apply transport theories to deep-inelastic collisions. [For a review see, ref. ‘).I The present work is concerned with the solution of a Fokker-Planck equation which couples relative motion, deformations and charge transfer. This equation can be obtained from a generalization ‘) of the brownian motion theory of ref. 3). It has the same structure as that derived from linear response theory 4, or in ref. ‘). Particular emphasis is placed on the treatment of dynamical deformations; these

”

’ Also at Fachbereich Physik (WE3). Freie Universitst. Present address: GSI, Darmstadt. 557

Berlin

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P. Frijhrich

et al. i Un(fied

description

have been previously treated by several authorsh-‘2), but except for the last two references no statistical fluctuations with respect to deformations have been taken into account in calculating cross sections. We aim also at a universal description of the Coulomb-like, focusing and orbiting behaviour of different systems, or of the same system at different bombarding energies. A good example for the latter case, which also is one of the experimentally most carefully analyzed systems, is the 13’Xe + “‘Bi reaction studied at Elab = 940 MeV [ref. “)I (Coulomb-like behaviour), 1130 MeV [ref. 14)] (focusing) and 1422 MeV [ref. “)I (beginning of orbiting). Therefore this system is studied in detail as representative example. The paper is organized as follows. In sect. 2 the multi-dimensional FokkerPlanck equation is established, and the quantities that enter in this equation are explained in detail. In sect. 3 the quasilinear approximation (moment expansion up to second order) to this equation is discussed, and the validity of this method for the Xe+Bi system is demonstrated. A detailed theoretical analysis of the experimental data for this system is performed in sect. 4. Finally, in sect. 5 some shortcomings of the present model are discussed and suggestions for further improvements are made.

2. The multi-dimensional

Fokker-Planck

equation

The Fokker-Planck equation which is studied in the present paper is derived 2, from a generalization of the brownian-motion theory of ref. 3, by including, besides the relative motion, shape deformations and the projectile charge as dynamical variables. A markovian approximation is made and an evaluation of the Liouville propagator by expansions in (m//l)* and (m/(DiR~))~ is performed (m = nucleon mass, p = reduced mass. Di = liquid drop mass parameter of the vibration i, Ri = nuclear radius), leading to microscopic expressions for the transport coefficients which are given by the corresponding force-force correlation functions. The resulting equation reads

(1)

559

P. Frfihrich et al. / Unified description

The distribution

function

d(R, P,, cp, I, xi, 7ti, Z,, t) depends

on the radial

distance

R, the radial momentum P,, the angular momentum I and its conjugate variable cp. the amplitudes cli and the momenta 7ci of the vibrational states and the charge of the projectile Z,. In the applications the index i runs only over the low-lying quadrupole states of target and projectile, and the dynamics is restricted to a noseto-nose geometry of the collision partners. The conservative potential V(R. z) consists of monopole-monopole and monopole-quadrupole Coulomb interactions, a centrifugal term, liquid drop harmonic potentials for the vibrators and a nuclear part which is obtained from the spherical folding potential of Gross and Kalinowski 16) modified to include curvature corrections as e.g. in ref. 9). The potential obtained in this way agrees very well I’) with that resulting from a folding procedure including only one deformation variable lo); however, it has the advantage of allowing for different deformations in target and projectile. No attempt is made to calculate the microscopic transport coefficients. Instead phenomenological values are used. In order to determine, besides the diagonal transport coefficients, also the nondiagonal frictional coefficients coupling relative motion and the vibrations and among the vibrations, we use the fact that with steep form factors as in refs. “.l’) friction is essentially a surface phenomenon. Then the Rayleigh dissipation function can be assumed to be proportional to the square of the time derivative of the distance d, between the surfaces of the colliding nuclei lo) : W = +K&

Here K, is the radial friction For nose-to-nose geometry

coefficient. and quadrupole

ri- R,;r,Y,,

ds =

(2)

deformations

one has

- R,Cr,Y,,,

(3)

where Ricn is the radius of the projectile (target), and Yzo a spherical harmonic. The Rayleigh dissipation function which leads to the same equations of motion as those derived from the Fokker-Planck equation (1) by taking its first moments, has the form 8’ = $K,d2+$~

K,,,hidrj+ i,j

c K,,,l&. I

Comparing eq. (2) with eq. (4) one can read off the friction with the deformation modes: K Ra,

=

coefficients

connected

-RiY2oKR,

K =,a,= RiRjY,6K,+6ijKzaJCiDi, where dij is the Kronecker

(4)

symbol.

(5)

560

P. Friihrich

In the applications deformed

nuclear

et al.

/

UtGfied description

we have used a radial friction

potential

according

form factor calculated

from the

to

where the strength Kg is reduced by about 10 ‘>,,compared to that of ref. lb). Such a renormali~dtion of the strength is pl~ysically reasonable because in the present calculation we take explicit account of the deformation degrees of freedom. which is not the case in ref. 16), We do not change the tangential friction strength of ref. I’):

In the diagonal part of K,<,, it is necessary to add a term KY=a[MeV set] (Ci, Di = liquid drop stiffness and mass parameters of the vibrations), which describes the intrinsic damping of the vibrators [cf. ref. “)I. For the strength we use Kg = 20, which leads to reasonable Wilsczynski plots (see figs. 2 and 3) and to realistic deformations of the collision partners. In all variables generalized Einstein relations have been used, and the temperature T = (gE*/A)’ (I?* = energy loss, A = mass of the compound system) is calculated along each trajectory. The charge transfer is treated as an overdamped motion. theoretical diffusion coefficients of Ayik H crl. ‘*) are used and correlated proton and neutron transfer is The drift term is calculated from assumed for the Xei- Bi reaction. where a local harmonic approximation to the driving l’z = -(D,/T)i;U/~Z,, potential U = (c/Z’)(Z,Z,)’ is made (2 = Z,+Zr, the driving potential). After having specified the input of the Fokker-Planck equation.

Zs = *Z, c is the stiffness of data we turn to the solution

3. The quasilinear approximation to the Fokker-Planck equation A full numerical solution of the multi-dimensional Fokker-Planck equation (1) is not feasible. Therefore we use the quasilinear approximation which corresponds to a moment expansion up to second order. We have performed calculations 17) where the variable Z, is taken into account in the reduced masses, the Coulomb and nuclear potentials, and find that this dynamical coupling of the charge transfer to the relative motion and also the correlations of the charge transfer to the variables of relative motion have only a small effect. This is in agreement with the findings of ref. 19) and justifies the approximation made in ref. “) where this coupling is neglected, as we shall also do in the following.

P. Friihrich

et al. / Unified

description

561

The cross section d3a/dQdEdZ, is then obtained by a factorization of the asymptotic distribution function in two gaussian distributions and by integrating over the initial impact parameters. For the calculations the following expression is used :

exp

x

c

(Q-
2(8-(0))(P-(P))o,Z,+(P-(P))2a,2,’,,

- ~~________

1.

4(%%& - (c&J)‘)

(8)

The double and single differential cross sections are obtained by integrating over the appropriate variables. The quantities in brackets ( ) are the mean values. and the 0: are the half-variances. The equations for the mean values are obtained by taking the first moments of eq. (1). These are the classical equations of motion and have the following form:

p

R

=

_dWA

-

dR

KRPRIP-

(I, = sticking

angular

momentum).

rci,lD. I' d V(R, a)

fii = - ___ dcci i,

KRz,niIDi, I

i = - Cl- l,)KJp ~2~=

1

= - +

C K,,,,njlDj - K+PRIPL, j

& ’

-

U(Z,). P

The summations run over projectile and target variables. From can read off the role played by the non-diagonal friction

these equations one coupling terms. In

particular, the term K,,,P,/p changes sign if the trajectory runs over the turning point, thus helping the collision partners to become oblate in the entrance channel, and prolate in the exit channel. As already discussed, the coupling of the charge transfer to the relative motion is neglected in the calculations of the variances, and because of the nose-to-nose geometry we do not take into account the coupling of 1 and cp to the vibrational degrees of freedom in the variances. The equations for the half-variances which have been used in the actual calculations are

(10)

P. Friihrich

562

et al. I Unified

description

with

Dij =

ti,.j = Pa, I, rr,, rcr ; all other Dij = 0), /

1

-_

0

0

P VRR

fij =

K apR

KK

DP

P

0

0

0

0

0

0

0

0

-

1 DP

f

K*R

0

fK,R

\

K apdp DP

K

a,R

P 0

0

K ITR

K 2 DP

P

Here expressions T/ii denote derivatives variables i, j. The equation for the charge do2 -z= dt

of the variance

4D,C - -0;+2D,. T

potential is

with

respect

to

the

(12)

Recently the validity of the moment expansion has been shown to break down I’) for orbiting systems at small impact parameters. This is due to the negative curvature of the potential around the scission point and is indicated in particular by the behaviour of the variances o&., G& and criR which tend to increase very rapidly with time in such a situation. It has been also demonstrated in ref. 21) by a comparison of an exact numerical solution of a two-dimensional

563

P. Friibrich et al. / lJn$ied description

Fokker-Planck

equation

with the results

of a moment

expansion,

that the latter

is

not applicable for trajectories close to fusion. The moment expansion is, however, reliable for trajectories which are far away from the top of the barrier. The Xe+Bi reaction, which is studied in detail in the present paper, does not show negative-angle scattering and does not run into fusion. Therefore a moment approach is expected to be valid. This is shown in fig. 1 where the ratios of the crucial variances & and C& to typical mean values (Pr)* and (R)* are plotted as functions of time. (We use (Pr)’ = (E,)2p as a typical momentum because the mean value of the radial momentum is zero at the turning point; (E,) is the energy of the relative motion along the trajectory.) Although these variances rise very quickly during the collision process (eventually they saturate), the ratios &,/(Pr)* and a&/(R)* remain bounded. They reach at most a value of about one for the deepest penetrating trajectory (/ = 0, final energy loss AE = 207 MeV), which practically does not contribute to the cross section, and stay below 0.1 for I = 100, AE = 180 MeV and below 0.001 for a grazing trajectory with I = 300, AE = 56 MeV. In the upper part of fig. 1 also the distance between the centers of mass (R) as well as the surface distance (0) (defined as the distance between the half-density radii) for these trajectories are shown. From the differences between these curves one can read off the amount of deformation during the reaction.

4. Analysis

of experimental

data: the 13’jXe + “‘Bi 1422 MeV

reaction

at Elab = 940,

1130,

In this section we present a detailed analysis of the 136Xe+ *“Bi reaction at Elab = 940, 1130, 1422 MeV, which is chosen as a representative example. The experimental data with which we confront the theoretical results are published in refs. l 3_ l 5). In Jig. 2 the theoretical mean scattering angles as a function of energy loss are entered in the experimental Wilczynski plots. A substantial improvement over calculations with spherical nuclei is found, in particular for the 1422 MeV case. For comparison, a calculation within the proximity model including a neck i5) is also shown in fig. 2. The universal parametrization discussed in sect. 2 also gives good results for systems with completely different angle behaviour, e.g. for the orbiting Kr + La reaction **), as shown in fig. 3. For the latter system successful trajectory calculations have also been reported by Blocki et al. ‘). We would like to mention that the present dynamical calculations lead also to very good capture cross sections 23) for heavy systems such as Pb+ Mg, Pb+Fe and U+Fe, without the necessity of introducing an “extra push”. The total deep-inelastic cross sections are in good agreement with experiment as shown in table 1.

P. Frijbrich et al.

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I/ Unified description

30

cm. distance 20

10

5

I

0

I

I

I

I

I

I

I

I

I

2 UPP

--

(P,Y

1

\ to-1

\ \

\ \ \ \

10-I

\ \ ! \

\ 10-j

I

I

1

2

3

4

5

6

7

I 8

I 9

I

\ \

10

t I 10-2’sl Fig. 1. For Xe+Bi at 940 MeV the quantities a&/(P,)’ and CT&/(R)~ are plotted as a function of time for the trajectories with 1 = 0, 100, 300. In the upper part the c.m. distance (R) and the surface distance between the collision partners (0) are shown for these trajectories.

P. FrBhrich

600

‘096i

et al. 1 Unified

+136Xe

565

description

E,,,=940MeV

&

$f&rm

7 500 : 1 E G 400 300 5,O

I

6,0

I

7,O

I

8,O

I

1

9,0

I

e,,i

I

o1 1.

700

- 600

800 700 600 500 400 3oc

Fig. 2. Experimental Wilsczynski plots for the Xe+ Bi system at E,,, = 940. 1130. 1422 MeV are compared with theoretical E-O correlations with deformations (solid lines) and for spherical nuclei (dashed lines). For comparison a proximity calculation including a neck IS) is shown (dotted lines).

566

P. FrGbrich

;

300

iz ,:

200

et al. 1 Unified

description

----__

100 300 -m-s200

100 0

20

40

60

80

100

120

Qc, (deg ) Fig. 3. The same as fig. 2, but for the Kr + La system at Eiab = 5f)S. 610. 710 MeV

TABLE Theoretical

total deep-inelastic

cross sections are compared

“) Ref. 13).

G$, for the Xe + Bi system at E,,, = 940, 1130. 1422 MeV with the experimental ones Experimental quasielastic cut-off dE (MeV)

E,,,

940 1130 1422

1

2.1 +0.1 “) 2.8 Y 3.11_0.4”)

b, Ref. 14).

“) Ref. I’).

25 35 50

zz 1.7”) 2.5 bf 2.8 ‘)

1.70 2.40 3.01

P. Friibrich et al. / Unified description

In fig. 4 the cross sections

do/da

are shown.

561

For the 940 MeV case the shifting

of the theoretical maximum to larger angles by about 5” is connected with the fact that the corresponding E -8 ridge line (see fig. 2) runs too quickly towards backward angles. This is also systematically the case for other Coulomb-like systems such as Pb + Pb at E,,, = 1575 MeV [ref. 24)] or 154Sm + ‘54Sm (E,,, = 970 MeV) and 144Sm + L44Sm (E,,, = 1000 MeV) [ref. ‘,)I. The magnitude of the cross section is of the correct order for 940 MeV. This is in contrast to the 1130 and 1422 MeV cases where the positions of the maxima are correct but the distributions are too narrow and too high. The latter fact indicates that quanta1 diffraction “j) pl a y s a role in cases where a rainbow occurs in the deep-inelastic region, as is the case for 1130 and 1422 MeV (see fig. 2); in the 940 MeV case there is no rainbow beyond the quasielastic region. Fig. 5 shows the cross sections do/dE. For the three incident energies the energy losses reach down sufficiently below the Coulomb barrier of spherical nuclei. The minima of the theoretical cross sections between the quasielastic and deep-inelastic region are not deep enough. In contrast, the spherical calculation of ref. ‘) (shown for 1130 MeV) does not show any structure at all. As further examples the cross sections d’a/dQdE at fixed final energies for the 1422 MeV case and d301dEdZdQ for fixed final energies and (Z) = 54 are shown in figs. 6 and 7. The theoretical cross sections are too narrow and too high in the quasielastic region. With increasing energy loss they become considerably broader but not sufficiently so; however, in contrast to spherical calculations [see e.g. ref. ‘)I, they reach the correct order of magnitude. In fig. 8 we compare the theoretical and experimental cross sections dc/dZ, which agree fairly well with one another. We should mention that the height of the cross section maxima is very sensitive to the quasielastic cut-off. (In the calculations the experimental cut-off is used.) Fig. 9 gives the cross sections d2a/dEdZ for different final energies as a function of Z. The improvement over a spherical calculation is demonstrated for the 1130 MeV case. In order to see more clearly the deviations of the theory from the experimental data it is more instructive to consider the mean values (Z) and the variances ai of the distributions as a function of energy loss. This is done in figs. 10 and 11. The calculations show a satisfactory agreement with experiment for the mean values and variances at 940 MeV, whereas the magnitude of the cross section is still too small for high energy losses (see fig. 8). The theoretical mean values for the 1130 MeV case are reasonable, whereas for the 1422 MeV case they do not exhibit the slight shift to asymmetry observed in experiment. For 1130 and 1422 MeV the variances become far too small for large energy losses; this is also the case in the proximity calculation of ref. 21) which is shown in fig.. 11 for comparison for 1422 MeV. In the figure we also indicate the maximum values attainable by the variances if the interaction times are very large ag(t + co) = *TZ’/c, where c is the stiffness of the driving potential. In contrast to the proposal of ref. 27) shell

568

P. Frijbrich et al. / Unified description

P. Friibrieh et al. / Unified description

(1INr-l Z AJWJS/qw)ZP3P"'UP/o,P

P. FrBbrich et al. / Unified description

570

IO3

-

T3, +‘36Xe

260 5 E = 546 MeV

EL= 940

to-

MeV

t

EL=1130

Fig. 8. Theoretical

Number

MeV

2605 Es810

cross sections

da/d2

30 50 Z (Atomic

38 46 5L 62 70 Z (Atomic Number 1

1

MeV

70 Number)

for Xe + Bi at E,,, = 940, 1130, 1422 MeV are compared the data.

25"*0, ,475"

with

EL=1422 MeV

EL=1130 MeV

E,: 940 MeV 40"s o,, -100"

180* o:,,-128'

t_,010

?O"

109‘

lOi -

E,=1422

IO3

Xe

Z (Atomic

MeV

30O"TKE'bSO

8, IF 1 IOO-

IO6

IO0 lOI

~ I_

Id

_

IO5

F

IO4

IO3 IO2

10'

1’

38

; ’

46

Z (Atomtc

1

54

’ 62

Number)

1 70

78

38 46 54 62 70 78 86

2

(Atomic Number)

’

’

’

’

’

30 40 50 60 70 Z (Atomic Number)

‘I

80

Fig. 9. Theoretica cross sections d20/dEdZ at different final energies are compared with experiment. For the 1130 MeV case the results of a calculation for spherical nuclei are indicated (dashed lines) for 325, 375 and 425 MeV energy loss; for smaller energy losses the results are close to the deformed case.

P. Frijbrich et al. / Unified description

500

571

-

Xe+Bi

+

m LOO :: 2 2300 L ,” w 200

1422MeV :

100

54

55 56 Mean

Fig. 10. Theoretical

and experimental

51

58 59 ”

53

54 55 56

51 58

Value
mean values (Z)

7

as a function

,

r-1

6007”-

1

1

of energy

I

I

loss are shown.

,

3

0 -

ZOSBi + 136xe . E,:940MeV o E,=1130MeV 0

0 E,=1422MeV

-

-----CORR.EXCHANGE PROXIMITYll422MeV) I-

Fig. 11. Variances of the present theory (solid lines) and of a proximity calculation 15) for the 1422 MeV case are compared with the data. The 0 are the asymptotic variances ai(t -B 00) = iTZ’/c for the highest energy losses.

P. Friibrich et al.

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I Unified description

effects in the driving potential do not explain the large variances with the interaction times resulting from the present dynamical calculations. Shell effects should have melted away for the large energy losses one observes for the Xe+Bi system. One could think that the discrepancy with experiment is due to a wrong energy dependence of the theoretical diffusion coefficient, but if one uses 17) a diffusion coefficient which gives the experimental variances one observes that the corresponding mean values run much too quickly to symmetry. This seems to be a general difficulty in a Fokker-Planck description of charge transfer, which has also been stressed in ref. ‘* ).

5. Conclusion

and discussion

In conclusion we can say that the present dynamical calculations reproduce correctly many of the gross features of deep-inelastic collisions. The universal parametrization is not expected to describe all reactions in detail, but it should nevertheless have a certain predictive power, because it can describe at least the rough features of systems with very different behaviour [see also refs. “~““~““)]. In addition the detailed calculations bring to light several problems of the present approach which demand further investigation : (i) Coulomb-like systems are not described too well; the theoretical E- 0 correlation runs too quickly to large angles. (ii) Quanta1 effects should be included in cases where the angular distributions are too narrow and too high. (iii) We should mention that in orbiting systems the theoretical cross sections are too low ‘7,29) at negative angles because too few trajectories lead into this region. This is an indication that higher multipole deformations and/or neck formation are needed in order to stabilize orbiting. (iv) The fluctuations are still too small at large energy losses (see figs. 6 and 7) where quanta1 diffraction should play a minor role. This might be improved by including the statistical fluctuations of additional degrees of freedom (see (iii)). (v) The impossibility of getting simultaneously correct mean values and variances for the charge transfer should be further investigated.

References 1) H. A. Weidenmiiller, 2) 3) 4) 5) 6) 7)

Progr. Nucl. Part. Phys. 3 (1980) 49 M. Durand and P. Frabrich, Proc. Int. Conf. on nuclear physics, D. H. E. Gross, Z. Phys. A291 (1979) 145 H. Hofmann and P. Siemens, Nucl. Phys. A275 (1977) 464 S. Ayik and W. Niirenberg, Z. Phys. A288 (1978) 175 H. H. Deubler and K. Dietrich, Nucl. Phys. A277 (1977) 493 C. M. Ko, Phys. Lett. 81B (1979) 299

Berkeley,

1980, p. 525

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573

8) J. Blocki, M. Dworzecka, F. Beck and H. Feldmeier, Phys. Lett. 99B (1981) 13 9) R. A. Broglia, C. H. Dasso and A. Winther, Proc. Int. School of Physics ‘Enrico Fermi’, Varenna Come 1979 (North-Holland, Amsterdam, 1981) 10) D. H. E. Gross, R. C. Nayak and L. Satapathy, Z. Phys. A299 (1981) 63 11) S. K. Samaddar, D. Sperber, M. Zielinska-Pfabe, M. I. Sobel and S. I. A. Garpman, Phys. Rev. C23 (1981) 760 12) H L. Yadav and W. Norenberg, Phys. Lett. 11SB (1982) 179 13) W. W. Wilcke, J. R. Birkelund, A. D. Hoover, J. R. Huizenga, W. U. Schriider, V. E. Viola, K. L. Wolf and A. C. Mignerey, Phys. Rev. C22 (1980) 128 14) W. U. Schroder, J. R. Birkelund, J. R. Huizenga, K. L. Wolf and V. E. Viola, Phys. Reports 45C (1978) 301 15) H. J. Wollersheim, W. W. Wilcke, J. R. Birkelund, J. R. Huizenga, W. U. Schroder, H. Freiesleben and D. Milscher, Phys. Rev. C24 (1981) 2114 16) D. H. E. Gross and H. Kalinowski. Phys. Reports 45 (1978) 175 17) B. Strack, diploma thesis, FU Berlin, 1981, (unpublished) 18) S. Ayik, G. Wolschin and W. Norenberg, Z. Phys. A286 (1978) 271 19) R. Schmidt and J. Teichert, J. of Phys. G7 (1981) 1523 20) C. Ngo and H. Hofmann, Z. Phys. A282 (1977) 83; M. Berlanger, P. Grange, H. Hofmann, C. NgB and J. Richert, Z. Phys. A286 (1978) 207; R. Schmidt, V. D. Toneev and G. Wolschin, Nucl. Phys. A311 (1978) 274 21) U. Brosa and W. Cassing, Z. Phys. A307 (1982) 167 22) R. Vandenbosch, M. P. Webb, P. Dyer, R. J. Puigh and R. Weislield, Phys. Rev. Cl7 (1978) 1642 23) P. Frbbrich, Phys. Lett. 122B (1983) 338 24) T. Tanabe, R. Bock, M. Dakowski, A. Gobbi, H. Sann, H. Stelzer, U. Lynen and A. Olmi, Nucl. Phys. A342 (1980) 194 25) K. D. Hildenbrand er al., GSI-Nachrichten 8-81, p. 5 26) K. Hartmann, Z. Phys. A294 (1980) 65; W. Cassing and H. Friedrich, Z. Phys. A299 (1981) 201; K. Dietrich and Ch. Leclercq-Villain. Nucl. Phys. A359 (1981) 201; Ch. Leclercq-Villain, M. Baus-Baghdikian and K. Dietrich, Nucl. Phys. A359 (1981) 237 27) S. Grossmann, Z. Phys. A296 (1980) 251 28) U. Brosa and S. Grossmann, preprint, 14th Summer School, Mikolajki, Poland 29) M. Durand and P. Friibrich, Fizika 13, suppl. 1 (1981) 74 30) H. Siekmann, B. Gebauer, H. G. Bohlen, H. Kluge, W. von Oertzen, P. Frobrich, B. Strack, K. D. Hildenbrand, H. Sann and U. Lynen, Z. Phys. A307 (1982) 113