Open problems in sub-barrier fusion reactions

Nucle

ytits A

(1993) 163c-1

O

4 rpop

Niels

BARRIER FUSION REACTIONS A. r Institute, n -2100

i

LIA rsity of Copenhagen, g", rk

magnitude and dependence on mass number displayed by subfusion cross sections is wrongly predicted by standard penetrability calculations . Qualitatively the observed unsystematic A-dependence can be correlated with specific two-nucleon transfer processes, while the large measured cross sections can be understood in terms of zero point fluctuations . indicate that the coupling of the entrance h tic model calculations inelastic and transfer channels act to enhance the transmission channel to rding energies and decrease it at energies through the barrier at l a the barrier . 7b become more quantitative one has to treat these couplings in detail i use cf the specific formfactors and taking care of the associated value effects .

Abstract : l.

1. rust

ion and t

t

one

of

most prolific reactions in collisions among heavy tfte the interest these reactions have attracted, nt l information available to date . prcjectile and target remain spherical as they approacl , when they reach some critical ansatz that they fu presstcon' ) th

ason alone ju

wealth of ex ri Assuming that th each other a tinq R lead to th distance

for the Coulo a R exa la

Overview

The quantity U(R) is the sum of the nuclear and fu ion cross section . the ntor of ss bombarding energy . Zb extract U(R) tentials and E t An f the data it is standard to plot O f as a function of 1/E . f . 2 fig . 4 .15) . t n in fig . 1 (cf . a

1500

A

160t 27AI

1000

b~

1 1 o02 004 1/Ecu (MW)

OA6

0:08

Figure 1 . Fusion cross sectiar. fcr the reaction 160 + 27A1 as a function of the inverse of the bombarding energy (cf . ref . 2) .

detaile hove to well be" important lysis of t challenge . In fach

for three

orders of magnitude . shown in fig . 2(b) . fusion is obse the change in the numbe

of

we Copenhagen . In Section 2a have on the charge and mass dens also mention the role they play

nt of view of the like the ow sl of the expression

reaction the data is reproduced ion is underestimated by for the other are found I of h

report on some of the mied out at s the effect zero-point ouations t Z FF} es predicted by Hartree-Fork ca: culations . We ion-ion potential . In Sections 2b and 2c study the effect of ZPF in sub-barrier fuon o f of populating transfer reach is discussed in Section 3 . In Section 4 a uniLned in Sections 2 and 3 is presented making use The conclusions are collected in Section 5 .

OPEN PROBLE S IN SUB-BARRIER FUSION REACTIONS

165c

Zero-point Fluctuations Collecti+ tates associated w th vibrations of the surface are rather comscribed by a model 6) which paramet°izes the n in nuclei . They are well collective variables and assumes the vihape of t nucleus king u of a f harmonic . The parameters of the model are calculated in terms of bratior,s to coupling of t nuclear surface to the single-particle motion . The zero-point fluctuations associated with these vibrations can have a sign-inn potential nificant influence in reactions between heavy ions where the i 3U) sY . They can nds on the instantaneous position of the nuclear surface 11 1 ' 2), and for the also important for the ground state charge densities value of the isotopic shifts ) . 2a_ . In the collective collective variables CL

ajj(

Charge and mass density

in terms of the model the radius is parametrized nXu -vibration according to n) associated with each R=R (®) +s

where

siR( ®)

(3)

j Ot u(n)Y X* (n) . X la

nXu

(4)

They are the multipolarity indices are needed to identify each mode . Three the magnetic quantum number ji, and the number n indicating whether one is dealThe etc ., excited state of given multipolarity . first, second, ing with the quantity R( ® ) is the effective radius of the nucleus in the absence of fluctuations . The collective ground state 1 Ô> associated with the variables aat,( nI ) is X (n)/2CX (n) z . with standard deviation (~fw functions each a product of Gaussian The Hartree-Fork density can be parametrized according to

where f (x)=C1+exp(x/a )

-1,

(6)

The dependence of p HF on the collective variables is a Fermi distribution . The average nuclear density in the correlated can be obtained making use of (3) . ground state 1Ô> is p(r)=lds g(s) PHF (r,s),

(7)

where g (s) =

<Ô

( 1

2no s

VI

1

s-R(0)

nXU exp

(

_

~u(n)Y~u(f»I~>

OL

s2 2Q

s

(8)

is :he distribution of radii s = (R - R (0 ) ) in 16 . From the knowledge of this function the associated standard deviation can be calculated leading to

2 s s g(s)

Wi ttf

(0)

iR

47r

The quantity ß(T =0 ; O-An) multipol~rity A associated The deformatioW parameter

(JOX tn) /2C, (n) )

.2

)

I

2

B(T=O ;

(X+2) A
X_I~"

(10)

is the isoscalar reduced trihsition probability of th with the excitation of the n collective mode . related to the zero point amplitude (n) is A

according 'to

=(2A+1) the one to be compared with experiment and not a fraction of the fluctuations measured by the parameter (9) are already contained in P,,, . They have thus to be subwhich has to be used tracted from 0s and at is the resulting Qß 2 =0 2_GzW 0 W (7) and (8) to. calculate the density p (r) . In ref . I I a study of the role of zero-point fluctuations on P (r) has been carried out along the lines discussed above . The results of this study for 40Ca are shown in fig . 3 . Changes of the order of 10% are observed in the tail Most of the, effect arises from the zero paint fluctuations associated ow-lying-octupole vibration, PIW . '

The

quantity

p {r}

is

It is however noted that

.a cs .0.04 We 3

4 r (fm)

Figure 3 . Charge density of 40Ca . The full drawn curve is from experiment while the dashed curve is obtained f rom P The dots are the result of HFo using 80 = 0 .62 fm . The difference between the two calculated densities also shown (dot-dashed curve) . For more details cf . ref . 11 . The result shown in fig. 3 has consequencIS f r the ion-ion potential 114C calculated as a double folding integral accord-ng to

OPEN PROBLEMAS IN SUB-BARRIER FUSION R

U aA (r)=IPa(r')pA

IO

(r")V i2 (r-r'+r"d)d

PA are the experimental where pa and densiti t with projectile . The relation between the bare potential V . & (r) l lat,7 A by PHF in (12) and the dilased potential A (r) at dists nct have like an exponential ïs

a being the diffusivity of ~(r) . discus below, 'give to be taken into account explicitly In the lcu atioi, sections at energies l t Coulo In such barrier . of VN (r) to avoid double counting . aft is noted that the relation (13) Implies t the force felt the iona due to the empirical potential is larger than that associated with t tential VN- A . This result e to in contradiction with the intuitive pi c fluctuations lead to a function that than ture where the zero-point with r less rapidly than the original function . A.lthou correct around the in this i flection point, it must remembered that . (13) is valid only asymptotically .

2b.

on

coons

e1') is that the coherent surface @Acitati two nuclei keep their identity throughout the lli-ion . e co-ordinate of re lative motion in thus one of t dynamical variabl of t problem. T other degrees of freedom are those associated with each of the two interacting nuclei . Surface vibrations a explicitly taken Into account . Because of the short-rare The

basic

ansatz

of t:

nature of the interaction between the nuclei it is important to follow the evolu-

of the nuclear surfaces throughout t collision . A central feature of the model is the possibility it has of dâaplaying a coherent response of the surface modes of the two interacting nuclei . The transfej 5if particles from one which starts taksystem to the other is described as an incoherent process

tion

ing place as soon as the surface are in contact . The equations controlling the time evolution of the relative motion, the nuclear surfaces and the particle flux are classical first order differential equations . The effect of quantal fluctuations assr-lated with the surface nodes is included as initial conditions for the man n and amplitudes of these modes . The validity of this procedure can be shown ) in terms of the properties of the "phase space" distribution obtained from a Wigner transformation of the density matrix of a harmonic oscillator . In the limit in which the coupling between relative motion and vibrations is linear in the amplitudes of the modes the treatment becomes equivalent to the quantal treatment . The quantal fluctuations treated in terms of the zero-point notion of the surfaces lead to large fluctuations in the energy and angular momentum loss, as well as in the scattering angle . Similarly, the fluctuation in the surface-sur face distance between the two colliding ions will lead to fluctuations in the One can thus expect that fusion barrier height of the effective interaction . cross sections at energies near and below the Coulomb barrier will be strongly In affected by the zero-point fluctuations (cf . e .g . refs . 3, 4, 5, 16 and 17) . what follows we discuss this effect in a simple model .

R,A, BROGl1A

1fi8c

The effective potential acting between t3ze

vN +U C

+T.x

crntr

{

t~ }

The assumption is made sum of the nuclear . Coulomb and centrifugal potentials . that the penetrabilities T~ agpearïng in eq, {2} are determined by the barrier associated with Ueff at r = r R , r g beïng the radius of the Coulomb barassur,~ption that the projectile a is a rigid rier . Zf one makes the further sphere while the target A_ has a Iow-1,~ ing quadrupole vibration {cf . fic~ . d~} of small amplitude ta i « a } one can write Ueff

~rhere

{r^r~3}

;EB+nUr

{ 15}

is the Co~i.omb barrier and

0

- R~ } ß.~l ~~ is proportional to ttae deformation paramet~~r the ~ruantity s `t~'r~e first term in { 16 } arises from the expansion of the ion-ion potmeraia2 it wr~i le the second term is ' the monopole-quadrupole Coulomb interaction at r ~_rS . The standard devïation of l~U around the average value ( t6 } i,s

~U=

~`i gore ~ . schematic representatï.on of a ro' Iision between two ions where t:~e is assum+~d to be a rigid sphere while the target ïs spherical and âisglays a low-lying quadrupole vibration . Snapshots of two e~"Ants with the same i et parameter are si:own, which display the effect of the zero point vibrati

OPEN PROBLEMS IN SUB-BARRIER r ljtision can be written as

between

169c

FUSION REACTIONS

medium and heavy nuclei the centrifugal potential

Ucentr=10-3  i

MeV .

(17)

The effective potential thus changes very slowl as a function of k , and only TjefY attain Z.'-33 does the contribution of Ucentr to for ,a value of 10% of the cal of t° Coulomb barrier . At low bombarding energies %'E < EB ) only the low waves contribute and for simple estimates one can use the results associated with the 0 partial wave . it is noted that in the estimates carried out below as well as in those of UN ref . 3,4,5,16 and 17, the empirical potential has been used . In fact the barn: potential was the one appropriate for these calculations (cf . Section 2a eq . (13)) " of the parabolic approximation for the barrier one can write the a trana issi coefficient as

T îj')exp (

E-(EB +AU)

where

(19)

The lu4ntit,y

frequency associated with the barrier.,

i .p .

(-0 2 v"/InaA ) ah ti' V°®-d" " ( r rb, s-0)), /dr y . The aver _A . ; "- : ,netrabt l t ty in the nor related ground state < Ô j To (s) noi , be calculat,~d . One obtains where

T

9

(20)

1 Ô>

can

ds g(s)T 0 (s)

^- exp

E-(EB -DS )

(21)

E

where

(0) R A rB

(22)

Thus, low the Coulomb barrier zero-Point fluctuations always lead for energies to an enhancement of the fusioaa cross section . We discusa this feature below (cf . Sect . 4 in particular the discussicn connected with fig . 7) .

R.A . BROGLIA

170C Sn

For the reaction

the parameters

ë

=

ire

(18)-(20)

107 . 5 MeV 11 .4 fm 0 .75 MeV(

the

WW

are equal to

`r-4 .7 MeV) .

B

experinent the quantity (C s one can nov extract from quantity carrying nuclear structure information . From fig . 2 one reads 7 Me

implies

(23) the

(24)

r

B

To be able to calculate this quantity one should know which vibrations as . Intuitively one would include the contribution of those contribute _0 states for which the projectile sees the target deformed . This condition is approximately satisfied by the low-lying quadrupole and octupole vibrationa . In this case and making use of the ßX -values taken from ref . 18 the predicted value of G,/rB is 5%' As saLisfactory as the agreement between theory and experiment might be it only implies that the model contains some of the relevant degrees of freedom . In fact, as shown h>Iow,many other features not taken into account in this model are expected to play an important role in the fusion process . 2c . Zero-Point Fluctuations Associated with Pair

Vibrations

Before ending this section we want to comment on the vole the zero-point motion associated with pairing vibrations may play in the sub-barrier fusion process Pairing vibrations 19) are vibrations which change by two the number of nucleo -as . They have been systematically identified throughout the mass table (cf . e .g .20) . A close parallel can be carried out between pairing vib surface modes as can be seen from table 1 . For the case of nuclei far away from closed shells the system becomes deformed in gauge space and the pairing plays the role of the static quadrupole deformation . For nuclei around closed shells the zero-point fluctuation amplitude associated with pairing vibrations is given by

(26) where

+

V>O and

r

.1

a Ma 1 (0)

The operator a + { V ) creates a particle in the orbital v while a ( O ) is + laced to it by the operation of time reversal . The quantitios X -J ar

(27)

(20)

OPEN PROBLEMS IN SUB-BAPH1 1255P,

171c

Table I

Quadrupole fluctuations of

the avera

field

which change t .lie number of particles by two

angular mmentum by two CL

2

=

<ô ~ (a+a )

.

,

1

,D

~à lead W I I

surfece vibrations

patrin

vibratiolic

The rwmber of a vibrational band are connected by large 0

inelastic

do dQ I

A-0

ne l

cross sections 2

s

1

two-nucleon transfer

0 ) "-x

(2C19

~,

ua2 P

proportional to the square of the zero point amplitudes .

bl card F

The parallel can be Analogy between surface and pairing vibrations . 1. quadrupole vibrations . is not restricted to multipolarity and out for any ails ct .ref .19 .

LIA

175

plitudes of the pair addition or of the pair subtraction modes and have to be calculated microscopically . Typical values for (28) are 4-8 (cf . table 2) . As in(~W,/W,) can also be extracted from the analysis dicated in table 1, of two-particle transfer data . -

Table 2 nZj

I

Xj

2g9/2

0 .820 0 .398 0 .314 0 .180 0 .085 0 .146 0 .102

11/2 15/2 3d 5/2 4s 1/2 29 7/2 3d 3/2

Wavefunction describing around state of 210 Pb {pa Pb) . The two particles are allowed to correlate in the single-particle sub.pace spanned by the bound single-particle orbitals above the N = 126 shell clo sure* the zero-point amplitude associated with this vibration is 0 + = 4 .5 (cf . eqs . (26) 2.

The distribution of charges in the correlated ground state of a nucleus Isplaying proton pairing vibrations is (cf . dig . 5) (Z_Z O ) 2

9(Z-Z °}= w he ra-

1

20Z

A" Z +

0

2

.

(29)

(29a)

that only the target nucleus vibrates the quantity D., i .e . the equivalent to D cf . eqs . (21) - (22)), associated with the pairing modes is

(30) The constant c is proportional to the square of the two nucleon transfer formfactor . Preliminary estimates indicate that DZ is similar in magnitude to D. (cf . eq . (22)) . A quantitative analysis of the data will require considering the effects of pairing modes on equal footing to those of the surface modes .

OPEN PROBLEMS IN SUB-BARRIER FUSION REACTIONS

f 73c

Vv*

Ev

E

Figure 5. Schematic representation of the effect of the zero-point fluctuations associated with pairing vibrations in the probability occupancy Vv of the single-particle orbitals of energy v . In the absence of pairing vibrations the Fermi distribution is sharp, V v being equal to 1 up to the Fermi surface and 0 afterwards . The proton pairing vibrations change this occupancy around E F , leading to fluctuations in EB .

3 . Two-Nucleon Transfer Channels The effects discussed in the previous Section are related to the backcoupling of reaction channels to the entrance channel (cf . also Section 4) . In the present Section we study the effect on O f of populating specific channels (cf . ref . 21) . For low bombarding energies, the Fusion rate in a transfer channel will be different from that in the entrance channel because of the Q-value of the reac-

R.A. SROG LIA

174c

Fusion will be fation and because of the difference in the Coulomb barrier. . voured in a transfer channel if Q + A E c is positive, where AE c is the dif ference between the heights of the barriers in the two channels . For the case of the Ni + Ni reactions code^ted in fig . 2 the values of Q + For the reaction 64 Ni + 64Ni display A Ec a rather conspicuous behaviour . 58 and Ni + 58Nî all Q + A E c values associated with one- and two-particle However, for the 58 Ni + 64 Ni reaction the transfer channels are negative . two-neutron pick-up and two-proton stripping display positive values of AEe+Q . It may also be noted that 56 Ni is the ground state of well-developed pairing vibrational bands whose members are the different Ni isotopes and N = 28 isotones (cf . e .g . ref . 20 and references therein) . At low bombarding energies both the Q + A Ec value and the size of the the 64Ni( SSNi, 60 Ni) 62Ni reaction . This two-nucleon formfactor favour allows us to make a simple analysis of the data as follows . We assume that the measured fusion cross section Q for 58Ni+ 64 Ni at the energy E is given by m om (E ) = ( 1-P(E))Q 0 (E)iP!E)Q,(E+Q+àEC )

,

(31)

. 64 Ni entrance channel, U is where ß is the fusion cross section in the 58r, 1+ 60Ni + 62- Ni transfer channel, and P the fusion cross section for the is l an effective probability of two-neutron transfer followed by fusion . That is, eq . (31) describes a process where 6 An the entrance channel 58Ni reaches out and 92 lls away two neutrons from Ni . Because these neutrons are attached to Ni the pull needed to break them loose (= 4 MeV) is transmitted to the relative motion. of the ions . In the new charnel the ions can go over the barrier even if they started in the entrance channel at an energy smaller than EH . This possibility enhances the fusion cross section so much that it totally overwhelms the fact that the neutrons are successfully pulled off in only a few percent of the events . To construct CY O and o f we take the measured cross section for 58Ni+ 58 Ni and shift it to account for the mass differences . The result for VE) is shown in fig . 6 . -We find that a fit to the data is rather insensitive to the energy dependence of P. We show by the solid curve in f ,-g . 6 the result of taking a constatt P(E) =0 .06 in eq .(31) .

a.

0.01

Figure 6 . Fusion cross section for 58 Ni + 64 Ni and 58Ni + 74Ge . The data is fr ref . 4 . The dashed curves are predictions btained by extrapolating the cured cross sections for 58 Ni + 58Ni and 64 Ni + p14Ge . The solid curves include the contribution to fusion which occurs due to two-neutron transfer cording y4to ea . (31), assuming a transfer probability of 6$ and 10% for the t and targets respectively .

OPEN PROBLEMS IN SUB-BARRIER FUSION REACTIONS

175c

Systematic studies of the effects described by eq . (31) were carried out in ref . 5 and are reported in the Proceedings of the contributed papers to this Conference . 4 . Coupled Channel Formalism In the previous Sections we have discussed the role zeropoint fluctuations and two-particle transfer reactions play on the fusion process . A unified picture can be achieved by discussing, within a coupled channel formal-ism, the ef fects which the coupling of reaction channels to the entrance channel has on the p netrability coefficient 22 ) (cf . also ref . 23) . Let us assume the entrance channel is coupled to a number N of channels, the total wavefunction being X (r ) Y Y r 'y Y

1

Y(7Y)

.

(32)

The index is the produ-_t of the inY=(C,c) labels a reaction channel, ~y trinsic wavefunctions of th6 two nuclei while XY is the relative motion wavefunction . The associated Schrödinger equation

(33) (H C +V)T = ET , is equivalent to a set of N coupled equations . Much of the physics contained in them can be obtained making the following assumptions : 1) the coupling V has constant matrix elements which only connect the entrance channel 1 with the reaction channels $ i .e . V, B =Vc and V ,=0, so 2) the effective Q-values are equal, that is

2118 (E ~!z

eff )= 2111 ~z

0-Vß

(E1-Veff) .

(34)

Because of this degeneracy (33) reduces to a two-channel problem described by the wavefunctions of relative motion X1 and X=JN X~ . Introducing the functions and Xdefined through the equations X+

X1

X

=

_

2 (X++X -) .

2

(X -X)

(35)

the two equations decouple leading to 2x+

d

dr2 with

+

211 (E-V+ ) X+ = (), x2 --

(36)

R .A . BROGLIA

17 6C

V+ = Vef f ± VIN VC

(3'1)

V°= OIVC$ 0

0

I/2

0

0

ES T

Ee

E

Figure 7. Schematic representation of the coupling of N degenerate reaction For VC = 0 the system displays a barchannels ß to the entrance channel 1 . rier of height EB and the class .Ical penetrability coefficient is 0 or 1 deis smaller or larger than EB . For V 0 0 the original pending on whether E EB ± VC . barrier splits into two of height Three classical regions can be distinguished . One for E < E B - VC where the penetrability is zero another for E>EB + VC where it is 1 and an intermediate region EB-VC
T = Tl +T =

2

(T+ +T - ) .

(38)

are displayed in fig . 7 . The main result of this calculation is that the coupling of the entrance channel to reaction channels increases the transmission coefficients at energies belovi the Coulomb barrier and decreases it at energies above it, independent of the sign of the coupling . Assuming N=Jr;Q2(n) and VC =R( ° )9U/ar the problem discussed above corresponds to that presented in Section 2b (cf . refs . 22 and 23) . Solving numerically egs . (36) with a realistic value for Veff(= EH ) and plotting the results in logarithmic stale one gets a function which mimics nicely the trend cf the experimental data (cf . fig . 2 and fig. 8) . From these results

OPEN PROBLEMS IN

SUB-BARRIER FUSION REACTIONS

177c

one can conclude that the main effect of the coupling is to produce an effective barrier such that classical penetrability is possible (cf . also fig . 9) .

ENERGY

tbl

T

Figure 8. Transmission coefficient in the degenerate two-level channel for different values of the coupling str ngth and using barrier parameters which simulate the s-wave potential for 8Ni+ 8Ni(note that F=V ; ;for details cf .ref .22) . In (b) we schematically show the intermediate slope region corresponding to the plateau in the linear plot . Assuming that the energy interval spanned by this region is deliminated by the values of E where the Gaussian function des cribinq the distribution~f Coulomb barrier corrections AU achieves 1/5 of its maximum value (i .e .2(2kn5) =12MeV) ore obtains aS/rB = 4 .5% . In the curve shown in fig. 8 we identify three slopes, associated with tiie events leading to fusion at EEH and E z r.8 . The part of the curve corresponding to reactions with E = E8 lead, in the linear scale, to a plateau (cf . fig. 7) . The associated interval of energy over which this plateau extends measures the strength of t-he couplinj ®and .~Ti be related to t1re quantity G(W) defined in (16a) . For the reaction Ar+ Sr. this energy interval is 12MeV. (cf . figs . 2(a) and 8(b)) . Making rise of the numerics of (23) one obtains a value of C / r E which is very similar to (25) (for details cf . caption to fig . 8) . s

R.A . BRQGLIA

122Sn . re g . Fusion cross section associated with the reaction 40Ar+ tiru ass curve has been obtained making use of eq . (2), calculating the T V B approximation t- king into account zero-point fluctuations . A torn in the tting king use of the standard ion-ion potential, of _cozen shapes and of transmission penetrabilitias leads to the dotted curve which follows classical the data frog high energies down to = 115 f.KeV, i .e . = 8 MeV above the Coulomb rrier . Region A is thus controlled by the geometrical extension of the nuclei rlas-=ical objects . Taking into account the zero point fluctuations but using classical transmission coefficients extends the agreement down to Thus in region B zero-point fluctua 100 .V i .e . about 8 MeV below the barrier . tions are ° rtant but not quantal tunneling . The zero =point oscillation allows crier to fluctuate so that a fraction of the events with E < EB can fly r. .ôs-r the effective barrier . In region C tunneling effects are important . For rare tails cf . ref . 3 . 5 . Conclusions coupling of the entrance channel to reaction channels can enhance the cross section by orders of magnitude . Unsystematic behaviour with neutron r of the 9-value of the reaction channels is strongly reflected by O f . ratio between the collect parameter and the Q-value of the reaction dewhich channels play a role in the fusion process at energies somewhat beCoulomb barrier EH . At these energies the effect of the coupling of entrance channel to the reaction channels is to reduce the Coulomb barrier to fectïve value where classical penetrability is possible . Discussions with C . H. Dasso, H . Esbensen, S . Landowne and A .Winther are at fully acknowledged .

futi

1?9c

OPEN PROBLEMS IN SUB-BARRIER FUSION REACTIONS

References 1. M. Lefort, Heavy Ion Collisions, Ed . R. Bock, Vol . 2, North Holland, Amsterdam (1981) 45. U. Mosel, ibid ., p. 275 . 2. 3. W. Rcissdorf, F. P . ssborger, ::. Hildenbrand, S. tofmann, G. zerberg . K. H. Schmidt, J. H. R. Schneider, W. F. W. Schneider, K. SchUnmerer, G . Wirth, J. V. Kratz and K. Schlitt, Phys . Red. lett . 49 (1982) 1e11 . 4. M. Beckerman, M. Sal a, A. Sperduto, J. D. Molit®ris, and A. Di Rienzo, Phys . Rev. C25 (1982) 837. 5. R. Pengo, D. Fvers, F. E. G. L5bner, U. Quade, K. Rudolph, S. J. Skorka, and I. in 5 si .-am on Detector. i n Heavy Ion Reactions, Berlin 1982 Weidl, (unpublished) ; International conference on heavy-Ion Physics and Nuclear Physics, Catania Procs . of contributed papers p . 170 ; R. Pengo 1983 .

6. 7.

8. 9.

10 . 11 . 12 . 13 . 14 .

15 .

16 . 17 . 18 . 19 . 20 . 21 . 22 . 23 .

(private communication) .

A. Bohr and 2 . R . ttelson, Nuclear Structure , Vol . II, Benjamin, Reading Mass . (1975) . H. sso and A . Winther, Procs . of the International R. A. Broglia, C. School of Physics Enrico Fermi, Course LXXVII, eds . R. A . Broglia, C . H . Dasso and R . kicri, North Polland, Amsterdam (1981) p . 327 . R. A. Broglia, C. H. s and H . Ea nsesi, Progress in Particle and Nuclear

Physics, Vol . 4 (1980) 345 . R. A. Broglia (1981) .

and A . Winther Heavy Ion React ions , Benjamin,

Reading Mass .

H . Esbensen, in Proc . of the International School of Physics Enrico Fermi

on Nuclear Structure and Heavy Ion Reactions, eds . R. A . Broglia, C . H . Dasso and R . Ricci, North Holland, stefdam, (1981) p . 572 . H . Esbensen and G .F . Bertsch # Phys . v . C ., to be published . R . A . Broglia, Nuclear Theory 1981, Procs . of the Nuclear Theory Workshop, Santa Barbara 1981, Ed . G . F . Bertsch, World Scientific Singapore (1991)p . *)4 . R . Satchler and W . G . IDve, Phys . p . 55 (1979)183 . R. U. Akyuz and A . Winther, Procs . o the International School in Physics Enrico Fermi or Nuclear Structure and Heavy Ion Collisions, Course LXXVII, eds . R. A. Broglia, C. H. s and R . Ricci, North-Holland, Amsterdam, (1981) P . 392 J . Randrup, Nucl . Phys . .1307 (1978) 319 H. Esbensen, Nucl . Phys . A352 (1981) 147. Nuclear Physics , Edd . C . H . Dasso, R . A . Broglia and A_ Winther, North-Holland, Amsterdam (1982) 645 . S . Landowne sand J . R . Nix, Nucl . Phys . A368 (1981) 352. P . H . SLtlson and L . Grogains, Nucl . Data, Vol . 1 (1969) ; P . M . Endt, Nucl . Data 23 (1979) 3, 1)47 ; 26 (1981)47 . !) . R. Bes and R . A . Broglia, Nucl . Phys . _80 (1966) 289 . R. A . Broglia, O . Hansen and C . Riedel, Adv .i n Nucl .Phys .Vol .6 (1973)p .287 . D . R . Bes, F. A . Broglia, O . Hansen and O . Nathan, Phys . ]Rep-34C (1977)1 . R. A. Broglia, C. H. Dasso, S . Landowne and A . Winther, Phys . Rev . C27

(1983)2433C . H . Dasso, S . Landowne and A . Winther, Nucl . Phys . (to be published) . P. M . Jacobs and U . Smilansky, preprint Weizmann Institute of Science, WIS13/33 March-Ph .