The initial production and dynamical expansion of hot spherical nuclei are examined as the first stage in the projectile-multifragment ation process. The initial temperatures, which are necessary for entering the adiabatic spinodal region, as well as the minimum temperatures and densities, which are reached in the expansion, significantly differ for hard and soft equations of state. Additional initial compression, occurring in central collisions leads most likely to a qualitatively different multifragmentation mechanism. Recent experimental data are discussed in relation to the results of the proposed model

Nörenberg, W

Papp, G

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The Path of Hot Nuclei Towards Multifragmentation
G. Papp
a;1
and W. Norenberg
a;b
a
Gesellschaft fur Schwerionenforschung, D-64220, Darmstadt, Germany
b
Institut fur Kernphysik, TH Darmstadt, D-64289, Darmstadt, Germany
Abstract
The initial production and dynamical expansion of hot spherical nuclei are ex-
amined as the rst stage in the projectile-multifragmentation process. The initial
temperatures, which are necessary for entering the adiabatic spinodal region, as well
as the minimum temperatures and densities, which are reached in the expansion,
signicantly dier for hard and soft equations of state. Additional initial compres-
sion, occurring in central collisions leads most likely to a qualitatively dierent
multifragmentation mechanism. Recent experimental data are discussed in relation
to the results of the proposed model.
Keywords:
PACS: 25.70Mn
1
On leave from the Institute for Theoretical Physics of Eotvos University, H-1088
Budapest, Hungary
GSI-Preprint 95-30, subm. to Heavy Ion Physics
1 Introduction
New results [1,2] on projectile multifragmentation in relativistic heavy-ion
collisions indicate relations to the liquid-gas phase transition of nuclear mat-
ter. Motivated by these experiments we like to substantiate this conjecture
by studying the evolution of the excited projectile residue into the region of
instability.
In the following we report results on the expansion dynamics of hot spherical
nuclei. We estimate the initial excitation energy from the abrasion model
[3,4] using modications introduced in [5]. The dynamical expansion of the
heated spherical residue is calculated taking the evaporation of nucleons into
account [6].
The expansion of the projectile residue can be regarded as the rst stage in the
multifragmentation process. For large enough initial excitation energies and
temperatures the nuclear matter of the residue reaches states of mechanical
volume instability, where small density uctuations grow exponentially [7{9].
For an illustrative analysis of the instability evolution we refer to ref. [10].
The fragmentation process itself has been studied within molecular dynam-
ics [11{14]. Studies by Friedman [15,16] within an expanding emitting source
model (EES) show that intermediate mass fragments (IMFs) are indeed cre-
ated within time intervals of about 50 fm/c indicating a simultaneous breakup
of the residue.
2 The Model
In this section we describe a few essential features of the model. A more
detailed description of the model will be published elsewhere [17].
We start out from a thermalized spherically symmetric projectile-like nucleus
at ground state density, which has been created in the peripheral high-energy
collision (cf. [18]). The typical temperature for such a hot nucleus is not too
high, such that the mean free path of nucleons is still comparable or even
larger than the size of the system [6]. Thus, we can approximately assume
homogeneity in density and temperature for the expanding nucleus.
During the expansion the hot nucleus evaporates particles. We consider only
neutron and proton evaporation. Deuterons may be considered to arise from
coalescence of nucleons.
If the excitation energy is high enough, the expansion leads into the instabil-
2
ity region of the equation of state. For not too high excitation energies the
expansion of the homogeneous system stops at a certain density. Around these
turning points the system has enough time ( 30 fm/c, [7,9]) to develop in-
homogeneities in density, and subsequently decay into many fragments. The
nal mass and charge spectra is not given by this model, but the calculated
turning-point parameters (temperature and density) can be applied to static
statistical models [19,20] and related to experimental results (cf. sect. 3). In
a complete dynamical treatment of multifragmentation the turning points are
the initial states for the decay into the nal fragments.
2.1 Abrasion
The starting point in our calculations is the determination of the projectile
residue (prefragment) which is produced in the heavy-ion collision. A simple
picture for relativistic collisions is obtained by assuming that the particles in
the overlap zone of the two colliding nuclei are stopped, while the spectators
are propagating with their initial velocities. In the original abrasion model
[4] the excitation energy of the residues is calculated from the change of the
surface. For this geometrical abrasion process simple expressions for the mass
and energy of the participants and spectators have been given.
Gaimard and Schmidt [5] have improved this model by calculating the ex-
citation energy of the residue from the hole-state creation with 13.3 MeV
excitation energy in average for each abrased particle:
E

=  A;  = 13:3MeV: (1)
This prescription is in good agreement with the results of BUU calculations
[21] , and yields the excitation energy of the projectile independent of the
target. At small values of the mass loss this picture underestimates the BUU
prediction of the excitation energy, and a higher value of  = 26:6 MeV was
suggested. Also recent calculations of Campi et. al. [22] suggest somewhat
higher value for the excitation energy. We have used both values in our calcu-
lations. The charge-to-mass ration is assumed not to change by the abrasion
process.
Furthermore we assume, that after the abrasion step the system equilibrates
fast. Estimating the equilibration times according to ref. [23], we nd indeed
values which are small compared to the expansion times.
3
2.2 Expansion
Due to the supposed homogeneity in density, the radial ow is of the form
~v(t; ~r) = a(t)~r with a = 0 initially. All the other variables { the density %(t),
the temperature T (t), the mass number A(t) and the charge Z(t) depend only
on time.
The evaporation of nucleons yields the time evolution of mass and charge,
and is the only source for loss of energy and entropy. The evaporation integral
is calculated according to [6], and evaluated numerically. In this evaporation
model we need to know the mean free path not only at nuclear matter density,
but at all densities reached during the expansion. We assume that the density
dependence of the in-medium cross-section in the range of 0:3%
0
to %
0
(normal
nuclear density) is essentially due to the Pauli-blocking factor. We found that
the contribution from the Pauli eect is inversely proportional to the density
for the temperatures, energies and densities in question. Therefore, the mean
free path is considered here as independ on the density.
The equation of motion is determined from energy and entropy conservations
with loss terms which are due to the evaporation of nucleons.
The expansion is studied for a soft and a hard equation of state using, respec-
tively, the Skyrme forces SkM

and SIII, cf. [24]. We follow the time evolution
up to the turning point, where the density reaches its minimum. The system
spends most of its time around the turning point, and hence the fragmentation
process is expected to start here. For too low excitations the system cannot
enter the %; T region of instability, and hence a heavy residue will remain.
For high excitations no turning point is encountered in the expansion and,
therefore, the system is expected to explode [25].
3 Results and Discussion
As an example of projectile multifragmentation we consider the reaction Au
(600 MeV/u { 1 GeV/u) +X with X=C,Al,Cu which has been studied exper-
imentally by the ALADIN group [1,2].
3.1 Expansion trajectories
According to the abrasion model (subsect. 2.1), the initial excitation energy
(temperature) is correlated with the charge number of the projectile residue.
4
r]
024681012
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
Temperature (MeV)
ρ/
ρ 0
47
.5
53
.5
57
60
.2
63
.3
68
.9
73
.4
75
.2 76
.7
30
.3
35
.9
39
.6
44
.7
48
.3
53
.9
57
.3
60
.5
63
.5
66
.3
68
.9
71
.3
73
.4
75
.2
76
.7
Fig. 1. Time evolution in the (T; %)-plane of the Au residue for a soft equation of
state (SkM

). The crosses on the trajectories indicate time steps of 5 fm/c. The
numbers give the charge numbers initially and at the turning points which are
connected by the heavy solid line. The adiabatic spinodal region is shadowed. The
open boxes denote the experimental data [2] (without error bars).
These numbers are given in Figs. 1 and 2. The calculated expansion trajec-
tories in the (T; %)-plane are signicantly dierent for a soft (Fig. 1) and a
sti (Fig. 2) equation of state (EOS). They deviate from adiabats due to the
evaporation of one to four nucleons. The number of evaporated nucleons is so
small because of the fast cooling by the expansion of the system.
We follow the trajectories up to their turning points if reached within 200 fm/c.
The turning points are indicated by the heavy solid lines. Turning points exist
for initial temperatures up to about 14 MeV. We notice that the turning points
for the same initial excitation lie one to two MeV in temperature and 0:1%
0
to
0:2%
0
in density higher for the sti EOS as compared to the soft one. Moreover,
we see that the turning points are located almost at the same temperatures
5
r]
024681012
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
Temperature (MeV)
ρ/
ρ 0
46
.4
50
.1
53
.6
57
.
60
.3
66
.2
71
.3 7
3.
4 75
.2 76
.7
30
.3
35
.9
39
.6
42
.5
46
.9
50
.5
53
.9
57
.3
60
.5
63
.5
66
.3
69
.0
71
.3
73
.4
75
.2
76
.7
Fig. 2. Same as Fig. 1 for a sti equation of state (SIII)
independent of the initial excitation, i.e. around 4 MeV and 5 MeV for the
soft and hard EOS, respectively. This plateau does not move if one changes
the initial excitation energy (the  parameter in (1)), the only eect is, that
the charge numbers become dierent initially and at the turning points.
3.2 Multifragmentation
A natural criterion for multifragmentation to occur, is whether the system
reaches the region of volume instability (adiabatic spinodal region), where the
derivative of the pressure with respect to the density along an adiabat becomes
negative. Since the system is closed we consider the adiabatic process to be the
relevant one. In Figs. 1 to 3 the region of instability is shadowed and bounded
by the spinodal. To enter the spinodal region and stay there for more than 30
fm/c one needs initial temperatures of about 8 MeV and 11 MeV for the soft
6
r]
024681012
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
Temperature (MeV)
ρ/
ρ 0
in
iti
al
 d
en
sit
y 
1.
5 
ρ 0
Fig. 3. Same as Fig. 1 with additional initial compression to 1.5 %
0
for the soft EOS
and hard EOS, respectively.
Because of the occurrence of turning points in the expansion, the multifrag-
mentation process of the projectile residue has a unique feature: the subsequent
decay into fragments is expected to be rather free from collective ow. This
is dierent from the fragmentation of compressed compound systems which
are formed in central collisions. As shown in Fig. 3, already for a small ini-
tial compression (1.5 %
0
) no turning points occur for realistic initial excitation
energies. Thus the fragmentation process becomes qualitatively dierent and
looks more like an explosion [25].
7
3.3 Comparison with experimental results
Pochodzalla et al. [2] have determined the temperature of the nal break-up
into fragments as function of the initial excitation energy. In Figs. 1 and 2 we
have plotted the experimental break-up temperatures on the trajectories (or
their extrapolations beyond the turning points) for the corresponding initial
temperature. The experimental errors in the excitation energy transform into
errors in the density of about 0:06%
0
. We observe the following features.
{ The onset of multifragmentation is around 8 MeV of initial temperature in
agreement with the soft EOS.
{ The experimental points, which are related to initial temperatures between
6 MeV and 8 MeV, correspond to large values for the maximum charge of
the fragments, and hence suggest an evaporation-like process.
{ For the sti EOS no reasonable picture is obtained which would be consis-
tent with the experimental results.
{ There is some indication from molecular dynamics [11,12] that after a rel-
atively fast expansion close to adiabats the system follows a path close to
T = const with some tendency to increase the temperature. Assuming that
this is correct we would also conclude that the soft EOS gives results which
are consistent with experiment, whereas the sti EOS does not t.
{ For initial temperatures larger than about 12 MeV to 14 MeV the system
reaches densities well below 0:3%
0
before approaching the turning point.
At these densities collisions between nucleons become rare, and hence this
temperature is expected to be observed in the nal fragments.
4 Summary and Conclusion
We have studied the expansion of hot nuclei for a soft and a sti equation
of state. Turning points are encountered for initial temperatures smaller than
about 14 MeV. Explosive events occurs for initial temperatures larger then 12
MeV to 14 MeV.
Projectile fragmentation appears to be the optimal process for the study of
the phase-transition region, because there is no compression involved in the
formation of initial state. Already small additional compression, as expected
in central collisions, lead to explosion.
The occurrence of turning points in projectile fragmentation suggests to divide
the multifragmentation process into two steps. The rst step is approximately
described by the expansion of a homogeneous nuclear drop, because the time
necessary for the development of instabilities is too large. The second step
8
starts from the turning point and leads to a relatively slow further expansion
by developing inhomogeneities. This slow evolution may be the reason why
equilibrium models [19,20,22] describe the fragment distribution quite well.
Comparison with experimental results, obtained by the ALADIN collabora-
tion [2], gives evidence that the soft equation of state is more realistic. In
particular, the onset of multifragmentation at around 8 MeV initial temper-
atures is only consistent with the soft equation of state. Furthermore, the
break-up into fragments around a freeze-out density of %=%
0
= 0:3 is indicated
for initial temperatures higher than about 12 MeV to 14 MeV.
We conclude that further dynamical studies of the fragmentation process and
comparison with experimental results may yield more precise information
about the equation of state in the low-density region. In particular dynami-
cal treatments of the fragmentation process starting at the turning points are
needed.
Acknowledgement
We would like to thank G. Bertsch, U. Lynen, W.F.J. Muller, J. Pochodzalla
and W. Trautmann for the fruitful discussions.
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The path of hot nuclei towards multifragmentation

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