# Isospin compositions of correlated sources in the Fermi energy domain

R. Ogul  Department of Physics, University of Selçuk, TR-42079 Konya, Turkey
GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
A. S. Botvina  Institute for Theoretical Physics, J. W. Goethe University,
D-60438 Frankfurt am Main, Germany  Helmholtz Research Academy Hesse for FAIR
(HFHF), Max-von-Laue-Str. 12, D-60438 Frankfurt am Main, Germany  M. Bleicher
GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
Institute for Theoretical Physics, J. W. Goethe University, D-60438 Frankfurt
am Main, Germany  Helmholtz Research Academy Hesse for FAIR (HFHF), Max-von-
Laue-Str. 12, D-60438 Frankfurt am Main, Germany  N. Buyukcizmeci  Department
of Physics, University of Selçuk, TR-42079 Konya, Turkey  A. Ergun  Department
of Physics, University of Selçuk, TR-42079 Konya, Turkey  H. Imal  Department
of Physics, University of Selçuk, TR-42079 Konya, Turkey  Y. Leifels  GSI
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  W.
Trautmann  GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291
Darmstadt, Germany

###### Abstract

Isotopic yield distributions of nuclei produced in peripheral collisions of
80Kr+40,48Ca at 35 MeV/nucleon are studied. Experimental results obtained by
the FAZIA Collaboration at the LNS facility in Catania are compared with
calculations performed with the statistical multifragmentation model (SMM).
The fragments with atomic number $Z=19-24$ observed at forward angles are
successfully described with the ensemble method previously established for
reactions at higher energy. Using the SMM results, the isotopic compositions
of the projectile residues are reconstructed. The results indicate a
significant isospin exchange between the projectile and target nuclei, not far
from isospin equilibrium, during the initial phase of the reaction. The two
groups of light fragments with $Z=1-4$, experimentally distinguished by their
velocities relative to coincident heavy projectile fragments, are found to
originate from different sources. The isotopic composition of the slower group
is consistent with emission from a low-density neck, enriched in neutrons, and
satisfactorily reproduced with SMM calculations for a corresponding neck
source of small mass.

###### pacs:

25.70.Mn,25.70.Pq,21.65.Ef

##  I INTRODUCTION

There is continuing high interest in investigating the nuclear fragment
production in heavy-ion collisions of intermediate energies dyntherm ; nusyme
. It is a typical many-body process in which the nuclear interaction can
manifest its ingredients and lead to important states of matter. The fragment
production mechanisms, the properties of nuclear matter at both high and low
densities, new reaction products, e.g., exotic isotopes, and their
correlations can be addressed nusyme ; exotic . The theoretical explanations
of these phenomena include various methods involving dynamical and statistical
approaches. In such complicated many-body processes, phenomena governed by
interactions of both, high and low energy coexist. Their description may go
beyond the capabilities of one specific model. It is, therefore, important to
determine correctly the regions of applicability for each model and to explore
alternatives to reach a comprehensive picture.

The technical achievements of the FAZIA Collaboration have widened the
possibilities for reaction studies at intermediate energies FAZIA21 ;
camaiani21 ; ciampi22 . By extending the limits of mass and charge resolution
of telescope-type detection units, arrays with large solid-angle coverage have
become available for new experiments exploring the role of isospin in heavy-
ion collisions bougault14 ; pastore17 . Isospin transfer between the reaction
partners reflects the action of the symmetry parts of the nuclear forces
nusyme ; baran05 ; lipr08 . Studying their effects requires isotopic
identification of the reaction products. It has been known for a long time
that product distributions respond to the isotopic properties of the primary
source, whose composition, however, may be concealed by secondary-decay
effects porile64 ; kukarol77 .

In a first experiment addressing isospin transport at intermediate energies,
the FAZIA Collaboration studied the fragment production in $\rm{}^{80}Kr$
collisions with $\rm{}^{48}Ca$ and $\rm{}^{40}Ca$ targets at 35 MeV/nucleon
beam energy FAZIA21 . Four detector cubes equipped with the FAZIA technology
were placed at forward angles. The chosen inverse kinematics had the effect of
focusing projectile fragments into forward directions efficiently covered by
the FAZIA modules.

The analysis of the data was aimed at extracting the density dependence of the
symmetry-energy term in the nuclear equation of state. For this purpose, the
antisymmetrized molecular dynamics (AMD) transport model and the statistical
decay code GEMINI were used and “a weak indication in favor of a stiff
symmetry energy” was reported. At densities below saturation, the stiff
solution corresponds to lower values for the symmetry energy. However,
comparison with the experimental data shows that the isotopic composition of
the observed fragments is not much influenced by the symmetry-energy term and
depends mainly on the statistical processes at the later reaction stages. For
this reason, we believe it is instructive to perform an alternative analysis
of the experimental results in order to clarify the role of the statistical
breakup and decay in the formation of the final products.

The present work attempts an interpretation of the same data aiming at the
isospin transfer during the contact time between the reaction partners. The
isotopic composition of the primary reaction products will be determined from
the observed products by modeling the deexcitation of the former with the
statistical multifragmentation model (SMM, Ref. SMM ). The SMM is capable of
following various reaction scenarios from compound evaporation through fission
and multifragmentation up to vaporization according to the statistical weights
given to these paths in the model description. It thus represents a rather
universal model which was previously successfully applied for the description
of the fragment production in peripheral and central heavy ion collisions at
relativistic energies Bot95 ; Xi97 ; Ogul11 ; Imal15  and in the Fermi energy
domain (Refs. MSU ; INDRA ; TAMU ; Bot06  and references given therein).

To describe the projectile fragments, the calculations start from an ensemble
of excited sources whose properties, mainly the isotopic composition, is
varied until an optimum description of the measured isotopic distributions for
projectile fragments with atomic numbers $Z=19-24$ is achieved. The peripheral
character of the data has encouraged us to attempt an extrapolation of the
method found useful for reactions at higher energies Bot95 ; Ogul11  to the
present case. It differs from an earlier theoretical study of peripheral 112Sn
+ 124Sn and 124Sn + 112Sn reactions in which sources with fixed mass and
charge were used as initial configurations for SMM calculations Imal20 .

Normalized relative yields of light fragments from the same experiment are
also presented in the FAZIA publication FAZIA21 . They were recorded in
coincidence with a heavy fragments of $Z\geq 12$ ($Z>18$ in the case of
$\rm{}^{80}Kr$ \+ $\rm{}^{40}Ca$) and sorted into two groups depending on
whether their longitudinal velocities are larger or smaller than those of the
coincident heavy fragment. Considering that fragments with $Z=3$ or 4 included
in the faster group are not typical evaporation products, their origin is not
immediately obvious. The slower group extends up to $Z=11$ and thus covers the
$Z$ interval of intermediate mass fragments abundantly produced in
multifragmentation processes. Their neutron content $\langle N\rangle/Z$ as a
function of $Z$ exhibits the odd-even staggering known from other reactions,
revealing a neutron richness exceeding that of the projectile sources. To
explain these fragments we assume the formation of a highly excited
statistical source forming a neck between the target and the projectile
ditoro06 . With SMM calculations based on this scenario, the observed isotopic
composition is very satisfactorily reproduced.

##  II FRAGMENT PRODUCTION IN THE STATISTICAL APPROACH

Presently there are two main statistical conceptions intending to describe the
physics of fragment production as a decay of excited finite nuclei. The most
popular one is that of the compound nucleus Bohr  which decays with the
sequential emission of light particles. There are several mathematical
formalisms according to this conception which can be used for practical
calculations. For example, GEMINI charity88 ; charity90  uses the Hauser-
Feshbach method, while SMM uses the Weisskopf method for the low-energy
emission. As was shown in the theoretical and experimental studies, the
compound nucleus conception works very well from low excitation energy of
nuclei up to values around 2 to 3 MeV/nucleon SMM ; pienkowski02 .

Another conception was mainly developed in the 1980s (see, e.g., reviews SMM ;
Gross90  and references therein), and we may call it the freeze-out-volume
conception. It was motivated by intensive experimental studies of the
multifragmentation process leading to the production of many intermediate mass
fragments from the decay of a highly excited nuclear source.

According to the freeze-out-volume conception, the compound nucleus has no
time to be formed, since the highly excited nucleus rapidly expands (on a time
scale of approximately 100 fm/$c$), and the system breaks-up (reforms itself)
into fragments during this expansion. The last moment at which the fragments
can still be considered as interacting and being under formation is the
statistical freeze-out time. Subsequently the fragments propagate in the
mutual Coulomb fields and undergo secondary deexcitation as usual for low-
excited nuclei. SMM includes this conception and thus provides an opportunity
to describe multifragmentation data. Since we are dealing with finite nuclear
systems, a weak point of the freeze-out-volume conception up to now was that
the upper limit of the excitation energy was not determined. In this case, the
very idea of applying statistical laws in the rapidly expanding matter is not
convincing from a theoretical view.

A method for solving this problem was suggested recently: By analyzing the
fragment production in central nucleus-nucleus collisions at beam energies of
more than 50–100 MeV/nucleon, it was established that the maximum excitation
energy of such statistical sources can not reach more than 8–10 MeV/nucleon
Bot21 ; Bot22 . This compares well with the previous maximum excitation energy
of sources found in peripheral and central collisions. It is consistent with
the binding energy of finite nuclei, providing the natural physical limit for
the application of the statistical theory. In our case of nucleus-nucleus
collisions at the Fermi energy, such energies correspond to the maximum energy
reached in the center of mass of the colliding nuclei.

According to the statistical hypothesis, initial dynamical interactions
between the colliding nuclei lead to a redistribution of the available energy
among many degrees of freedom. The nuclear system evolves towards equilibrium.
In a general consideration, the process may be subdivided into two stages: (1)
a dynamical stage leading to the formation of an equilibrated nuclear system,
and (2) the statistical disassembly of the system into individual primary
fragments, followed by their deexcitation and the formation of the final
product distribution observed in experiments.

Dynamical models indicate that individual reactions at intermediate energies
are not evolving as equilibrium processes furuta09 . However, because of the
complexity of the dynamical processes, the intermediate reaction states
populate a phase space that can be described with very few parameters, and
whose disassembly according to statistical laws leads to the observed
asymptotic state of the reaction raduta07 . By applying the SMM, a scenario is
adopted that places the equilibrium state at an early stage of the reaction,
before the partitioning of the excited system occurs. The parameters
describing the equilibrium state thus contain information on global properties
of the initial reaction stages forming it. The present study aims at
identifying the isospin transport during this early part of the collision.

Sequences of statistical decay processes are expected to proceed towards the
valley of stability at which binding energies reach their maximum. More
precisely, the expected asymptotic destination is represented by the
evaporation attractor line (EAL), whose location is mainly determined by the
competition between proton and neutron evaporation charity98 . Often, and also
in the present case, the decay sequences are not long enough, so that the EAL
is approached but not fully reached.

The emission of fragments is treated in the SMM by expanding the excited
sources to the statistical freeze-out volume. This is assumed to be caused by
thermal pressure or as a decompression process after an initial dynamical
collective compression. However, a similar low-density freeze-out state of
nuclear matter may also be created through the stochastic knock-out of
nucleons produced by initial nucleon-nucleon collisions. The freeze-out
density is expected to be around (0.1–0.3)$\rho_{0}$, where $\rho_{0}\approx
0.16$ fm-3 is the normal nuclear density. The value of 0.3$\rho_{0}$ has been
adopted for the projectile sources as in previous analyses Bot95 ; Xi97 ;
Ogul11 .

Peripheral reactions at the present intermediate energies are additionally
characterized by a low-density third source of nucleons and light fragments
located at intermediate velocities ditoro06 ; montoya94 ; dempsey96 ;
lukasik97 ; piantelli02 ; baran04 ; hudan04 . It may contain products of
nucleon-nucleon collisions removed from the main residues as well as remnants
of necklike structures joining projectile and target residues before they
separate. Interpreted as a continuation of the multifragmentation mechanism
towards intermediate impact parameters, the processes at intermediate velocity
seem suitable for a statistical description with the SMM, an approach rarely
attempted up to now Erg15 . The neutron content of these sources is of
specific interest because of the expected neutron drift toward regions of
lower density caused by the density dependence of the symmetry energy Bot01 ;
Colonna05 .

##  III Method

In Ref. FAZIA21 , the FAZIA Collaboration reports the relative isotope
distributions for elements with $Z=19-24$, individually normalized for each
atomic number $Z$. They were measured with four FAZIA detector blocks placed
at forward angles between $\theta_{\rm lab}=2.4^{\circ}$ and $17.4^{\circ}$,
covering approximately 14% of the solid angle up to $\theta_{\rm
lab}=17.4^{\circ}$ (see Ref. FAZIA20  for a layout of the experiment). In the
present analysis, we assume that the fragment distributions recorded within
the covered solid angle are representative for projectile fragmentation in the
studied reactions. The same is assumed for the coincident light fragments with
atomic numbers $Z\leq 4$ and $Z\leq 11$ recorded with the same setup for two
velocity regimes.

The velocities of the $Z=19-24$ fragments are shown to be only slightly lower
than the initial velocity of the $\rm{}^{80}Kr$ projectiles. This, together
with their masses of more than half of the projectile mass, identifies them as
resulting from peripheral processes. One may thus expect that their masses are
correlated and their excitation energies inversely correlated with the impact
parameter. Such correlations are reminiscent of the participant-spectator
scenario established for reactions at higher energies. It is, therefore,
further assumed that the equilibrated intermediate state can be described with
an ensemble of sources with properties similar to those determined in the
analysis of projectile fragmentation at energies up to 1 GeV/nucleon Bot95 ;
Xi97 ; Ogul11 ; Imal15 .

The ensemble consists of a distribution of sources extending from systems of
low excitation and masses near the projectile mass to highly excited systems
of small mass expected to be formed in central collisions, and, in this form,
is universally applicable (Fig. 1). The form of its peripheral branch with
excitation energies up to $\approx 4$ MeV/nucleon compares well with AMD
predictions for comparable reactions hudan06  and the general energy-mass
correlation is consistent with transport model predictions up to much higher
energies Bot17 . The fragments of interest and their distributions are
selected from the output configurations obtained from the eventwise performed
calculations for the full ensemble.

Figure 1: Ensemble of hot thermal sources represented in a scatter plot of
reduced mass number $A/A_{0}$ versus excitation energy $E_{x}/A$, as used in
the SMM calculations. The frequency of the individual sources is proportional
to the area of the squares (reprinted with permission from Ref. Ogul11 ;
copyright © 2011 by the American Physical Society).

Altogether, seven parameters are required to define the ensemble of excited
sources. In the analyses performed with the ALADIN data for projectile
fragmentation at energies up to 1 GeV/nucleon, the experimental data permitted
a determination of these parameters with very little ambiguity as shown in
Refs. Bot95 ; Ogul11 . This is not possible here, and the general form of the
ensemble was, therefore, adopted from the earlier work. The maximum mass
$A_{0}=80$ and atomic number $Z_{0}=36$ are those of the projectile, assuming
that pickup processes are rare at the present energies. Also the maximum
excitation energy $E_{x}/A=8$ MeV was chosen as in previous studies and as
supported by recent work Bot21 ; Bot22 . It is not crucial, however, because
the event selection requiring a fragment with $Z\geq 12$ and $Z\geq 19$ for
the two reactions implies that the excited projectile residues compose at
least about half of the original projectile. The excitation energies of such
sources with $A/A_{0}\geq 0.5$ are limited to values near or below 4
MeV/nucleon (Fig. 1). As in previous studies (see, e.g., Refs. MSU ; INDRA ;
TAMU ; Bot06 ; Imal20 ), angular momentum is not considered as an independent
property of the source because its effects are small and, regarding the
isotopic fragment distributions, indistinguishable from the much larger
effects of the excitation energy.

Figure 2: Mean multiplicity $\langle M_{\rm imf}\rangle$ of fragments of
intermediate mass with $Z\geq 3$ in coincidence with the fragment with the
largest atomic number $Z_{\rm max}$ found in an event as a function of $Z_{\rm
max}$, as obtained with the SMM for the fragmentation of the projectile in
81Kr + 48Ca (solid circles) and 78Kr + 48Ca (open circles).

The experimental data FAZIA21  were collected without a particular impact-
parameter selection, even though peripheral collisions are favored by the
detection at forward angles. Correspondingly, the whole ensemble of excited
projectile residues is taken into account in the analysis, and the element
range of interest is selected from the calculated set of final products. For
exploring the isospin transfer during the initial reaction stage, the mean
composition $\langle N/Z\rangle$ was varied by varying the maximum mass
$A_{0}$ without changing $Z_{0}$. It is assumed that the associated small
change in $E_{x}/A$ vs $A$, caused by the construction of the ensemble, is
negligible for the narrow interval of fragments with $Z=19-24$.

The isotope yields depend also on the properties of the primary hot nuclei in
the statistical freeze-out. In this respect the most important ingredient is
the symmetry term of their liquid-drop description and its coefficient
$\gamma$. The symmetry term governs the widths of isobaric or isotopic
fragment distributions botvina85 ; Botvina02 . The standard value used in SMM
applications is $\gamma\approx 25$ MeV SMM ; Buy05 . Similar but also lower
values have been determined from statistical interpretations of reaction data
at lower souliotis07 ; hudan09  and higher energy. The isoscaling analysis of
fragmentation reactions with Sn targets and light-ion beams of up to 15 GeV
incident energy resulted in $\gamma=22.5$ MeV Botvina02 . From the isoscaling
analyses performed for 12C + 197Au reactions at 300 and 600 MeV/nucleon
LeFevre05  and for the fragmentation of 107,124Sn and 124La projectiles on Sn
targets at 600 MeV/nucleon Ogul11  values of $\gamma=15$ MeV and lower were
deduced. The lowest values, however, were found for the most violent
collisions and for fragments with $Z\leq 10$. A similar scenario is not
expected in the present case for the production of fragments with $Z=19-24$ in
peripheral collisions at 35 MeV/nucleon. As will be shown below, $\gamma$ can
be determined rather precisely from the widths of the measured isotope
distributions.

In a first step, the calculations were used to confirm that the chosen
procedure permits satisfactory reproductions of the experimental isotope
distributions and to obtain global properties of the reaction. As an example,
the multiplicity of fragments of intermediate mass produced in coincidence
with a largest fragment with atomic number $Z_{\rm max}$ is shown in Fig. 2.
The chosen threshold is $Z\geq 3$ and the upper limit is either $Z_{\rm max}$
or $36-Z_{\rm max}$, whichever is smaller. Fragments in this interval are not
typical for evaporation but, according to the calculations, are produced with
mean multiplicities close to 1 for $Z_{\rm max}$ around 10. In the interval
covered by the experiment, $Z_{\rm max}\geq 12$ and $Z_{\rm max}\geq 19$ for
the two reactions, the mean multiplicity of associated fragments decreases
from about 0.8 to very small values at the upper end of the range of $Z_{\rm
max}$. The multiplicity widths are probably large and the production of
intermediate mass fragments cannot be expected to be negligible.

In a second step, the sensitivity of the mean mass number of the produced
projectile fragments to the isotopic compositions of the chosen ensembles was
explored. The compositions reveal the results of isospin transport phenomena
during the initial phases of the collision, the main goal of the present
study. The capabilities of the SMM SMM  to describe the statistical
disassembly of the intermediate system and the subsequent modes of
deexcitation permits a view back on to these early reaction stages through
backtracing from the detected fragmentation patterns. In a third step, the
origin of the coincident light fragments was investigated by comparing their
mean neutron-over-proton ratios to the results obtained for 107Sn and 124Sn
fragmentations in experiment S254 conducted at the GSI laboratory Ogul11 .

Figure 3: Comparison of the predicted isotopic distributions of fragments with
$Z=19-24$ with the experimental data measured in the 80Kr + 48Ca reaction. The
solid red circles represent the experimental data from Ref. FAZIA21  and open
circles the results of the SMM-ensemble calculations for the same reaction.

Figure 4: Comparison of the predicted isotopic distributions of fragments with
$Z=19-24$ with the experimental data measured in the 80Kr + 40Ca reaction. The
solid red circles represent the experimental data from Ref. FAZIA21  and open
circles the results of the SMM-ensemble calculations for the same reaction.

##  IV Comparison with the experiment

The results obtained in the first step are presented in Figs. 3 to 6. The
isotopic distributions of nuclei in the interval of atomic numbers $Z=19-24$
for collisions of 80Kr + 48Ca and 80Kr + 40Ca are shown in Figs. 3 and 4,
respectively. The normalized isotopic yield distributions measured in the
FAZIA experiments FAZIA21  are represented by solid circles. For the SMM
calculations represented by the open circles, it was assumed that the
projectile 80Kr picks up an extra neutron from the 48Ca target and loses two
neutrons in interactions with the 40Ca target, or that an equivalent exchange
of nucleons takes place. With the corresponding parameters for the maximum
mass of the projectile $A_{0}=81$ and 78 and with $Z_{0}=36$, average
compositions $\langle N/Z\rangle=1.25$ and 1.17 were thus assumed for the two
cases, respectively.

Figure 5: Isotopic yield distributions for light nuclei with atomic number
$Z=1-4$, emitted forward with respect to the coincident projectile residue in
the 80Kr + 48Ca reaction. Red solid circles represent the experimental data of
Ref. FAZIA21 , and open circles represent the results of the SMM-ensemble
calculations.

Other parameters describing the ensemble of sources were fixed as in Ref.
Ogul11 . Specifically, for the coefficient $\gamma$ of the symmetry term,
values of $\gamma=18$ MeV for $Z=19-20$, $\gamma=19$ MeV for $Z=21-22$, and
$\gamma=20$ MeV for $Z=23-24$ were found to provide the results closest to the
experimental data for 80Kr + 48Ca and values larger by 1 MeV for 80Kr + 40Ca.
The figures demonstrate that the chosen procedure permits a rather
satisfactory reproduction of the experimental isotope distributions for the
products of mid-peripheral collisions with atomic number $Z=19-24$.

Figure 6: Isotopic yield distributions for light nuclei with atomic number
$Z=1-4$, emitted forward with respect to the coincident projectile residue in
the 80Kr + 40Ca reaction. Red solid circles represent the experimental data of
Ref. FAZIA21 , and open circles represent the results of the SMM-ensemble
calculations.

Figure 7: Experimental and theoretical results for the $\langle N\rangle/Z$
ratios of light fragments, emitted forward with respect to the coincident
projectile residue, obtained in the 80Kr + 48Ca and 80Kr + 40Ca reactions as
presented in Figs. 5 and 6. Red solid circles represent the experimental data
and open circles represent the results of the SMM-ensemble calculations.

The same ensemble nuclei were also used to calculate isotopic distributions of
light fragments with $Z=1-4$. The distinction made in the experiment between
particles emitted into forward or backward directions with respect to the
coincident heavier fragment cannot be applied in the model calculations but
the coincidence with a heavier fragment was required. In the case of 80Kr +
48Ca, only events containing a fragment with atomic number $Z$ between 12 and
36 and, for 80Kr + 40Ca, only events containing a fragment with $Z$ between 19
and 36 were selected.

Figures 5 and 6 show the calculated relative isotopic yields of light nuclei
with $Z=1-4$ in comparison with the experimental yields for fragments with
longitudinal velocities larger than those of the coincident heavier fragment.
Protons and $\alpha$ particles obviously dominate the distributions of
hydrogen and helium isotopes. In both reactions, 7Li is the strongest lithium
isotope while 9Be is the strongest beryllium isotope only in the 80Kr + 48Ca
reaction and is equally abundant as 7Be in the 80Kr + 40Ca reaction. Overall,
the observed differences between the two reactions appear to be small. The
results of the SMM-ensemble calculations represent the experimental data
rather well. On a logarithmic scale, deviations are essentially only visible
in the cases of helium and lithium isotopes. Some of the predictions for weak
isotopes slightly overestimate the experimental yields. These yields play a
minor role for the mean neutron-over-proton ratios shown for the $Z=1-4$
products in Fig. 7. The agreement is best for $Z=2$, obviously so because of
the dominance of 4He, and overall very satisfactory.

##  V Symmetry coefficient $\gamma$

Within the SMM, the statistical disintegration of highly excited nuclei
includes the expansion of the nuclear system into a freeze-out volume within
which the primary hot fragments are formed and the chemical equilibrium (i.e.,
the detailed-balance interaction) between them is established. The properties
of these hot fragments can be different from those of cold stable nuclei. In
particular, their symmetry-term coefficients $\gamma$ can be lower than the
standard value of 25 MeV that is taken in the liquid-drop Bethe-Weizsäcker
formula describing masses of normal nuclei (see, e.g., Refs. SMM ; Botvina02 ;
Buy05 ; souliotis07 ; hudan09 ; LeFevre05 ). The reason is that the fragments
are interacting until freeze-out, they can be slightly expanded, and their
surface parts can be extended. Subsequently, the standard value of the
symmetry-term coefficient is restored during the secondary deexcitation of the
hot fragments Buy05 .

The coefficient $\gamma$ controls the widths of the calculated isotope
distributions botvina85 ; Botvina02 . This is illustrated in Fig. 8, which
shows the examples of isotopic yields for $Z=19$ calculated for $A_{0}=78$ and
values of $\gamma=8$, 19, and 25 MeV in comparison with the experimental data
obtained in 80Kr + 40Ca. It is evident that the widths change significantly
with $\gamma$ and that the value $\gamma=19$ MeV chosen for this case leads to
the good reproduction of the data seen in Fig. 4. It is also evident that the
Gaussian fits included in the figure permit a quantitative analysis of the
distribution widths. The dependence of the Gaussian standard deviations on
$\gamma$ is illustrated in the top panel of Fig. 9. It confirms that
$\gamma=19$ MeV for $Z=19$ is rather precisely, with an uncertainty of around
$\pm 1$ MeV, determined by the measured data. The dependence of the mean
neutron number $\langle N\rangle$ on $\gamma$, equally evident in Fig. 8, is
given in the bottom panel of Fig. 9.

The widths of some of the isotope distributions appear to be slightly
underpredicted by the calculations (Figs. 3 and 4). An adjustment of $\gamma$
in these cases requires the simultaneous modification of the mean $\langle
N\rangle/Z$ that is to be expected. The variations of the widths and mean
neutron numbers with $\gamma$, shown for $Z=19$ in Fig. 9, are rather
universal and very similar for the studied elements with $Z=19-24$. Further
individual optimizations of specific parameters may lead to a slightly
different ensemble of excited sources representing the intermediate reaction
systems.

Figure 8: Relative yield distributions for $Z=19$ and the results of Gaussian
fits (blue lines) as measured in 80Kr + 40Ca (top left panel, red solid
circles) and as calculated with $A_{0}=78$ and the indicated values of the
symmetry-term coefficient $\gamma$ (open circles). The errors of the
experimental data are mostly smaller than the symbol size.

Figure 9: The standard deviation $\sigma$ of the Gaussian functions fitted to
the measured and calculated isotope distributions in units of the neutron
number (top) and the mean neutron number $\langle N\rangle$ (bottom) as a
function of the symmetry-term coefficient $\gamma$ used in calculations for
$Z=19$ with $A_{0}=78$ and 81 (filled circles). The horizontal lines indicate
the results deduced from the experimental data including their uncertainties
which, for the case of $\langle N\rangle$, are smaller than the line widths.

The values $\gamma=18-21$ MeV applied in the present calculations had already
been found useful for describing projectile fragments of similar magnitude
from 112,124Sn + 112,124Sn collisions at 1 GeV/nucleon Imal15 . They are also
close to the $Z$ dependent symmetry-term coefficients used in the
microcanonical multifragmentation model raduta07b  and in the AMD model ono04
. The surface term included in the AMD leads to $\gamma=20.4$ MeV and 21.2 MeV
for $Z=19$ and 24, respectively, assuming $A=2Z$. The AMD coefficients were
determined by fitting the ground state binding energies for $A\leq 40$ nuclei
calculated with the AMD model. In the present case, the symmetry term is part
of the description of the produced nuclear fragments with excitation energies
up to about 4 MeV/nucleon. The difference of the coefficients is, however,
marginal.

##  VI Initial compositions

The initial compositions defined by choosing $A_{0}=78$ and 81 indicate a
considerable amount of isotopic equilibration between projectile and target
during the early stages of the two reactions, more than 60% of full isospin
equilibrium. For a more quantitative analysis, the $N/Z$ compositions of the
ensembles of excited sources were investigated by varying the $A_{0}$
parameter from 75 to 83 while keeping $Z_{0}=36$ fixed. It covers the interval
of initial compositions $\langle N/Z\rangle_{\rm ini}=A_{0}/36-1$ from 1.083
to 1.306. The subscript ”ini” is used here and in the following to indicate
the initial isotopic composition at the beginning of the statistical decay
modeled with the SMM.

Figure 10: Mean quadratic deviations ($\chi^{2}$ per degree of freedom) of the
relative isotopic yields for $Z=20$ projectile fragments with mass numbers
$A=40-46$ as a function of the $A_{0}$ parameter used for varying the initial
composition $\langle N/Z\rangle_{\rm ini}$ of the ensemble of sources for the
two reactions 80Kr + 48Ca (top) and 80Kr + 40Ca (bottom). The lines are the
results of parabolic fits performed for three different intervals of $A_{0}$
as indicated in each panel. The uncertainty intervals obtained for the fits
extending over seven data points (blue lines) corresponding to $4\sigma$ in
the upper and $2\sigma$ in the lower panel, are shown for illustration.

The calculated fragment yields were normalized, as in the experimental report
FAZIA21 , and the sum of the quadratic deviations from the measured values in
units of the experimental errors was determined for each of the sets obtained
with a given $A_{0}$. The so obtained $\chi^{2}$ distribution was divided by
the number of isotopes considered for a given $Z$, reduced by one because of
the normalization. The result for the example of $Z=20$ fragments is shown in
Fig. 10 for the two studied reactions. The displayed distributions of
$\chi^{2}$ per degree of freedom exhibit the expected parabolic behavior.
Their absolute values are rather large because the quoted experimental errors
are very small FAZIA21 , considerably smaller for the data obtained with the
48Ca target than for the data obtained with the 40Ca target.

The minima of the parabolic fits (Fig. 10) determine the optimum composition
of the ensemble of sources as approximately $N/Z_{\rm ini}=1.28$, obtained
with $A_{0}=82$, for the fragmentation on the 48Ca target and $N/Z_{\rm
ini}=1.11$ obtained with $A_{0}=76$ for the 40Ca case. They are similar to the
original assumptions but indicate an even larger degree of equilibration. By
analyzing the fits for the intervals extending over seven data points
(represented by the blue lines), $A_{0}=77-83$ for 48Ca and $A_{0}=75-81$ for
40Ca, the compositions $N/Z_{\rm ini}=1.280\pm 0.004$ and $N/Z_{\rm
ini}=1.110\pm 0.016$ are obtained for the two cases, respectively. The
corresponding 4$\sigma$ and 2$\sigma$ widths are indicated in the figure.

Table 1:  Summary of the results obtained for the initial configurations $\langle N/Z\rangle_{\rm ini}$ by comparing the calculated mean neutron numbers $\langle N\rangle$ with the experimental value (fourth column) or by searching for the minimum of the $\chi^{2}$ distribution obtained with the experimental errors (last column). The prediction of the evaporation attractor line charity98  is shown in the third column.  48Ca |  |  | $\langle N/Z\rangle_{\rm ini}$ | $\langle N/Z\rangle_{\rm ini}$
---|---|---|---|---
$Z$ | $\langle N\rangle_{\rm exp}$ | $N/Z_{\rm EAL}$ | from $\langle N\rangle_{\rm exp}$ | from $\chi^{2}$
19 | $21.78\pm 0.01$ | 1.116 | $1.278\pm 0.003$ | $1.300\pm 0.005$
20 | $23.11\pm 0.01$ | 1.118 | $1.292\pm 0.002$ | $1.280\pm 0.004$
21 | $24.10\pm 0.01$ | 1.121 | $1.209\pm 0.002$ | $1.215\pm 0.007$
22 | $25.23\pm 0.01$ | 1.123 | $1.206\pm 0.002$ | $1.198\pm 0.006$
23 | $26.39\pm 0.01$ | 1.125 | $1.210\pm 0.002$ | $1.207\pm 0.007$
24 | $27.49\pm 0.01$ | 1.128 | $1.220\pm 0.002$ | $1.244\pm 0.009$
| weighted mean |  | $1.232\pm 0.001$ | $1.254\pm 0.002$
40Ca |  |  | $\langle N/Z\rangle_{\rm ini}$ | $\langle N/Z\rangle_{\rm ini}$
$Z$ | $\langle N\rangle_{\rm exp}$ | $N/Z_{\rm EAL}$ | from $\langle N\rangle_{\rm exp}$ | from $\chi^{2}$
19 | $21.42\pm 0.02$ | 1.116 | $1.207\pm 0.011$ | $1.169\pm 0.017$
20 | $22.53\pm 0.02$ | 1.118 | $1.098\pm 0.008$ | $1.110\pm 0.016$
21 | $23.77\pm 0.02$ | 1.121 | $1.067\pm 0.010$ | $1.126\pm 0.013$
22 | $24.92\pm 0.02$ | 1.123 | $1.060\pm 0.011$ | $1.115\pm 0.014$
23 | $26.10\pm 0.02$ | 1.125 | $1.072\pm 0.006$ | $1.113\pm 0.016$
24 | $27.21\pm 0.02$ | 1.128 | $1.103\pm 0.007$ | $1.113\pm 0.023$
| weighted mean |  | $1.101\pm 0.003$ | $1.124\pm 0.006$

The results obtained for the six elements are given in the last column of
Table 1. The reconstructed intermediate configurations of the excited
projectile systems are fairly neutron rich for 80Kr + 48Ca and fairly neutron
poor for 80Kr + 40Ca. The weighted mean values are 1.254 and 1.124,
respectively, with a difference much larger than naively expected from the
fairly similar experimental isotope distributions (Figs. 3 and 4; see also
Fig. 6 in Ref. FAZIA21 ). It is obvious that the individual errors obtained
for each element are not representative for the element-to-element variations.
For further use, we therefore chose to give equal weight to each of the
elements for determining the mean values and to adopt the mean squared
deviations from the mean divided by $\sqrt{5}$ as their uncertainties. With
this choice, the compositions obtained with the $\chi^{2}$ method are $\langle
N/Z\rangle_{\rm ini}=1.241\pm 0.017$ and $1.124\pm 0.009$ for the two
reactions. Their difference amounts to $82\pm 14\%$ of the difference between
the fully equilibrated compositions.

In addition to the $\chi^{2}$ analysis employing the experimental errors, the
measured and calculated mean mass numbers for each of the six elements were
used to confirm the obtained results. The example for the same $Z=20$ case and
for the two reactions is shown in Fig. 11. The figure, first of all, documents
the strong attenuation of the initial $N/Z$ ratios caused by the breakup and
deexcitation of the excited sources as modeled with the SMM. Significantly
different compositions of the hot sources (red symbols) are required for
obtaining the small differences of the experimental mean fragment masses. The
comparison of the experimental $\langle N\rangle/Z$ with the values for cold
fragments obtained with the SMM (blue symbols), identifies $A_{0}=82$ and
$A_{0}=76$ as $A_{0}$ parameters coming closest to reproducing the
experimental mean fragment masses for $Z=20$. The corresponding compositions
$\langle N\rangle/Z_{\rm ini}=1.278$ and 1.111 are nearly the same as those of
the $\chi^{2}$ analysis (Table 1).

Figure 11: Mean isotopic compositions $\langle N/Z\rangle_{\rm ini}$ of the
initial systems (hot, red symbols) and the $\langle N\rangle/Z$ of the final
$Z=20$ isotope distributions obtained with the SMM (cold, blue symbols) as a
function of the maximum mass $A_{0}$ of the chosen ensembles. The calculated
results are compared with the experimental mean neutron-over-proton ratios
(labeled “exp”, full black lines) for both reactions and the EAL prediction
(dashed purple line) for $Z=20$ fragments. The compositions corresponding to
the isotopically equilibrated projectile and target systems (labeled “eq”,
dashed black lines) are shown in addition.

To find the optimum composition for each element, linear fits to the three
calculated mean $\langle N\rangle/Z$ closest to the experimental value were
used and the final result determined with a linear interpolation. The results
obtained in this way for the six elements are also shown in Table 1 (fourth
column). The displayed uncertainties are very small, reflecting again the
precision of the experimental mean neutron numbers. For obtaining more
realistic uncertainties, the method used in the $\chi^{2}$ analysis was
adopted. Equal weights were given to the results for each of the elements, and
the uncertainty of the mean composition determined by dividing the mean
squared deviations of the six individual values by $\sqrt{5}$. The obtained
compositions $\langle N/Z\rangle_{\rm ini}=1.236\pm 0.016$ and $1.101\pm
0.022$ for the two reactions are, within their errors, compatible with the
results obtained with the $\chi^{2}$ method. The difference amounts to $94\pm
17\%$ of the difference between the equilibrated systems. The weighted average
of the results obtained with the two methods corresponds to $87\pm 11\%$ of
the difference between the equilibrated systems.

To arrive at a lower limit for the degree of equilibration following from the
present study, a possible effect of the symmetry-term coefficient $\gamma$ has
to be included. On average, we find that $\Delta\gamma=+1$ MeV causes $\langle
N\rangle$ to decrease by approximately 0.10 and vice versa (cf. Fig. 9).
Following Fig. 11, it translates into an uncertainty $\Delta\langle
N\rangle/Z\approx 0.005$ and $\Delta\langle N/Z\rangle_{\rm ini}\approx
0.031$, equivalent to 22% of the difference between the equilibrated systems.
This uncertainty is larger but is only effective if changes of $\gamma$ are
different in the two reactions by a certain amount, for example additionally
increased by 1 MeV only in the case of 80Kr + 40Ca. A global change of
$\gamma$ will only change the vertical position of the line describing the
calculated final compositions but not its slope determining the separation of
the two experimental values on this line. Adopting this possibility as cause
for an additional uncertainty leads to a lower bound of 87% -11% -22% = 54%
for the degree of equilibration reached in the two reactions.

According to the analysis presented here, rapid $N/Z$ equilibration during the
initial reaction stage seems necessary for reaching the observed final
isotopic distributions. This conclusion depends crucially on the rate with
which the mean $\langle N\rangle/Z$ of fragments (blue symbols in Fig. 11)
increases with $\langle N/Z\rangle_{\rm ini}$, a result of the treatment of
secondary decay within the SMM. We also note that, for both reactions, the
initial compositions are less neutron rich than the equilibrated systems. As
we will argue below, it may be caused by the drift of neutrons into a neck
region of low density which is no longer involved in the disintegration of the
projectile residues.

As evident from Fig. 11, the breakup and deexcitation of the hot sources
proceeds very differently in the two cases. There is very little variation of
the isotopic composition in the 80Kr + 40Ca case. The hot and cold SMM values
of $\langle N/Z\rangle$ and the experimental value are all very close to the
equilibrium composition and not far from the value of the evaporation
attractor line charity98 . In the case of 80Kr + 48Ca, the neutron richness of
the excited system disappears to a large extent but, also here, the
evaporation attractor line is not fully reached. The valley of stability
attracts the decay chains but they are not long enough to end on the
evaporation attractor line. Apparently, since the multifragmentation mechanism
favors the production of neutron rich fragments, the deposited energy is
primarily used for the disintegration of the system into smaller fragments
instead of neutron evaporation.

##  VII Neck Emissions

In Ref. FAZIA21 , the group of coincident light fragments with longitudinal
velocities smaller than those of the coincident heavy fragments is interpreted
as either being emitted in backward direction from the excited projectile
residue or as representing emissions from the intermediate velocity source,
the latter containing products of nucleon-nucleon collisions and from the
decay of the neck forming as the main residues separate ditoro06 . In the data
analysis, the additional condition was required that the velocities had to be
larger than the center-of-mass velocity.

Figure 12: Experimental $\langle N\rangle/Z$ values of neck fragments as a function of their charge number $Z=2-11$ for the 80Kr + 48Ca reaction, in Ref. FAZIA21  defined as fragments emitted backward with respect to the projectile residue (solid red circles), and results of SMM calculations simulating the decay of a neck-type source with $A=19$ and properties given in the text for the interval $Z=2-6$ (solid green triangles) in comparison with the results for $Z=2-11$ of the projectile source (PS) with $A=81$ and $Z=36$, selected with the requirement of a coincident fragment with $Z\geq 12$ (open circles).  Table 2:  Reconstruction of the composition of the emitting sources obtained with the measured $\langle N\rangle/Z$ ratios of $Z=3,4$ fragments presented in Figs. 8, 11, and 12(a) of Ref. FAZIA21  (third and fifth columns) and based on the inclusive $\langle N\rangle/Z$ ratios measured for 107,124Sn fragmentations in experiment S254 (first two lines of Table 3). For beryllium, the uncorrected $\langle N\rangle/Z$ ratios have been extracted from the ALADIN data base. The source compositions are obtained by linear interpolations and extrapolations (fourth and sixth columns). Their errors represent the effects of the displayed experimental errors reported in Ref. FAZIA21 .  |  |  $Z=3$ (Expt.) | Source |  $Z=4$ (Expt.) | Source
---|---|---|---|---|---
group | target | $\langle N\rangle/Z$ | $\langle N/Z\rangle$ | $\langle N\rangle/Z$ | $\langle N/Z\rangle$
forward |  48Ca | $1.255\pm 0.003$ | $1.29\pm 0.02$ | $1.180\pm 0.006$ | $1.38\pm 0.01$
backward |  48Ca | $1.300\pm 0.002$ | $1.49\pm 0.01$ | $1.273\pm 0.003$ | $1.53\pm 0.01$
forward |  40Ca | $1.224\pm 0.009$ | $1.15\pm 0.04$ | $1.109\pm 0.020$ | $1.27\pm 0.03$
backward |  40Ca | $1.274\pm 0.016$ | $1.38\pm 0.07$ | $1.170\pm 0.032$ | $1.37\pm 0.05$

The $\langle N\rangle/Z$ ratios for fragments with $Z=2-11$ are among the data
reported for this group of fragments and, for the 80Kr + 48Ca reaction, shown
in Fig. 12. As the authors note, the observed pattern as a function of $Z$ is
well known from other studies and, as an example, the work of Barlini et al.
FAZIA13  is cited in FAZIA21 . The same characteristic odd-even staggering has
also been observed in projectile fragmentation at higher energies. The
systematics presented in Ref. traut07  for projectiles from nickel to uranium
and energies of 600 MeV/nucleon and 1 GeV/nucleon shows that the pattern is
invariant and that only its location on the $N/Z$ scale depends on the
projectile, apparently linearly correlated with the $N/Z$ ratio of the
projectile. In Ref. traut07 , a correction for the missing yield of unstable
8Be was applied to the values for $Z=4$ by including an estimate for it
obtained from a smooth interpolation over the identified yields of 7,9-11Be.
This correction has a negligible effect for the case of 107Sn but lowers the
$\langle N\rangle/Z$ of beryllium for 124Sn fragmentations from 1.24 to 1.18
which makes the systematic odd-even staggering more clearly visible for the
neutron rich case.

For the interpretation of the fragmentation reactions at relativistic
energies, it may be assumed that the excited sources inherit the neutron-over-
proton ratio of the original projectiles. It is motivated by the reduced
nucleon wave lengths, the dominance of nucleon-nucleon dynamics during the
initial reaction stages, and was shown to be successful in applications as,
e.g., the SMM analysis of the 107,124Sn and 124La fragmentations at 600
MeV/nucleon Ogul11 . It implies that the systematics obtained for light
fragments may be used for calibrating the isotopic compositions of the excited
sources.

In the following, we will assume that the observed correlation between sources
and fragments holds for the present type of reactions as well. The statistical
decay of excited sources should only depend on their properties and not on how
they were formed. With the measured $\langle N\rangle/Z$ values for products
with $3\leq Z\leq 9$ from the 107Sn and 124Sn fragmentations displayed in Fig.
3 of Ref. traut07 , a source reconstruction is attempted, not only for the so-
called backward emitted group of light fragments (Fig. 12) but also for the
forward emitted fragments whose $\langle N\rangle/Z$ ratios for $Z=1-4$ are
shown in Fig. 8 of FAZIA21  and reproduced in the present Fig. 7. In the
latter case, the overlap in atomic number with the 107,124Sn fragmentations is
restricted to $Z=3$ and 4. The corresponding $\langle N\rangle/Z$ ratios for
backward emitted fragments from 80Kr + 40Ca are obtained by adding the
backward-forward differences shown in Fig. 11 of FAZIA21 .

The results for the two elements $Z=3$ and 4 are listed in Table 2. They are
obtained by assuming that the values measured for the 107Sn and 124Sn
fragmentations, given in the first two lines of Table 3, are representative
for source compositions $\langle N/Z\rangle=1.16$ and 1.48, respectively, and
that a linear interpolation between (or extrapolation from) these values is
appropriate. The value $\langle N/Z\rangle=1.16$ for the case of 107Sn refers
to the actual radioactive beams used in the experiment which, besides the
desired 107Sn, contained contributions of neighboring isotopes whose
intensities had been determined Ogul11 ; lukasik08 .

As a main conclusion, we observe that the sources responsible for the so-
called backward-emitted fragments in both reactions are significantly more
neutron rich than the sources responsible for the forward-emitted fragments.
Simply on the basis of the obtained neutron-over-proton ratios, it can thus be
excluded that both groups of fragments have the same origin. In the case of
80Kr + 48Ca, the average composition $\langle N/Z\rangle=1.51$ of the
backward-emitted fragments is still significantly larger than the value 1.31
expected for a mixed source consisting of equal numbers of nucleons from the
projectile and from the target. The same holds for 80Kr + 40Ca. Also here the
average composition $\langle N/Z\rangle=1.37$ largely exceeds the value 1.11
of an equally mixed source. The values obtained for the forward-emitted
particles, on the other hand, are in approximate agreement with the results
for the fragments with $Z=19-24$ shown in Table 1. It permits the conclusion
that the latter two groups of fragments result from the breakup of the same
excited intermediate sources.

The errors displayed in Table 2 are those of the experimental data reported in
Ref. FAZIA21  and their effect on the reconstructed source compositions. These
errors are very small. To obtain an estimate of a potential systematic
uncertainty associated with the method, $\langle N\rangle/Z$ ratios for
$Z=5-9$ from the fragmentation of 124,136Xe projectiles on Pb targets are used
which had been measured at the GSI Fragment Separator FRS henzlova08 .
Reconstructing the source compositions on the basis of the 107,124Sn results
for the same $Z=5-9$ fragments (Table 3) leads to values consistently higher
by $\Delta N/Z=0.09$ than the $N/Z$ ratios of the 124,136Xe projectiles,
corresponding to almost five additional neutrons. The difference $\Delta
N/Z=0.23$ between the results for the two Xe projectiles, however, is very
close to the actual difference $\Delta N/Z=0.22$ of their $N/Z$ ratios.
Relative differences are preserved but the absolute values, according to this
test, are shifted by $\Delta N/Z\approx 0.1$. For the following discussion,
the source compositions listed in the tables will be reduced by $\Delta
N/Z=0.05$ and an error of equal magnitude will be assigned to them. It is
equivalent to giving equal weights to the Sn and Xe fragmentations in their
role for the intended reconstruction of source compositions.

Table 3:  Reconstruction of the neutron-over-proton ratios of the emitting sources (fifth column) obtained with the $\langle N\rangle/Z$ ratios of $Z=3-9$ fragments presented in Fig. 12 of Ref. FAZIA21  (second column) and based on the inclusive $\langle N\rangle/Z$ ratios measured for 107,124Sn fragmentations in experiment S254 as reported in Ref. traut07  (third and fourth columns). For beryllium, the uncorrected $\langle N\rangle/Z$ ratios have been extracted from the ALADIN data base. The errors of the sources represent the effects of the displayed experimental errors reported in Ref. FAZIA21 .  Z | Expt. |  107Sn |  124Sn | Source
---|---|---|---|---
3 | $1.300\pm 0.002$ | 1.227 | 1.297 | $1.494\pm 0.009$
4 | $1.273\pm 0.003$ | 1.038 | 1.243 | $1.527\pm 0.005$
5 | $1.196\pm 0.001$ | 1.134 | 1.200 | $1.461\pm 0.005$
6 | $1.122\pm 0.001$ | 1.037 | 1.138 | $1.429\pm 0.003$
7 | $1.137\pm 0.001$ | 1.103 | 1.153 | $1.378\pm 0.006$
8 | $1.103\pm 0.001$ | 1.048 | 1.120 | $1.404\pm 0.004$
9 | $1.158\pm 0.002$ | 1.132 | 1.190 | $1.303\pm 0.011$

This uncertainty does not affect the resulting implications. The so-called
backward emitted particles are unlikely to originate from the same source as
the forward emitted particles and light fragments. This was also concluded by
the FAZIA Collaboration and reported in their paper FAZIA21 . The neutron
richness of the backward sources is enhanced even in comparison to those of
the equally mixed sources by corrected values $\Delta\langle N/Z\rangle\approx
0.15$ in the case of 80Kr + 48Ca and $\approx 0.21$ in the case of 80Kr +
40Ca. If this is caused by neutrons drifting into these sources their
densities will have to be low, a property expected for the neck region located
between the separating projectile and target residues ditoro06 . Even its mass
may be estimated by taking into account that the compositions of the
projectile sources were, on average, lower than those of the equilibrated
systems by $\Delta\langle N/Z\rangle\approx 0.035$, a change achieved by the
drift of two neutrons out of a system with together 56 protons. The same two
neutrons will raise the neutron-over-proton ratio by $\approx 0.18$, the
average enhancement observed for the two reactions, in a smaller system with
only about one fifth the number of protons. The mass of the neck region,
according to this estimate, should thus be of the order of $(\langle
N/Z\rangle+1)\times 56\times 0.035/0.18$, i.e., about 25 nucleons.

The obtained source properties are very consistent. Going more into detail,
one observes that the values for the sources of $Z=4$ fragments are
systematically larger than those for the sources of $Z=3$ fragments. On the
other hand, the calibration for $Z=3$ amplifies small differences and thus may
be associated with a larger uncertainty (cf. Table 2). Elements with odd $Z$
generally respond less strongly to the source composition, according to the
data published in Ref. traut07  which extend up to $Z=9$. It is thus possible
to extend the calibration to heavier fragments of the so-called backward
emitted group in 80Kr + 48Ca whose composition is presented in the FAZIA
report FAZIA21 .

The results, again obtained with linear interpolations and extrapolations, are
given in Table 3. The obtained compositions are all neutron rich, starting
with values $\langle N\rangle/Z\approx 1.5$ for $Z=3$ and 4 and slowly
decreasing with $Z$ to values 1.4 and 1.3 for the $Z=8$ and 9 fragments,
respectively. A dependence on $Z$ of this kind is not unexpected considering
the so-called hierarchy effect according to which the lightest fragments are
expected from the regions of lowest density in between the projectile and
target rapidities ditoro06 ; montoya94 ; colin03 . The scatter of the
individual results may be used as indication of their systematic uncertainty.

It will finally be shown that SMM calculations of the deexcitation of a neck-
type source with properties corresponding to these results will lead to light
fragment yields with $\langle N\rangle/Z$ ratios in good agreement with the
FAZIA measurement for the 80Kr + 48Ca reaction FAZIA21 . Following a series of
test calculations with varying parameters for the masses and excitation
energies of the source as well as for its $\langle N/Z\rangle$ composition,
the following scenario was assumed: a fixed mass $A=19$ and atomic number
$Z=8$ of the source, corresponding to $N/Z=1.375$, an excitation energy of 5
MeV/nucleon, a reduced breakup density $\rho=\rho_{0}/6$, and a reduced
symmetry-term coefficient $\gamma=14$ MeV were considered as appropriate for
the small masses of the produced fragments. Similar parameters were found
useful in previous SMM studies of multifragmentation reactions SMM ; MSU ;
INDRA . For example, in central 197Au + 197Au collisions at the same beam
energy of 35 MeV per nucleon, a statistical source with the same density and
excitation energy per nucleon was extracted MSU .

The obtained $\langle N\rangle/Z$ ratios for neck fragments with $Z\leq 6$ are
shown in Fig. 12, and a very satisfactory agreement with the experimental
results is observed. For comparison, the results for the projectile source
(PS) with $A=81$ and $Z=36$ ($N/Z=1.25$) is shown as well. They are obtained
from the same calculation as the results shown in Figs. 5 and 7 and seen there
to agree well with the experimental results for the forward emitted coincident
fragments. The characteristic odd-even staggering is apparent in both
calculations as nuclear structure effects are taken into account in the SMM
for the ground-state masses which are restored at the end of the secondary
deexcitation.

The significant difference in $\langle N\rangle/Z$ with respect to the
experimental values for the class of slower fragments, documented for $Z=3,4$
in Table 2, is found to extend to larger $Z$. The modest reduction of the
difference toward $Z=11$ is not unexpected. Neck fragments are predominantly
small as already mentioned in the discussion related to Table 3. It thus
appears that the definition of two groups by the FAZIA Collaboration
represented a very useful choice, giving evidence for the existence of
different sources of coincident light fragments. The sorting according to the
sign of the relative velocity is clearly not sufficient for precisely
distinguishing particles and fragments according to their various origins, as
discussed in detail in the paper FAZIA21 .

##  VIII Discussion

The indicated isotopic exchange between projectile and target seems somewhat
unexpected on the basis of the measured isotope distributions exhibiting
rather small differences. The latter are the consequence of the attraction of
the valley of stability which causes the initial neutron richness of the
excited sources in 80Kr + 48Ca to quickly disappear (Fig. 11). Large
contributions of the target to measured isotopic ratios of light fragments
have already been observed in the earlier experiment of the FAZIA
Collaboration FAZIA13 . At that time, a three-element telescope placed at
$\theta_{\rm lab}=5.4^{\circ}$ was used to measure inclusive fragment
production in the 84Kr + 112,124Sn reactions at 35 MeV/nucleon. The observed
$\langle N\rangle/Z$ values of light fragments depend significantly on the
target. For lithium and beryllium fragments, values of $\approx 1.3$ are
reported with differences of $\approx 0.03$ for the two target cases.

Applying the same method of source reconstruction as for the present reaction
leads to rather neutron rich sources with $\langle N/Z\rangle_{\rm ini}=1.63$
and 1.46 for the reactions with 124Sn and 112Sn targets, respectively. These
values are higher than in the present reactions (Tables 2 or 3) as expected
for the more neutron rich projectile. They are also higher than the $\langle
N/Z\rangle_{\rm ini}=1.42$ and 1.28 of the equilibrated 84Kr + 124,112Sn
systems and their difference 0.17 exceeds even slightly the difference 0.14 of
the latter. Following the interpretation offered in the previous section, one
would, therefore, conclude that these light fragments are primarily emitted
from low-density regions of the reaction volume whose neutron content has been
enhanced. According to the analysis of the slightly heavier Xe+Sn reaction by
Łukasik et al. lukasik97 , the very light fragments have their origin
predominantly in the midrapidity regime.

Isotopic equilibrium is not unusual in the Fermi energy domain. It was, for
example, observed by Johnston et al. for inclusive reactions of Ar and Ca
beams of mass $A=40$ on Fe and Ni targets of mass $A=58$ and incident energies
33 MeV/nucleon johnston96 . As shown in that work, when the energy is
increased to 45 MeV/nucleon full equilibrium is no longer achieved. Partial
equilibrium was observed also in the cross bombardment of 112Sn and 124Sn
beams and targets at the slightly higher beam energy of 50 MeV/nucleon by
Tsang et al. tsang04 . Partial isospin equilibrium of $\approx 53\%$ was
reported by Keksis et al. for 40,48Ca on 112,124Sn reactions at the slightly
lower energy 32 MeV/nucleon keksis10 .

In the most recent experiment of the FAZIA Collaboration, isospin transport
ratios were determined for cross bombardments of 58Ni and 64Ni projectiles and
targets at 32 and 52 MeV/nucleon incident energy ciampi22 . Increasing isospin
equilibration with decreasing impact parameter is reported but full
equilibration is not observed, even for the most central collisions. A very
similar conclusion was presented very recently for the 40,48Ca + 40,48Ca
system at 35 MeV/nucleon by Fable et al. who also applied the isospin
transport-ratio technique fable23 . These results emphasize the transitional
nature of the Fermi energy domain with the effect that small differences in
the reaction systems and the experimental conditions may manifest themselves
in measurable differences of the observed isospin transport coupland11 .

Contact times, if assumed to correspond to the sum of the diameters of the
collision partners divided by the velocity of the projectile, are of the order
of 70 fm/$c$ in the present reaction and, depending on the impact parameter
and dynamical details, may fall between 50 and 100 fm/$c$. This interval
represents the timescale on which the observed isospin equilibration
apparently occurs. It is also consistent with the time characterizing the
statistical nucleation process in the neck region which is described within
the freeze-out volume conception. Similar times of about 100 fm/$c$ were found
by Tsang et al. tsang04  to be needed for the observed isospin diffusion in
the 112,124Sn + 112,124Sn collisions at 50 MeV/nucleon according to Boltzmann-
Uehling-Uhlenbeck (BUU) calculations performed for these reactions. For the
similar Xe + Sn system at 32 MeV/nucleon, isospin equilibration times of less
than 200 fm/$c$ were deduced from calculations with the stochastic mean field
(SMF) model ademard14 .

Equilibration times of very similar magnitude were deduced from an experiment
of different type by Jedele et al. jedele17 . There, the authors followed the
isospin equilibration in a rotating dinuclear system formed by the decay of
deformed projectile fragments produced in 70Zn + 70Zn collisions at 35
MeV/nucleon. For selected systems, the obtained equilibration times were of
the order of 0.3 zs, corresponding to $\approx 100$ fm/$c$, whereas the
process up to full isospin equilibration may be followed for somewhat longer
times brown13 . It thus seems that, according to the presented examples, $N/Z$
equilibration proceeds fast enough to be observed in the 80Kr + 48,40Ca
experiments at 35 MeV/nucleon. It should be interesting to study the full
charge range of projectile fragments and connect it with the evolution of the
isospin transport. The measurements of correlated fragments, both in the neck
and projectile regions, are important in this case.

The neutron richness of neck emissions has been reported by many authors
lukasik97 ; fable23 ; theriault06 ; kohley11 ; Filip12 ; bougault18  and
deduced from theoretical work baran04 . Thériault et al. have studied the
symmetric 64Zn + 64Zn reaction at 45 MeV/nucleon theriault06 . They found that
the completely reconstructed midrapidity material is more neutron rich than
the overall reaction system whereas the reconstructed $N/Z$ ratio for the
projectile system remains below that value. This result was presented by the
authors as the most complete evidence, at that time, of neutron enrichment of
midrapidity nuclear matter at the expense of the remaining reaction system, a
phenomenon related to the density dependence of the nuclear symmetry energy.
The reconstruction of the compositions of the projectile and neck source
presented here for the 80Kr + 48,40Ca reactions is in full agreement with
these earlier findings.

##  IX Summary

The isospin transport in the 80Kr + 48Ca and 80Kr + 40Ca reactions at 35
MeV/nucleon was studied with calculations using the statistical
multifragmentation model (SMM, Ref. SMM ) in comparison with the data reported
by the FAZIA Collaboration FAZIA21 . Regarding the experimental data, it was
assumed that the events recorded with four FAZIA blocks are representative for
the mid-peripheral collisions of the studied reactions. In the analysis, the
ensembles of excited sources formulated and verified in previous studies of
ALADIN and other experimental data were considered as applicable for the
present reactions, an assumption supported by the obtained very satisfactory
representation of the measured isotopic distributions. The odd-even staggered
$\langle N\rangle/Z$ pattern of light fragments and its relation to the
composition of the source was assumed to be universally valid.

With these assumptions, the intermediate reaction system at the time of the
transition from its dynamical creation to the assumed statistical decay was
reconstructed by identifying it with the ensemble of sources whose decay is
treated with the SMM. It was found that the isospin exchange leads the
intermediate system to cover at least half of the way to equilibrium. The most
probable value of $87\pm 11\%$, in fact, does not exclude full equilibrium.
The reconstructed compositions were found to be slightly reduced in neutrons
with respect to the equilibrium compositions by amounts corresponding to a
loss of about two neutrons, similar for both reactions.

The compositions of the sources responsible for the production of the two
groups of light fragments with $Z\leq 4$ were found to be significantly
different. For the group with longitudinal velocities larger than those of the
coincident heavier fragments, the source composition is similar to that of the
heavier fragments, indicating that both, the heavier and the lighter
fragments, originate from the breakup of the same intermediate system. The
sources of light fragments with longitudinal velocities smaller than those of
the coincident heavier fragments were found to be enriched in neutrons beyond
the values expected for mixed systems consisting of equal numbers of nucleons
from the projectile and from the target. The assumption of a low-density neck
region, enriched in neutrons and with a mass of the order of 25 nucleons,
provided a consistent explanation.

These conclusions are based on SMM calculations using ensembles of excited
sources as initial configurations whose decay is assumed to be statistical and
followed according to the model. The simultaneous consideration of two
correlated statistical sources, one representing the projectile residues and
one describing the excited low-density matter in the neck region, represents a
novel theoretical development. The applicability of the method was
demonstrated by satisfactorily reproducing the reported isotope distributions
with reasonable assumptions regarding the effect of the target composition in
forming the excited projectile residues.

In the second part of the analysis, the isotopic composition of the ensemble
was systematically varied and the agreement of the calculated and experimental
results quantitatively evaluated. Two methods were used, one of comparing the
experimental and theoretical mean $\langle N\rangle/Z$ values, and the other
of minimizing the $\chi^{2}$ of the reproduction of the measured isotope
distributions by using the experimental errors. Both methods were found to
yield consistent results. The $\langle N\rangle/Z$ pattern as a function of
$Z$ of fragments from the neck-type source was successfully described by
assuming a lower density and higher excitation energy. These parameters are
similar to those found in the statistical analyses of central nucleus-nucleus
collisions at similar energies. Altogether, the chosen statistical approach
using two correlated sources, was found applicable for studying isospin
transport phenomena in mid-peripheral collisions in the Fermi energy domain.

###### Acknowledgements.

The authors are grateful to S. Piantelli and the FAZIA Collaboration for
providing published data in numerical form and to the ALADIN Collaboration for
giving access to the S254 experimental data. R.O. gratefully acknowledges the
support by TUBITAK (Turkish Scientific and Technological Research Council)
under Project No. BIDEB-1059B192001096 and the warm hospitality of GSI. N.B.,
A.S.B., and M.B. acknowledge support from the DAAD/TUBITAK exchange program.

## References

  * (1) Dynamics and Thermodynamics with Nuclear Degrees of Freedom, edited by Ph. Chomaz, F. Gulminelli, W. Trautmann, and S. J. Yennello (Springer, Berlin, 2006); Eur. Phys. J. A 30, 1 (2006).
  * (2) Topical Issue on Nuclear Symmetry Energy, edited by Bao-An Li, À. Ramos, G. Verde, and I. Vidaña (Springer, Berlin, 2014); Eur. Phys. J. A 50, 9 (2014).
  * (3) For examples see, e.g., B. Blank, T. Goigoux, P. Ascher, M. Gerbaux, J. Giovinazzo, S. Grévy et al., Phys. Rev. C 93, 061301 (2016); K. Schmidt, X. Cao, E. J. Kim, K. Hagel, M. Barbui, J. Gauthier et al., ibid. 95, 054618 (2017).
  * (4) S. Piantelli, G. Casini, A. Ono, G. Poggi, G. Pastore, S. Barlini et al., Phys. Rev. C 103, 014603 (2021).
  * (5) A. Camaiani, G. Casini, S. Piantelli, A. Ono, E. Bonnet, R. Alba et al., Phys. Rev. C 103, 014605 (2021).
  * (6) C. Ciampi, S. Piantelli, G. Casini, G. Pasquali, J. Quicray, L. Baldesi et al., Phys. Rev. C 106, 024603 (2022).
  * (7) R. Bougault, G. Poggi, S. Barlini, B. Borderie, G. Casini, A. Chbihi et al., Eur. Phys. J. A 50, 47 (2014).
  * (8) G. Pastore, D. Gruyer, P. Ottanelli, N. Le Neindre, G. Pasquali, R. Alba et al., Nucl. Instrum. Methods Phys. Res. A 860, 42 (2017).
  * (9) V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rep. 410, 335 (2005).
  * (10) Bao-An Li, Lie-Wen Chen, and Che Ming Ko, Phys. Rep. 464, 113 (2008).
  * (11) N. T. Porile and L. B. Church, Phys. Rev. 133, B310 (1964).
  * (12) T. H. Ku and P. J. Karol, Phys. Rev. C 16, 1984 (1977).
  * (13) J. P. Bondorf, A. S. Botvina, A. S. Iljinov, I. N. Mishustin, and K. Sneppen, Phys. Rep. 257, 133 (1995).
  * (14) A. S. Botvina, I. N. Mishustin, M. Begemann-Blaich, J. Hubele, G. Imme, I. Iori et al., Nucl. Phys. A 584, 737 (1995).
  * (15) Hongfei Xi, T. Odeh, R. Bassini, M. Begemann-Blaich, A. S. Botvina, S. Fritz et al., Z. Phys. A 359, 397 (1997); H. Xi et al., Eur. Phys. J. A 1, 235(E) (1998).
  * (16) R. Ogul, A. S. Botvina, U. Atav, N. Buyukcizmeci, I. N. Mishustin, P. Adrich et al., Phys. Rev. C 83, 024608 (2011); 85, 019903(E) (2012).
  * (17) H. Imal, A. Ergun, N. Buyukcizmeci, R. Ogul, A. S. Botvina, and W. Trautmann, Phys. Rev. C 91, 034605 (2015).
  * (18) M. D’Agostino, A. S. Botvina, P. M. Milazzo, M. Bruno, G. J. Kunde, D. R. Bowman et al., Phys. Lett. B 371, 175 (1996).
  * (19) N. Bellaize, O. Lopez, J. P. Wieleczko, D. Cussol, G. Auger, Ch. O. Bacri et al., Nucl. Phys. A 709, 367 (2002).
  * (20) J. Iglio, D. V. Shetty, S. J. Yennello, G. A. Souliotis, M. Jandel, A. L. Keksis, S. N. Soisson, B. C. Stein, S. Wuenschel, and A. S. Botvina, Phys. Rev. C 74, 024605 (2006).
  * (21) A. S. Botvina and I. N. Mishustin, Eur. Phys. J. A 30, 121 (2006).
  * (22) H. Imal, N. Buyukcizmeci, R. Ogul, and A. S. Botvina, Eur. Phys. J. A 56, 110 (2020).
  * (23) M. Di Toro, A. Olmi, and R. Roy, Eur. Phys. J. A 30, 65 (2006).
  * (24) N. Bohr, Nature 137, 344 (1936).
  * (25) R. J. Charity, M. A. McMahan, G. J. Wozniak, R. J. McDonald, L. G. Moretto, D. G. Sarantites et al., Nucl. Phys. A 483, 371 (1988).
  * (26) R. J. Charity, K. X. Jing, D. R. Bowman, M. A. McMahan, G. J. Wozniak, L. G. Moretto et al., Nucl. Phys. A 511, 59 (1990).
  * (27) L. Pienkowski, K. Kwiatkowski, T. Lefort, W.-c. Hsi, L. Beaulieu, V. E. Viola et al., Phys. Rev. C 65, 064606 (2002).
  * (28) D. H. E. Gross, Rep. Prog. Phys. 53, 605 (1990).
  * (29) A. S. Botvina, N. Buyukcizmeci, and M. Bleicher, Phys. Rev. C 103, 064602 (2021).
  * (30) A. S. Botvina, N. Buyukcizmeci, and M. Bleicher, Phys. Rev. C 106, 014607 (2022).
  * (31) T. Furuta and A. Ono, Phys. Rev. C 79, 014608 (2009).
  * (32) A. H. Raduta, M. Colonna, and M. Di Toro, Phys. Rev. C 76, 024602 (2007).
  * (33) R. J. Charity, Phys. Rev. C 58, 1073 (1998).
  * (34) C. P. Montoya, W. G. Lynch, D. R. Bowman, G. F. Peaslee, N. Carlin, R. T. deSouza et al., Phys. Rev. Lett. 73, 3070 (1994).
  * (35) J. F. Dempsey, R. J. Charity, L. G. Sobotka, G. J. Kunde, S. Gaff, C. K. Gelbke et al., Phys. Rev. C 54, 1710 (1996).
  * (36) J. Łukasik, J. Benlliure, V. Métivier, E. Plagnol, B. Tamain, M. Assenard et al., Phys. Rev. C 55, 1906 (1997).
  * (37) S. Piantelli, L. Bidini, G. Poggi, M. Bini, G. Casini, P. R. Maurenzig, A. Olmi, G. Pasquali, A. A. Stefanini, and N. Taccetti, Phys. Rev. Lett. 88, 052701 (2002).
  * (38) V. Baran, M. Colonna, and M. Di Toro, Nucl. Phys. A 730, 329 (2004).
  * (39) S. Hudan, R. Alfaro, L. Beaulieu, B. Davin, Y. Larochelle, T. Lefort et al., Phys. Rev. C 70, 031601(R) (2004).
  * (40) A. Ergun, H. Imal, N. Buyukcizmeci, R. Ogul, and A. S. Botvina, Phys. Rev. C 92, 014610 (2015).
  * (41) A. S. Botvina and I. N. Mishustin, Phys. Rev. C 63, 061601(R) (2001).
  * (42) V. Baran, M. Colonna, M. Di Toro, M. Zielinska-Pfabé, and H. H. Wolter, Phys. Rev. C 72, 064620 (2005).
  * (43) S. Piantelli, G. Casini, A. Ono, G. Poggi, G. Pastore, S. Barlini et al., Phys. Rev. C 101, 034613 (2020).
  * (44) S. Hudan, R. T. de Souza, and A. Ono, Phys. Rev. C 73, 054602 (2006).
  * (45) A. S. Botvina, K. K. Gudima, J. Steinheimer, M. Bleicher, and J. Pochodzalla, Phys. Rev. C 95, 014902 (2017).
  * (46) A. S. Botvina, A. S. Iljinov, and I. N. Mishustin, Yad. Fiz. 42, 1127 (1985) [Sov. J. Nucl. Phys. 42, 712 (1985)].
  * (47) A. S. Botvina, O. V. Lozhkin, and W. Trautmann, Phys. Rev. C 65, 044610 (2002).
  * (48) N. Buyukcizmeci, R. Ogul, and A. S. Botvina, Eur. Phys. J. A 25, 57 (2005).
  * (49) G. A. Souliotis, A. S. Botvina, D. V. Shetty, A. L. Keksis, M. Jandel, M. Veselsky, and S. J. Yennello, Phys. Rev. C 75, 011601(R) (2007).
  * (50) S. Hudan, A. B. McIntosh, J. Black, D. Mercier, C. J. Metelko, R. Yanez et al., Phys. Rev. C 80, 064611 (2009).
  * (51) A. Le Fèvre, G. Auger, M. L. Begemann-Blaich, N. Bellaize, R. Bittiger, F. Bocage et al., Phys. Rev. Lett. 94, 162701 (2005).
  * (52) Ad. R. Raduta and F. Gulminelli, Phys. Rev. C 75, 024605 (2007).
  * (53) A. Ono, P. Danielewicz, W. A. Friedman, W. G. Lynch, and M. B. Tsang, Phys. Rev. C 70, 041604(R) (2004).
  * (54) S. Barlini, S. Piantelli, G. Casini, P. R. Maurenzig, A. Olmi, M. Bini et al., Phys. Rev. C 87, 054607 (2013).
  * (55) W. Trautmann, Nucl. Phys. A 787, 575c (2007).
  * (56) J. Łukasik, P. Adrich, T. Aumann, C. O. Bacri, T. Barczyk, R. Bassini et al., Nucl. Instrum. Methods Phys. Res. A 587, 413 (2008).
  * (57) D. Henzlova, K. H. Schmidt, M. V. Ricciardi, A. Kelić, V. Henzl, P. Napolitani et al., Phys. Rev. C 78, 044616 (2008).
  * (58) J. Colin, D. Cussol, J. Normand, N. Bellaize, R. Bougault, A. M. Buta et al., Phys. Rev. C 67, 064603 (2003).
  * (59) H. Johnston, T. White, J. Winger, D. Rowland, B. Hurst, F. Gimeno-Nogues, D. O’Kelly, and S. J. Yennello, Phys. Lett. B 371, 186 (1996).
  * (60) M. B. Tsang, T. X. Liu, L. Shi, P. Danielewicz, C. K. Gelbke, X. D. Liu et al., Phys. Rev. Lett. 92, 062701 (2004).
  * (61) A. L. Keksis, L. W. May, G. A. Souliotis, M. Veselsky, S. Galanopoulos, Z. Kohley et al., Phys. Rev. C 81, 054602 (2010).
  * (62) Q. Fable, A. Chbihi, J. D. Frankland, P. Napolitani, G. Verde, E. Bonnet et al., Phys. Rev. C 107, 014604 (2023).
  * (63) D. D. S. Coupland, W. G. Lynch, M. B. Tsang, P. Danielewicz, and Y. Zhang, Phys. Rev. C 84, 054603 (2011).
  * (64) G. Ademard, B. Borderie, A. Chbihi, O. Lopez, P. Napolitani, M. F. Rivet et al., Eur. Phys. J. A 50, 33 (2014).
  * (65) A. Jedele, A. B. McIntosh, K. Hagel, M. Huang, L. Heilborn, Z. Kohley et al., Phys. Rev. Lett. 118, 062501 (2017).
  * (66) K. Brown, S. Hudan, R. T. deSouza, J. Gauthier, R. Roy, D. V. Shetty, G. A. Souliotis, and S. J. Yennello, Phys. Rev. C 87, 061601(R) (2013).
  * (67) D. Thériault, J. Gauthier, F. Grenier, F. Moisan, C. St-Pierre, R. Roy et al., Phys. Rev. C 74, 051602(R) (2006).
  * (68) Z. Kohley, L. W. May, S. Wuenschel, M. Colonna, M. Di Toro, M. Zielinska-Pfabe et al., Phys. Rev. C 83, 044601 (2011).
  * (69) E. De Filippo, A. Pagano, P. Russotto, F. Amorini, A. Anzalone, L. Auditore et al., Phys. Rev. C 86, 014610 (2012).
  * (70) R. Bougault, E. Bonnet, B. Borderie, A. Chbihi, D. Dell’Aquila, Q. Fable et al., Phys. Rev. C 97, 024612 (2018).