Charge correlations as a probe of nuclear disassembly

NUCLEAR PHYSICS A

Nuclear Physics A556 (1993) 672-696 No~~-Ho~la~d

Charge correlations

as a probe of nuclear disassembly

P. Kre&‘, J.C. Adloffb, M. Begemann-Blaich’, P. Bouissoud, J. Wubele”, G. Imme’, I. Iorif, G.J. Kunde’, S. Lerayd, V. Lindenstruth”, Z, Liu”, U. Lynen’, R.J. Meijer’, U. Milkau”, A. Moronif, W.F.J, Miiller”, C. NgGg, CA. Ogilvie”, J. ~o~hod~a~~aa, G. Raciti”, 6. Rudolf”, I-I. Sann”, A. S~h~tta~~~ W. Seidel”, L. Stuttgeb, W. ~ra~trnann’ and A. ~~~~olski~

’

’ ~~5~~tuf~~~ ~e~n~~y~~~ ~nivers~t~t Frankfurt, D-6U~~ ~runkfurt, Germany h Centre de Recherches Nu&aires, Strasbourg, France ’ Gesellschaft fir Schwerionenforschung, D-6100 Darmstadt, Germany ’ Laboratoire National Saturne, CEN Saclay, F-91191 GiS_sur-Yvette, Frunce e Dipartimento di Fisica dell’ Universitri, I-95129 Catania, Xta1.v lstituto di Scienze Fisiche, Universita degli Srudi di Milano, I-20133 IWilano, Ita1.v g DEIN/SFE, F-91 191 Gif-sur- Yvette, France h FZ Rossendorf, Dresden, Germany

Received 1 April 1992 (Revised 14 November 1992)

We have studied multi-fragment decays of Au projectiles after collisions with C, Al, Cu and Fb targets at a bombarding energy of 600 MeV/nucleon. We examine the correlations between the charges emitted in these reactions. These correlations are given as a function of the total charge in bound fragments, Zboundr at forward angles, which is a measure of the violence of the collision and can be related to the impact parameter. The charge distributions have been fitted by a power law and the extracted T parameter exhibits a minimum as a function of Gound. We observe a strong reduction in the maximum charge, Z,,,, of the event with decreasing Gound. For those events where Z,,, is less than half Z&,undr the relative sizes of the two largest charges within the event cover the full spectrum of possibilities. The charge-Dal&z plots indicate that the rn~~t~-fragmentation events are not an extension of s~mmetrjc fission reactions. The event-by-event charge moments are examined to measure the size of the charge fluctuations. All of the charge correlations are i~depende~t of the target when plotted as a function of qounJ The results are compared to both nuclear statistical and percolation calculations. The model predictions differ from each other, establishing that the observables are sensitive to bow the available phase space is populated. The sequential nuclear model predicts too as~rnrngtrjc a decay, while the simultaneous model predicts too symmetric a break-up. The percolation model, which was adjusted to reproduce the size of Z,,,,,, , correctly predicts the mean behaviour and the fluctuations of the lighter fragments.

Abstract:

E

NUCLEAR REACTIONS C, Al, Cu, Pb(Au, X), E = 600 MeV/nucleon; cross-sections, charge asymmetries, event-by-event charge moments.

Correspondence to: Dr. C. Ogilvie, ~~~a~rn~~t Room 26-402, MA ~2139-43~7 Camb~d~e, USA. ~37~-9474/93/$~6.~#

@ 1993 - Etseviet

Science

of Physics,

Publishers

measured

MIT, Laboratory

B.V. Ail rights reserved

fragment

for Nuclear

Science,

613

F? Kreutz et al. / Charge correlations 1.

There

has been

fragmentation’-‘).

considerable These

onset of multi-fragment that decay by emitting is not yet known

progress

experiments emission

several

Introduction recently

in the study

have provided

the connection

6-8), and the formation intermediate

is the exact nature

of excited

mass fragments

of these decays

of nuclear

(IMFs)

and whether

multi-

between nuclear

the

systems

9*10). But what the new results

place strong restraints on the alternative physical scenarios that have been suggested in the literature. These range from the formation of fragments due to large fluctuations in regions of mechanical instability ‘1,‘2), to statistical multifragmentation decay processes where equilibrium is re-established models ‘3*‘4), and sequential after each binary nuclear liquid-gas

decay 15*16).There is also the prospect that the physics of the phase transition may be probed by these multi-fragment exit

channels. Two methods have been recently successful in forming nuclear systems that decay into multi-fragment states. In near-symmetric collisions between large nuclei at 50 MeV/nucleon, Bowman et al. ‘) have found that several IMFs are produced in the most central collisions. However it is possible that these fragments come from a system that still resembles the projectile and target rather than a combined A = 300 nucleus. This has been established for the most violent collisions at the lower beam energy of 30 MeV/nucleon “). For asymmetric reactions of Au on light targets at 600 MeV/nucleon, we have previously reported the “Rise and Fall” of multi-fragment emission ‘). In these reactions, the fragments come from the projectile spectator4) which decays into very few IMFs at low excitation energy, produces a mean multiplicity of three to four IMFs at an estimated excitation energy of about 8 MeV/nucleon, and at still higher excitation the projectile totally disassembles into lighter fragments. A similar evolution of the charges emitted in symmetric reactions was found by Piasecki et al. “) who measured the charge distribution as a function of neutron multiplicity in Pb + Au at 29 MeV/nucleon. Most of the above research emphasized the number of fragments that are emitted in each reaction and, to some extent, the charge distribution of these fragments. It is the purpose of this paper to probe multi-fragmentation at the next level of charge observables,

namely

the charge

correlations

within

the event.

These

correlations

reveal details about the relevant phase space for the decaying system, i.e. which exit channels are most probable, and therefore provide a probe of nuclear disassembly. The observables that we develop in this paper progress in complexity from impact parameter selected charge an event, and finally to an observables evolve with the tive views of the same data of multi-fragment events.

distributions, to the relative sizes of the charges within event-by-event moment analysis. We describe how the violence of the collision. These observables offer altemaset and hence provide different insights into the nature

Our choice impact

of reactions

parameter

mid-rapidity

source. For collisions

in near-central This changing function

means

of the collision collisions

that the size of the spectator decreases.

decreases

as the

is lost in the formation

of the

on C, Al, and 0.1 the size of the projectile

is estimated

size of the source

Charge

is taken

of the sum of the charges

remnant

3*4) to be A = 150, 120 and 80, respectively. into account

in bound

by plotting

fragments

(22

our results

2) detected

as a in the

time-of-flight wall, _Zbound[ref. “) J. Many models of heavy-ion reactions, e.g. fireball model and transport models (Boltzmann-Uehling-Uhlenbeck), predict that the charge that is contained in the projectile spectator, ZProj, monotonously decreases with decreasing

impact

parameter‘

We use Abound as a substitute for Zr,,,s primarily because the time-of-~igbt wall did not have the ynamic range to measure Z = I fragments s~multa~~o~sly with heavy remnants. The quantity Gound is Z,,,j minus the multiplicity of Z = 1 fragments lost by the spectator. This multiplicity depends on the excitation energy of a system, hence Zbound is expected to be influenced by the excitation energy as well as the size of the projectile spectator. As a measure of the violence of the collision, Zhound also provides an easy comparison across the range of targets. This is because Zbound is related to the properties of the projectile spectator, and the targets perform the secondary role of depositing energy into the projectile spectator. For the different targets, a given on average, to a different reaction geometry, e.g. central vaiue of Zbound corresponds, collisions on C targets lead to Z;round - 35 whereas this value of Zbound is populated for more peripheral collisions on the heavier targets “1. In our previous work 4), we found that the correlation between the I~Fmultiplicity from evaporation type processes (large Z&& to the total and Zbound, extending of the target. This disassembly of the projectile (small Zbound) is independent universal behaviour may indicate an - at least partial - equilibration of the projectile spectator prior to its decay. The use of Zbound does lead to some restrictions in the range of the charge distributions and correlations; discussed in sect. 3.

mathematical these will be

We describe here the pertinent aspects of the experiment, for more details the reader is referred to ref. ‘). The experiment was performed with the ALADIN forward spectrometer as SIS, using a gold beam with an energy of 600 MeV/nueleon that bombarded G, Al, Cu, and Pb targets. The acceptance of the ALADIN spectrometer for beam velocity fragments is sufficient to detect greater than 95% [ref. “)I of the heavier fragments originating from the decay of the gold projectile, since in inverse kinematics these particles are strongly forward focused. The charge and multiplicity of nuclear fragments were determined by means of the time-of-flight (t.o.f.) wall. The discriminator threshold

R Kreutz

set to be below alpha

was

the observables were resolved,

et al. / Charge

particles.

Particles

in the next section. while the charge

heavier fragments.

correlations

with 2 = 1 are not included

For fragments

resolution detector

predominantly

originating

in any of

with 2 < 8 individual

deteriorated

The TP-MUSIC

675

to a f.w.h.m.

was used to calibrate

charges

of k2.0 for

the charge response

of the t.o.f.-wall. Light particles,

from the mid-rapidity

source,

were

detected by the 64 elements of a Si-CsI(Ti) target hodoscope. A minimum multiplicity of one particle in the target hodoscope was required in the off-line analysis. The on-line trigger condition was a beam particle in the start detector and no fragment larger than 2 - 70 in the central t.o.f. strips. Scaled-down beam events were also recorded to provide the normalization for the cross section.

3. Results

3.1. CHARGE

DISTRIBUTIONS

AND

ASYMMETRIES

In fig. 1 we plot the charge distributions and Pb at 600 MeV/nucleon for different

measured in Au reactions on C, Al, Cu cuts in Gound. The width of each Gound

bin is indicated on the figure. These distributions have been corrected for the loss of efficiency due to our finite coverage, with the correction being of the order 5%

Z bound 276 60-76 50-59 40-49 30-39 20-29

Fig. and For 104,

1. The measured charge distributions from Z = 2 for Au 600 MeV/nucleon collisions on C, Pb targets. The data have been gated by the values of Z,,,,,, that are listed in the right-hand each target the data have been multiplied by the following factors (in decreasing order of lo*, 1, lo-‘, 10e2, 10m3, 10e4, and 10-s. The error bars are in most cases smaller than the the symbols.

Al, Cu panel. Gound; size of

676

P. Kreutz

et al. / Charge

correlations

for 2 = 3 fragments and less than 2% for 2 > 6. Reactions from the scaled-down beam events that satisfied our off-line requirement of at least one particle in the target hodoscope are also included in this figure. The shapes of the charge distributions are very similar for all targets as a function of Gound. In the most peripheral collisions, one observes a Z -40 fission peak for all targets. For lower values of z, ound, the charge distribution exhibits a U-shape. The large charges are the residue of the lowly-excited projectile spectator after it has evaporated light fragments and nucleons. For fragments evaporated from the heavy residue, the charge distribution is quite steep. In more central reactions, i.e. smaller values of Gound, the distribution broadens and exhibits a power-law behaviour over nearly the full charge distribution. Note that near the limit of each Zbound cut there is reduced phase space for the charge distribution and hence there is a sharp fall-off in this region. For still lower values of Z&,&, the charge distribution becomes steep. This is consistent with the system disassembling into predominantly lighter fragments 3*18). The t.o.f. spectra for all fragments “) is peaked near the beam velocity, indicating that the fragments in fig. 1 originate from the projectile spectator. For the most central collisions on the Cu target, less than 10% of the measured alpha particles have a slow, fireball-like velocity. The fraction is smaller for the heavier fragments and for less violent collisions. This also implies that the quantity Z&,& is calculated from projectile fragments and is not strongly contaminated by mid-rapidity fragments. We have fitted the charge distributions of fig. 1 with a power law parameterization a(Z) K z-’

.

(1)

The fitting range was 2 = 3, 5 s 2 s 15. For our data, this form gives better fits than an exponential function, though the latter can not be ruled out for the most violent collisions. Our results focus on how the parameter of our chosen form evolves as a function of the violence of the collision. In fig. 2, we plot the parameter 7 as a function of Z&d with the width of the z, ound selection being ten charge units. The T parameter for all four targets lies on a universal curve and shows a minimum near Gound = 35. The error bars on this figure include statistical uncertainties, and systematic uncertainties due to uncertainties in both the choice of fitting range and efficiency correction. Also shown on this figure (and figs. 3-15) are the predictions from the GEMINI 16) (dotted line), COPENHAGEN 14) (dashed line) and percolation (solid line) models. A brief characterization of these models and the significance of the comparison are discussed in sect. 4. The lowest bin in this figure corresponds to events with 0 S &,,und < 10. By definition, this bin can only contain light fragments and hence there is an auto-correlation that forces 7 to be large. This datum is however useful since there is still some, albeit small, variety of possible charge distributions within this bin. This is demonstrated by the different values for 7 that are predicted by the models for this bin of Zr,.,und.

0

25

50

75 Z bound

Fig. 2. The extracted T parameters as a function of Z&,undfor Au 6QQMeV/nucleon collisions on C (circles), Al (triangles), Cu {squares) and Pb (stars). The lines are COPENHAGEN (dashed), GEMlNl (datted) and percolation (full) predictions.

The maximum charge in each event, 2 max, provides one way to characterize the exit channel of each coflision. Fig. 3 is a scatter plot of Z,,, versas Zbaund for C, Al, Cu [ref. “)f and Pb targets. In peripheral reactions, the largest fragment co~t~i~~ most of the total bound charge. fn these collisions .ZmaXmay be ~de~t~~ed as the heavy residue after evaporation. A small symrnetr~~~~ss~on group can be seen in each of the plots of fig. 3, where Z,,, - 40 and Z&aund- 79. For more central reactions, .&,,, becomes a smaller fraction of Z&,nd. In fig. 4, the average value of the largest 80 60 40 20 0 60 40 20 0 0

20

40

60

0

20

40

60

80

Z bound Fig- 3, The measured correlation between Z,,, and 2 bDundfor Au 600 M~~~nucleon collusions on C, Al, Cu and Pb targets.

6%

P. Kreutz et al. / Charge correlations A g

60

E F

50 40 30 20

0

0

20

40

60 Z bound

collisions on C (circles), Al Fig. 4. The average Z,,,,, as a function of Zhound for Au 600 MeV/nucleon (triangles), Cu (squares) and Pb (stars). The error bars are in most cases smaller than the size of the symbols. The lines are COPENHAGEN (dashed), GEMINI (dotted) and percolation (full) predictions.

charge ((Z,,,)) falls sharply with decreasing .Z&und. The decrease is than the trivial restriction Z,,, < Gound. The resulting correlation all four targets. Since -Gax/Zbound decreases for smaller impact parameters, then a of charge is contained in the other fragments. This was emphasized works 3Y4)where we studied the Rise and Fall of IMF emission. In

’

0.6

g

0.4

q GN

is more rapid very similar for greater fraction in our previous fig. 5 we show

0.2 0

0.2 0 0

20

40

60

0

20

40

60

80

Z bound Fig. 5. The relative asymmetry between the two largest fragments in the event as a function of Zhound for Au 600 MeV/nucleon collisions on C, Al, Cu and Pb targets. Events with at least two fragments with Z 2 2 have been selected.

J? Kreutz et al. / Charge correlations

a scatter

plot of the relative

asymmetry,

az, between

679

the two largest

charges

in the

event (Z,,,,, , Z,), where Z mnx-Z* a2=z~.,+z2*

(2)

at least two fragments with Z 3 2 have been selected. For small the non-linearity of this asymmetry and the integer arithmetic values of Gound, causes an increase in the density of events at the simplest rational numbers. At large to events with one values of Gound and a2 further bands are seen that correspond light fragment, e.g. alpha or lithium fragment emitted in association with a heavy Events containing

fragment. For those

with large Gound, we see that a2 is near its upper limit of 3 i.e. one alpha particle and a large fragment. However, for In this region the lower values of Zbound there is a broad range of asymmetries. breakup can range from being very asymmetric, to exit channels where the two largest fragments are of equal size. The distribution in asymmetry is better illustrated in fig. 6, where for reactions on the Cu target, we plot four projections of a,. Each projection was taken for a limited range of Zbound; the limits of which are given in Q2 =

Gound

reactions

-4)/G,,,,

the figure caption. Panel (a) is from moderately central reactions, panels just below and above the level where (Z,,,,,) dominates

(b) and (c) are respectively the event, i.e. when (Z,,,)

600

0

0.4

0.8

0

0.4

0.8

(Zmax-Z2)/(Zmax+Z2) Fig. 6. The projection of the relative asymmetry between the two largest fragments over a limited range of Z,,ound for Au 600 MeV/nucleon collisions on the Cu target. The cuts were 36~ i&,und~40, 41 g zh0”“d s 45, 46 s Z&und s 50, and 51 s .Z&und ~55, for panels (a), (b), (c) and (d), respectively.

680

I? Kreutz

et al. / Charge

correlaiions

is 50% of Zbound, while panel (d) contains reactions value. The asymmetric peak that is clearly present significance asymmetry

from panel

(d) to (a). In panel

is approximately

In order to facilitate

uniform

the comparison

with &,und well above this in panel (d) reduces in its

(b) the distribution

of the two-body

and covers the full range of possibilities. with theory,

we plot the mean value of this

asymmetry

as a function of Zbound in fig. 7. The two-body asymmetry monotonously decreases with more violent collisions and the data are similar for all four targets. A method to investigate the correlations between the three largest fragments event-wise is a charge-Dalitz plot. The Dalitz plots are shown in fig. 8 for each target C, Al, Cu and Pb as a function of cuts in Zbound. In these plots, the distance from the three sides is da=

za

where the third distance

zb

db=

z,+z,+z,’

d,=

z,+z,+z,’ is redundant.

zc

z,+z,+z,’

The fragments

(3)

Z,, Zb, and Z, are the three

largest fragments (Z,,, , Z,, Z,) in the event. A different nomenclature has been used to indicate that their order has been randomized, e.g. in one event Z, might correspond to Z,, Zb to Z,,, and Z, to Z,. All events containing at least one fragment with Z 2 2 are included in these plots. A stronger restriction on the fragment multiplicity eliminates many of the evaporation and fission events. The size of each square is proportional to the number of events and the normalization for each Dalitz plot has been adjusted to display the features of each plot. Note also that in the

+ 2

0.7

k

0.6

s

0.5

A,

0.4

::

0.3

E c

0.2

0

20

40

60 Z

bound

Fig. 7. The average value of the relative asymmetry for the two largest fragments as a function of Zhound for Au 600 MeV/nucleon collisions on C (circles), Al (triangles), Co (squares) and Pb (stars). Events with at least two fragments with Z 3 2 have been selected. The error bars are in most cases smaller than the size of the symbols. The lines are COPENHAGEN (dashed), GEMINI (dotted) and percolation (full) predictions.

P. Kreurz et al. / Charge correla!ions

681

very peripheral bin, our trigger is inefficient for processes that lead to a heavy residue, hence this impact bin has relatively more fission events. When G,, is large and the other fragments are small, then the events lie at one of the apexes. This corresponds to events with a heavy residue and such reactions are clearly seen in fig. 8 for large values of &,ound. When two fragments are large and the third, if existing, is small, i.e. symmetric fission, then the event will lie in the middle of each side of the triangle. Fission events are observed for the largest values of Gound. When one fragment is large and the other two fragments are of similar size, then the event lies on one of the diagonal spokes of the Dalitz plot. As Zbounddecreases, i.e. as the reaction becomes more central, it is this branch of the Dalitz plot that is populated. As one moves along a diagonal spoke, then the second and third fragments are becoming comparable in size to the largest fragment. The region near the center of the triangle is dominant for the most central collisions. For reactions with &,und < 10 this is the only region that is mathematically possible. For values of .Z&und35-40 the mean multiplicity of IMFs reaches its peak of three to four “). The spokes on which these multi-fragment exit channels lie, extend C

Al

cu

Pb

Z bound

60-76

collisions Fig. 8. The charge Dalitz plots for different cuts in Z,,ound for Au 600 MeV/nucleon Al, Cu and Pb targets. The definition for Dalitz plot is given in the main text.

on C,

P. Kreutz

682

et al. [ Charge

c~rre~~t~~~s

from the apexes of the Dalitz plots. Hence they are an extension of those events that leave a heavy residue. They are not related to symmetric-fission events, since the spokes do not evolve as a function of Zbound from the sides of the Dali& triangles. A decay

scenario

of the daughters fragments.

of a symmetric would

However

fission

followed

lead to our observing

if this was a significant

by the symmetric

one large fragment process,

fission of one and two smaller

then the second

fission would

be in competition with evaporation and hence there would be some yield of two heavy fragments and a light fragment in the same Zbound bin. This exit channel of symmetric fission followed by evaporation is not strongly observed for &ound < 60. In order

to be more quantitative,

we define

the three-body

asymmetry~

a3 = JCZnl,X -(Z>)z+(z~-(2})2*(z~-(z))z

(41

AU> where

(9

’

(Z>=f(ZIn,,+&+Z3)

The quantity a3 has a maximum value near one when there is a heavy residue a value of zero when the three fragments are of equal size.

and

We plot the mean value of a3 as a function of Gownd for C, Al, Cu and Pb targets in fig. 9. This observable has been calculated with the restriction that there are at least three fragments in t e event with Z b 2. There is a smooth trend from asymmetry to symmetry as a function of Gound and the data are similar for all four targets.

0.6

0

20

40

60

Z bound Fig. 9. The average value of the three-body asymmetry, (a,), as 600 MeV/nucleon collisions on C (circles), Al (triangles), Cu (squares) least three fragments with 2 2 2 have been selected. The error bars are size of the symbols. The lines are COPENHAGEN (dashed), GEMINI predictions.

a function of &,untl for Au and Pb (stars). Events with at in most cases smaller than the (dotted) and percolation {full)

E? Kreutz et al. / Charge correlations

683

In fig. 10, we quantify the ~bs~~ati~~ that tbe second and third largest fragments in each event are of similar size by plotting the relative difference between these charges. The relative difference is somewhat constant at a value of 0.25 for a wide range of Zbound.

3.2. CHARGE

MOMENT

ANALYSIS

It has been suggested by Campi 19V20 ) that charge distributions can be studied by the event-by-event moments. The kth charge moment is defined by

where the sum runs over all the fragments in the event, except the largest fragment. The largest fragment is excluded in analogy with the liquid condensate and the percolating cluster in the liquid-gas and percolation phase transitions, respectively. The following combination of moments 20)

Y2 =

can be re-expressed

as

m2xm0 2 ml

the normalized charge variance

0

0””

20

40

60 Z bound

Fig. 10. The average value of the relative asymmetry between the second and third largest charges in collisions on C (circles), Al {triangles), Cu rhe event as a function of Lund for Au 600 MeV/nucleon (squares) and Pb (stars). Events with at least three fragments with 2 3 2 have been selected. The error bars are in most cases smaller than the size of the symbols. The lines are COPENHAGEN (dashed}, ~~~1~1 (dotted) and ~r~oiat~on {fulti) predictions.

F! Kreutz

684

where o: is the variance mean

charge

et al. / Charge

correlations

of the charge distribution

of the event.

If all the charges

within

the event, and (Z), is the

in the event are the same size then y2

will reach

its lower limit of y2 = 1. This limit will be approached

reactions,

those

with light evaporated

and also for those reactions value of y2 indicates

fragments

(the heavy

that lead to a total disassembly

that the charges

for two types of

residue

is excluded)

of the system.

in the event are widely distributed.

A large

A measure

of this distribution, i.e. the size of the charge fluctuations, is the mean value of y2. In the scaling theory of critical phenomena, yz diverges in an infinite system near a critical point, at a rate that depends on the critical indices of the transition 19). Due to the finite size of a nuclear

system,

y2 is predicted

to show only a smooth

peak 19). In fig. 11 we plot ( y2) versus Zbound for the targets C, Al, Cu and Pb. For values This indicates that the of Zbound - 50, we observe a peak in the ( y2) distribution. have the largest normalized charge charges emitted in events with Gound -50 variance.

Again the data show similar behaviour for all four targets. For the lowest mathematically to be near a value of one. In our z, ound bins (y2) is constrained experiment, the height is reduced by our experimental threshold of Z 3 2. Jakobsson et al. ‘) have also observed a peaked ( y2) distribution for I60 induced reactions on stacks of emulsion. A suggested measure of critical behaviour in nuclei is the correlation between Z,,,,, and the moments of eq. (6) [refs. “32’)]. In fig. 12, we show a scatter plot of ln(Z,,,,,) versus ln(mz/m,), where critical events are expected to occur at the largest values of m?/ m, . The distribution of this correlation is quite broad and fills most

c

A y.

1.4

V 1.3

0

20

40

60

Z bound Fig. 11. The average value of yz as a function of Zhoun,, for Au 600 MeV/nucleon collisions on C (circles), Al (triangles), Cu (squares) and Pb (stars). Events with at least two fragments with 282 have been selected. The error bars are in most cases smaller than the size of the symbols. The lines are COPENHAGEN (dashed), GEMINI (dotted) and percolation (full) predictions.

I? G4 E q3 2 1

4 3 2 1 123

123

In(mZ/ml) Fig. 12. The measured correlation between In(&,,,) and In(m,/m,) for Au 600 MeV/nucleon collisions on C, Al, Cu and Pb targets. Events with at least two fragments with 222 have been selected and the solid lines represent the mathematical limits for a Z = 79 system.

of the available phase space witbi~ the mathematical fimits (salid lines in fig. 12). Similar correlations have been obse~ed for “0 induced reactions “) on emulsion and Au reactions on the plastic detector CR-39 “). We have calculated the average (In{ mz/ m,)) as a function of ln(Z,,,,,) for different values of sound. This is shown in fig. 13 where there are two reasonably well-defined branches in each of the plots. Jaqaman and Gross *‘) emphasize the need to remove symmetric fission events before calculating these averages. Our observation is that symmetric fission occurs mainly in the peripheral collisions and has little effect on the results obtained for Zbound< 70. The upper and lower ~~~n~h~~ of the correlations in fig. 13 have been described as containing sub- and super-critical events, r~s~ect~v~~y 20S2t). We wilt use this classi~cation~ though the reader should be aware that this language has been used mainly to describe percolation calculations and the terms may not be appropriate for a disassembling nuclear system. The slopes of these two branches have been related to the critical indices of a phase transition ‘OY2’).In our measurement, the slopes of these branches are affected by the lower charge threshold of 2 b 2; omitting the 2 = 1 particles leads to a larger value of na,/ m, for every event. Prom our data in fig, 13, the slopes of the two branches are similar, for a given cut in Gound, for each of the four targets. As a representative example, we list the extracted slopes for the Pb target in table 1. The errors are dominated by different e slopes of both choices of the range used to fit a straight line to the branches.

686

k? Kreutz et al. / Charge correlations

Z bound 0 A 0 0 v A W 0

1

2

3

1

2

70-79 60-69 50-59 40-49 30-39 20-29 10-19 (10

3

Fig. 13. The average value of In(m,/m,) as a function of In(Z,,,,, ) for Au 600 MeV/nucleon collisions on C, Al, Cu and Pb targets for the cuts in Z,,houndthat are listed on the right-hand side. Events with at least two fragments with Z 3 2 have been selected.

branches steepen to the decreasing A given cut in lower branches of of Z&,, , we have and super-critical value of Z,,, as sub-critical value.

for smaller values of Z bound, part of this effect is likely to be due size of the system 2’). Zbound contains exit channels that populate both the upper and the correlations. To illustrate the changing population as a function plotted in fig. 14 the ratio of the number of events in the sub-critical branches. This is achieved, for each cut in .&,und, by finding the

that produces the largest value of (ln(m,/m,)). if they have a larger Z,,,, and super-critical

For large values

of Zbound most of the exit channels TABLE 1

Extracted slopes of the In(Z,,,,,) versus (In(mJm,)) correlation for Au 600 MeV/nucleon on the Pb target

z,

“““d

70-79 60-69 50-59 40-49 30-39 20-29 10-19 o-9

Upper

branch

-0.3 ‘: : -0.4::; -0.6’: f -0.7? 0. I -1.11:,; -1.2:;; -3.1::: -5.3:::::

Lower branch 0.7::.; 1.0:: ; 1.3:;,; 1.9Y 0.2 2.1 +“.s 2.8’11:: 0.Z 4.0:::: 4.7::::

Events if Z,,,

are classified is below this

can be identified

as

687

P. Kreutz et al. / Charge correlations

lo2

,/,,

0

I

/I

25

,

,

50

,

,

,

75

Z bound Fig. 14. The ratio of sub- to super-critical events as a function of Z,,ound for Au 600 MeV/nucleon collisions on C (circles), Al (triangles), Cu (squares) and Pb (stars). The error bars are in most cases smaller than the size of the symbols. The lines are COPENHAGEN (dashed), GEMINI (dotted) and percolation (full) predictions.

sub-critical, whereas the most violent collisions lead to those labeled The ratio crosses the value of one at .Z,,ound-30.

super-critical.

4. Statistical and percolation models

4.1. MODEL

DESCRIPTIONS

Both the measured

velocity

distributions

of the fragments

4,23) and the fact that

many observables are independent of the target, indicate that a certain degree of equilibrium may have been obtained in the collisions. This has prompted us to explore a set of hybrid dynamical-statistical calculations where the BoltzmannUehling-Uhlenbeck model (BUU) 24S25)is used to describe the first stage of the collision. This provides an estimate of the size and the amount of energy deposited into the projectile spectator, which is then used as input to two statistical fragmentation models. Such hybrid calculations have been performed for other reactions 26-29). In this paper, it is our aim to show that the new observables are sensitive to the physics of nuclear disassembly and offer the possibility to discriminate between theoretical descriptions. The problem with this comparison is that there is a dependence on the initial conditions predicted by the dynamical model and hence we cannot, with certainty, conclude that one statistical model describes the decay of hot nuclei better than the others. We can however establish two results; (1) If the model predictions of our observables differ from each other, then the observables

688

P. Kreutz et al. / Charge correlations

are sensitive

to how each theoretical

i.e. the competition range of initial whether,

between

conditions

the measured

charge

different

predicted

with these initial

model

conditions,

populates

exit channels.

the available

phase

(2) We can explore

by the first stage of the hybrid either of the statistical

models

model

space,

a limited and test

can reproduce

correlations.

There are microscopic

dynamical

models

of fragment

formation,

for example

the

quantum molecular dynamics model 30). These models are qualitatively different than the statistical models used in this paper and their comparison with our data will be published in a future work. In the BUU calculations of this paper ‘5), a projectile spectator is defined as all the nucleons within a sphere in coordinate space. The position and size of this sphere is such as to include all the projectile nucleons that have yet to undergo a nucleon-nucleon collision. The deposited energy is calculated from the momentum and potential energy of all the nucleons within this sphere minus an estimate of the ground-state energy. The deposited energy/nucleon varies by less than 10% between 60 and 100 fm/c after the onset of the reaction. By 60 fm/c most of the fireball-like nucleons have left the reaction zone. An average deposited energy was calculated during this time interval (60-100 fm/c) for all impact parameters. Representative calculations were performed for Au+Cu at 600 MeV/nucleon for 1 fm intervals of impact parameter and the results of these are listed in table 2. The A/Z ratio for the spectator was taken to be the same as the gold projectile. Interpolations were made to provide a 0.1 fm impact parameter grid for the input to the statistical models, with the number of events calculated at each grid point increasing linearly with impact parameter. We have not fully explored how different parameters within the BUU model, e.g. compressibility, cross section, affect the predicted initial conditions. We have partially explored the target dependence (Cu, Al and C) of TABLE

2

The calculated size and deposited energy of the projectile spectator from the BUU model in Au + Cu collisions at 600 MeV/n’ucleon b

Spectator

(fm)

A, 2

10 9 8 7 6 5 4 3 2 1

189,75 181.73 166,67 150,60 133,53 113,45 98,39 84,34 13,29 67.27

size

Energy deposited (MeV/nucl.) 1.6 2.5 4.0 5.0 7.1 9.2 11.4 15.0 18.8 22.5

689

P. Kreutz et al. / Charge correlations

the predicted

spectator

properties.

For collisions

the BUU model predicts

that the deposited

for the different

Since the initial

would

predict

targets.

an approximate

We have taken

one model

Other formulations

of nuclear

that leave the same size spectator,

energy/nucleon conditions

target independence

each hybrid

10%

model

for the exit channels.

from each of the two major disassembly

is similar to within

are similar,

views of fragmentation.

will most likely have different

predictions.

It is also clear that well-founded modifications can be made to each of the models included in this paper that may improve the agreement with our measurements. The purpose of our calculations is to demonstrate the ability of the charge observables to discriminate between different models, and to investigate how well the current formulations reproduce our results. The GEMINI model 16) is a sequential nuclear decay and the COPENHAGEN model 14) is a simultaneous nuclear fragmentation. In order to compare with the data, the predicted events were filtered by a simple efficiency function ~(2)=1--2e-~, &(Z)=O,

222,

(9)

ZSl.

(10)

This is an approximate parameterization of the geometrical acceptance of the t.o.f.-wall ‘) which ignores any dynamical correlations. The major effect of this filter is to reduce the multiplicity of a-particles by 30% and hence to reduce the value of Gound by several units. The filter also reduces the value of the event-by-event charge moments, e.g. (yz) is smaller by approximately 10%. The COPENHAGEN model treats nuclear decay as a fast process, i.e. it evaluates the statistical phase space only once for a nucleus of given size and excitation energy. The partition function for this decay is calculated using a hot liquid-drop model. The density at the breakup is controlled by choosing a “cracking distance” parameter (d,,), and for simplicity we used the same cracking distance for all impact parameters. Our choice of d, = 0.4 fm corresponds approximately to a breakup density of p = 0.1 fm-3, which is similar to the density predicted by the BUU model for collision times between 60 and 100 fm/c. The cracking distance is smaller than previous values 14) and experimental distribution do = 1.2 fm leads

was chosen so that the predictions came close to the of (Z,,,) as a function of Gound. The normal choice of

to lower

values

of (Z,,,,,)

and

a larger

disagreement

with the

measured data. The primary distribution of fragments are allowed to decay on a slower timescale by evaporation, The version of the code is called CRACKER and includes a new treatment of the evaporation stage 31). The GEMINI code calculates the decay chain of a compound nucleus via sequential binary decays. The conditional barriers used to calculate the decay widths for heavy nuclei are taken from the finite-range liquid-drop model. The 5.0 version of GEMINI was used in all calculations with its default parameters. The value of the

P. Kreutz et al. / Charge correlations

690

level-density angular model,

parameter

momentum.

since the latter model

tions were also performed without

was a = A/8.5

This last condition

rotation

and

the decaying

was chosen

system

does not allow for rotating

systems.

at L = 50h. The results were qualitatively

with, for example,

was given

zero

to match the COPENHAGEN

the (MiMF) prediction

GEMINI similar

calculato those

being larger in the rotating

case by approximately 15%. The GEMINI model assumes that the decaying system has normal nuclear density. However the prediction of the BUU model is that the spectator at 60-100 fm/c has a lower density. Recent statistical calculations 32) have shown that the sequential decay is sensitive to the density of the system. One way to overcome this inconsistency is to match the GEMINI model to an earlier time in the collision. This has the problem that the fireball-like nucleons are in the same region of coordinate space as the projectile spectator and leads to a transient and, therefore, artificially high energy in the spectator region. For this paper, we will use the same matching conditions for the two statistical models, i.e. both COPENHAGEN and GEMINI have the same excitation and size input parameters. There are other difficulties in the matching between the dynamical and statistical models. Foremost is that the model dependency of the dynamical step implies that the conclusions on the success or failure of the statistical model cannot be too strong. In order to minimize this problem, all calculations will be plotted as functions in the excitation energy as predicted by the of Zbound. This reduces the uncertainty BUU model. We tested this by performing statistical calculations with input excitation energies that were 20% lower than the BUU prediction. The first order change in the observables in figs. 2-15 is a 10% shift to lower values of Zbound. This provides an estimate of the level to which quantitative comparisons can be made between the statistical models and the data. Fluctuations are also expected to occur around the mean values of the deposited energy and spectator size that are predicted by the BUU model. To test for the impact of these fluctuations, we performed statistical calculations

with

input

excitation

energy

that

varied

randomly

from

the BUU

predictions by up to f 10%. The results in figs. 2-15 changed less than 5%. Calculations have also been performed using a percolation model 33,34). On a three-dimensional lattice nucleons are represented as sites and excitation energy is modeled by allowing the bonds between the sites to be broken. For our experiment, the size of the system decreases with the violence of the collision. To account for this, sites were occupied randomly within a sphere that contained 197 sites in a 9 x 9 x 9 lattice. The probability to occupy a site (p,) varied between 0 and 1; the events with Zbound > 10 come from pS > 0.4. The lattice contains some clusters of occupied sites, but it was found that additional disturbance was needed to obtain similar numbers of clusters as the data. The value of the bond-existence parameter (qb) was fitted to the (Z,,,,,) distribution as a function of Zbound (fig. 4). The optimum fit was obtained for qb = 0.45. Other choices such as qb decreasing for smaller values

P. Kreurz er al. / Charge

“0

20

40

correlations

691

60

Z bound collisions Fig. 15. The average multiplicity of IMFs as a function of Z,,ound for Au 600 MeV/nucleon on C (circles), Al (triangles), Cu (squares) and Pb (stars). The error bars are in most cases smaller than the size of the symbols. The lines are COPENHAGEN (dashed), GEMINI (dotted) and percolation (full) predictions.

It is not our aim to claim any of pS also reproduce the measured Z,,,,, distribution. uniqueness of our choice. In converting the predicted masses of percolation to charges, we have taken a semi-empirical mass formula for A 3 4. For A = 3 fragments we take a 3H/3He ratio of 0.2 [ref. “)I. The fragments were passed through the same efficiency algorithm that was used for the statistical models.

4.2. COMPARISON

TO DATA

The GEMINI model predicts a steeper charge distribution extracted T parameter in fig. 2 is larger than the experimental

than our data and the result.for &_, > 25.

Related

to this is the observation from fig. 4 that GEMINI overpredicts (Z,,,,,) for the same range of Gound. It appears that the decay sequence leaves a too large residual nucleus. This is confirmed when the asymmetry between the two largest predicts that very charges in the event is plotted versus Zbound. In fig. 7, GEMINI asymmetric exit channels are most probable for a wide range of collisions. This is in contrast to the more symmetric exit channels seen in the data. GEMINI also predicts a larger three-body asymmetry than is present in the data (fig. 9), and too small a relative asymmetry between the second and third largest charges in the event (fig. 10). The calculated values for ( yz) are significantly too low for all values of &ound, i.e. the emitted charges show too small a normalized variance within each event. There is also no indication of a peak within the range of Gound shown in this distribution (fig. 11). Over a wide range of Go_, values GEMINI

692

predicts

p. Krelltz ef al. / Charge ~ar~~~~~j~~s

more sub- than super-~~ti~a~

a small IMF rn~ltipl~~ity

events as is seen in fig. 14, ~E~l~~

(fig. 15) which is a similar

result to that found

predicts by Bowman

et al. “1. The COPENHAGEN fragments

model produces

in the mid-peripheral

too many of the heavier

collisions.

intermediate-mass

This is seen from a low value

of the

predicted T parameter at qound > 35 seen in fig. 2. The maximum charge in the event is underpredicted by the model for Gound ~45. It should be noted that increasing the cracking distance in the model leads to a smaller value of (Z,,,). Because of a larger probability for heavier IMFs and a small maximum charge in the event, the COPENHAGEN model predicts too small an asymmetry between the largest two fragments. This is seen in fig. 7, where the predicted mean asymmetry drops rapidly with decreasing values of GoLInd. A similar behaviour is seen for the three-body asymmetry shown in fig. 9, namely that the model predicts decays that are too symmetric. The opposite occurs for the highest values of Z&,d, namely the calculations predict very asymmetric exit channels, possibly because the model calculates an insufficient number of fission-like events. The COPENHAGEN model produces a peak in the (y2) distribution at nearly the right position and height but slightly underpredicts the width of this peak. The model predicts a very rapid change in the ratio of sub- to super-critical events as is seen from fig. 14. For the more central collisions there are too many events that lie on the s~p~r-~rit~~al branch. The p~rcolat~o~ model was tuned function of Zbound. The predictions

to reproduce for the other

the distr~b~tio~ of (Zm,,) as a charge correlations were then

compared to data, The predicted charge distribution is very similar to the data, as is seen from the extracted 7 values in fig. 2, The two- and three-body asymmetries of figs. 7, 9 and 10 agree with the data over the full range of Zbound. At the next level of complexity, the predicted peak of (yJ is slightly lower than the measured data, and the agreement with the ratio of sub- to super-critical events is reasonable. The percolation model reproduces very well the shape and magnitude of the (MlMF) distribution.

We have measured the multi-fragment decays of Au projectiles after collisions with C, Al, Cu, and Pb targets at a bombarding energy of 600 MeVfnucleon. In this paper, we have developed a series of observables that progressed in complexity from charge distributions, to the relative sizes of the charges, and finally to an event-by-event moment analysis. These observables were examined as a function of the violence of the collision and span the region from evaporation to total disassembly of the nuclear system. One of the striking results of this paper is the successful ordering of the data with z, ound ’ This quantity is related to both the size and excitation energy of the decaying

f? Kreutz et al. / Charge correlations

system, and all our observables

show a behaviour

693

which is independent

of the target

by Hubele et al. ‘) this indicates that the when plotted versus Zbound. As mentioned decay of the system is independent of how it is formed, and this is a necessary but not sufficient requirement that the system information on the question of equilibrium energy

can only be obtained

is thermalized before and the delocalisation

from the velocity

observables.

it decays. Further of the deposited

This is an important

topic for future research. The measured charge distributions resemble a power law distribution and the extracted T parameters are large for both evaporation-like events and for total disassembly of the system. The r parameter exhibits a minimum at Gound - 35. The exit channel is further characterized by the maximum charge and the two- and three-body asymmetries. These correlations

of the event, Z,,,,,, quantify how the

breakup evolves from a heavy-residue and evaporated fragments in peripheral reactions to more symmetric exit channels in the most violent collisions. The charge-Dalitz plots indicate that the multi-fragmentation events are an extension of evaporation-residue exit channels and are not an evolution of symmetric fission events. Several of our observables are motivated by studies of critical phenomenon in finite-sized systems. The ( yz) distribution has a clear peak at values near &,und - 50, which indicates that the variation of the emitted charges is greatest for this class of events. The correlation between ln(Z,,,,,) and (ln( mZ/ m,)) has two clear branches, which is used to label the events as sub- and super-critical exit channels. For a given value of Gound there are both sub- and super-critical events, with an equal number of these two types occurring at its minimum.

at Zbound -35.

This is similar

to the value where

The predictions of two nuclear models (GEMINI a sequential decay, HAGEN a simultaneous breakup) differ from each other, establishing

7 is

COPENthat the

observables are sensitive to how the available phase space is populated. The charge correlation observables offer different insights into the nature of a multi-fragment event, though they are not necessarily independent of each other. The usefulness of the full set of observables

lies not only in increasing

our understanding,

but also

in the fact that a given model may reproduce some observables but not others. This is illustrated in the case of the COPENHAGEN model in the region of &,und - 35. The agreement with the measured charge distribution of light fragments, the normalized variance ( y2) and the multiplicity of IMFs is reasonable, however the prediction of (Z,,,,,) is 50% too small. This example also illustrates that a given disagreement affects other observables, e.g. the prediction of a small (Z,,,,,) implies that the asymmetry observables are smaller than the data. The GEMINI model fails to reproduce many of our charge correlation measurements, and the agreement with data for the most peripheral collisions is only fair. In general, the model predicts highly asymmetric decay channels. The GEMINI model predicts too low a multiplicity of IMFs. The model dependence of the

694

P. Kreutz et al. / Charge correlations

matching mately

to the dynamical a 15-20%

model,

uncertainty

than the level of disagreement tively independent GEMINI decay.

energy,

charge

with the data, therefore

of the matching

is one implementation

It is not possible

e.g. excitation

in the predicted

corresponds

correlations.

to approxi-

This is smaller

our conclusions

are qualita-

conditions. of the class of models

to rule out this whole

that assume

class of models

a sequential

on the basis of the

calculations presented in this paper. However, any other sequential model must solve the current problem that too large a heavy residue survives the decay chain. The COPENHAGEN model comes closer to reproducing the data. In the midperipheral collisions, most of the disagreement comes from the prediction of a higher probability for the heavier IMFs. For more central reactions, the model predicts too high a probability for the emission of light fragments and alpha particles. In this region, highly symmetric decays are predicted. The COPENHAGEN mode1 predicts a maximum in the (~3 distribution and provides a reasonable reproduction of the IMF multiplicity. These conclusions are qualitatively independent of the uncertainties in the matching to the BUU calculations. It is intriguing that the percolation model, after parameter adjustment, shows nearly perfect agreement with the data. This model contains a well-defined phase transition for large systems and for smaller systems is still expected to show critical behaviour. The percolation parameters were chosen to reproduce the average value then of Z,,, as a function of Zhound. It seems that when this quantity is reproduced, the finite-size percolation model correctly predicts the mean and fluctuating behaviour of the light and intermediate mass fragments. All our measured data, especially the minimum of the 7 parameter, the peak in the (yJ distribution, and the changing population of sub- to super-critical events are consistent with the observation of a smoothed, percolation-like critical behaviour langugage, as we span the at moderate values of Z;lnund. In more thermodynamic region from evaporation to complete vaporization, one might expect to find some manifestation of a nuclear liquid-gas phase transition. At this stage, it is not clear what constitutes

definitive

evidence

for such a transition.

The strongest

is that the peak in the (yJ distribution is reasonably reproduced HAGEN model which contains the physics of a phase transition,

observation

by the COPENwhilst no peak is

predicted by the GEMINI model. This latter model does not have a clearly defined phase transition. Our measurements quantify which charges are emitted in multi-fragment reactions and how these charges are correlated with each other. The observables provide the possibility to discriminate between models that have different treatments of nuclear disassembly. The level of disagreement with the current nuclear statistical models may stimulate improvements to these models, or may lead to the necessity of a full dynamical description of the production of fragments. In either case, charge correlations probe the available phase space in nuclear disassembly and provide a challenge for both future experiments and theoretical predictions.

P. Kreutz et al. / Charge correlations

695

The authors would like to thank W. Bauer for the use of his BUU code, H.W. Barz and H. Schulz for the COPENHAGEN code and R. Charity for the GEMINI code. This work was partially supported by the Bundesministerium fiir Forschung and Technologie.

References 1) B. Jakobsson, G. Jonsson, L. Karlsson, V. Kopljar, B. Noren, K. Soderstrom, F. Schlusser, E. Monnand, H. Nifenecker, G. Fai, J. P. Bondorf and K. Sneppen, Nucl. Phys. A 509 (1990) 195 2) J.W. Harris, B.V. Jacak, K.-H. Kampert, G. Claesson, K.G.R. Doss, R. Ferguson, A.I. Gavron, H.-A. Gustafsson, H. Gutbrod, B. Kolb, F. Lefebvries, A.M. Poskanzer, H.G. Ritter, H.R. Schmidt, L. Teitlebaum, M. Tincknel, S. Weiss, H. Wieman and J. Wilhelmy, Nucl. Phys. A 471 (1987) 241~ 3) C.A. Ogilvie, J.C. Adloff, M. Begemann-Blaich, P. Bouissou, J. Hubele, G. Imme, 1. lori, P. Kreutz, G.J. Kunde, S. Leray, V. Lindenstruth, Z. Liu, U. Lynen, R.J. Meijer, U. Milkau, W.F.J. Miiller, C. Ng8, J. Pochodzalla, G. Raciti, G. Rudolf, H. Sann, A. Schiittauf, W. Seidel, L. Stuttge, W. Trautmann and A. Tucholski, Phys. Rev. Lett. 67 (1991) 1214 4) J. Hubele, P. Kreutz, J.C. Adloff, M. Begemann-Blaich, P. Bouissou, G. Imme, I. Iori, G.J. Kunde, S. Leray, V. Lindenstruth, Z. Liu, U. Lynen, R.J. Meijer, U. Milkau, A. Moroni, W.F.J. Miiller, C. Ng8, C.A. Ogilvie, J. Pochodzalla, G. Raciti, G. Rudolf, H. Sann, A. Schiittauf, W. Seidel, L. Stuttge, W. Trautmann and A. Tucholski, Z. Phys. A340 (1991) 263 5) D. Bowman, G.F. Peaslee, R.T. de Souza, N. Carlin, C.K. Gelbke, W. G. Gong, Y. D. Kim, M.A. Lisa, W.G. Lynch, L. Phair, M.B. Tsang, C. Williams, N. Colonna, K. Hanold, M.A. McMahan, G.J. Wozniak, L.G. Moretto and W.A. Friedman, Phys. Rev. Lett. 67 (1991) 1527 6) R. Trockel, K.D. Hildenbrand, U. Lynen, W.F.J. Miiller, H.J. Rabe, H. Sann, H. Stelzer, W. Trautmann, R. Wada, E. Eckert, P. Kreutz, A. Kiihmichel, J. Pochodzalla and D. Pelte, Phys. Rev C39 (1989) 729 7) Y. Blumenfeld, N. Colonna, P. Roussel-Chomaz, D.N. Delis, K. Hanold, J.C. Meng, G.F. Peaslee, Q.C. Sui, G.J. Wozniak, L.G. Morreto, B. Libby, A.C. Mignerey, G. Guarino, N. Santoruvo and I. Iori, Phys. Rev. Lett. 66 (1991) 576 8) S.J. Yennello, E.C. Pollacco, K. Kwiatkowski, C. Volant, R. Dayras, Y. Cassagnou, R. Legrain, E. Norbeck, V.E. Viola, J.L. Wile and N.R. Yoder, Phys. Rev. Lett. 67 (1991) 671 9) J. Aichelin and X. Campi, Phys. Rev. C34 (1986) 1643 10) H.H. Gutbrod, A.M. Poskanzer and H.G. Ritter, Rep. Prog. Phys. 52 (1989) 1267 and references therein 11) G. Bertsch and P.J. Siemens, Phys. Lett. B126 (1983) 9 12) C.J. Pethick, and D.G. Ravenhall, Nucl. Phys. A471 (1987) 19~ 13) D.H.E. Gross, Rep. Prog. Phys. 53 (1990) 605 and references therein 14) J.P. Bondorf, R. Donangelo, I.N. Mishustin, C.J. Pethick, H. Schulz and K. Sneppen, Nucl. Phys. A443 (1985) 321 15) W.A. Friedman and W.G. Lynch, Phys. Rev. C28 (1983) 16 16) R.J. Charity, M.A. McMahan, G.Z. Wozniak, R.J. McDonald, L.G. Moretto, D.G. Sarantites, L.G. Sobotka, G. Guarino, A. Pantaleo, L. Fiore, A. Gobbi and K.D. Hildenbrand, Nucl. Phys. A483 (1988) 371 17) B. Lott, S.P. Baldwin, B.M. Szabo, B.M. Quednau, W.U. Schriider, J. Toke, L.G. Sobotka, J. Barreto, R.J. Charity, L. Gallamore, D.G. Sarantites, D.W. Stracener and R.T. de Souza, Phys. Rev. Lett. 68 (1992) 3141 18) E. Piasecki, S. Bresson, B. Lott, R. Bougault, J. Colin, E. Crema, J. Galin, B. Gatty, A. Genoux-Lubain, D. Guerreau, D. Horn, D. Jacquet, U. Jahnke, J. Jastrzebski, A. Kordyasz, C. Le Brun, J. F. Lecolley, M. Louvel, M. Morjean, C. Paulot, L. Pienkowski, J. Pouthas, B. Quednau, W.U. Schroder, E. Schwinn, W. Skulski and J. Take, Phys. Rev. Lett. 66 (1991) 1291 19) X. Campi, J. of Phys. Al9 (1986) L917 20) X. Campi, Phys. Lett. B208 (1988) 351

696

p. Kreutz &i ai. / Charge ~~r~~~~~~o~~

21) H.R. Jaqaman and DIKE. Gross, Nucl. Phys. A524 (1991) 321 225 C. Lewenkapf, J_ Dreute, A. Abdul-Magd, J. Aj~h~Iin~ W. Heinrich, J. Hiifner, G. Rusch and B. Wiegel, Phys. Rev C44 (1991) 1065 23) J. Hubele, P. Kreutz, V. Lindenstruth, J.C. Adloff, M. Begemann-Blaich, P. Bouissou, G. Imme, I. Iori, G.J. Kunde, S. Leray, Z. Liu, U. Lynen, R.J. Meijer, U. Milkau, A. Moroni, W.F.J. Miiller, C. Ng6, C.A. Ogilvie, J. Pochodzalla, G. Raciti, G. Rudolf, I-I. Sann, A. Schiittauf, W. Seidel, L. Stuttge, W. Trautmann, A. Tucholski, R. Heck, A.R. DeAngelis, D.H.E. Gross, H.R. Jaqaman, H.W. Batz, H. Schulz, W.A. Friedman and R.J. Charity, submitted for publication 24) C.F. Bertsch and S. das Gupta, Phys. Reports 160 (1988) 189 25) W. Bauer, private communication (1991) 26) K. Sneppen and L. Vinet, Nucl. Phys. A 480 (1988) 342 27) A.S. Botvina, AS. Iljinov and I.N. Mishustin, NucI. Phys A507 (1990) 649 28) S. Leray, C. Ngci, M.E. Spina, B. Remaud and F. SebilIe, Nucl. Fbys A511 (1990) 414 29) M. Colonna, P. mousses-Cboma2, N. Colonna, M. Di Tore, L.G. Moretto and G.J. Wozniak Phys. 30) 31) 32) 33) 34) 35)

Lett. 283 (1992) 180 J. Aicbelin, Phys. Reports 202 (1991) 233, and references therein H.W. Barz and II. Schulz, private communication (1991) W.A. Friedman, Phys. Rev. C42 (1990) 667 J. Desbois Nucl. Phys. A466 (1987) 724 W. Bauer, Phys. Rev. C38 (1988) 1297 K.G.R. Doss, W.A. Gustafsson, H.H. Gutbrod, D. Hahn, K.H. Kampert, B. Kolb, H. Lohner, Poskanzer, H.G. Ritter, H.R. Schmidt and H. Stijcker, Phys. Rev. C37 (1988) 163

A.M.