11institutetext:  University of Wroclaw, 50-137 Wroclaw, Poland
22institutetext: Bogoliubov Laboratory of Theoretical Physics, JINR Dubna,
141980 Dubna, Russia 33institutetext: National Research Nuclear University
(MEPhI), 115409 Moscow, Russia

# Locating the special point of hybrid neutron stars

Mateusz Cierniak 11<EMAIL_ADDRESS>David Blaschke 112233
<EMAIL_ADDRESS>
###### Abstract

The special point is a feature unique to models of hybrid neutron stars. It
represents a location on their mass–radius sequences that is insensitive to
the phase transition density. We consider hybrid neutron stars with a core of
deconfined quark matter that obeys a constant–sound–speed (CSS) equation of
state model and provide a fit formula for the coordinates of the special point
as functions of the squared sound speed ($c_{s}^{2}$) and pressure scale ($A$)
parameters. Using the special point mass as a proxy for the maximum mass of
the hybrid stars we derive limits for the CSS model parameters based on the
recent NICER constraint on mass and radius of pulsar PSR J0740+6620,
$0.36<{c^{2}_{s}}_{\rm min}<0.43$ and $80<A[{\rm MeV/fm}^{3}]<160$. The upper
limit for the maximum mass of hybrid stars depends on the upper limit for
$c_{s}^{2}$ so that choosing $c^{2}_{s,max}=0.6$ results in $M_{\rm
max}<2.7\leavevmode\nobreak\ M_{\odot}$, within the mass range of GW190814.

##  1 Introduction

The recent observation of gravitational waves originating from the inspiral of
a binary system of neutron stars has resulted in a significant step forward in
the field of nuclear astrophysics. The event, labeled GW170817
LIGOScientific:2017vwq , along with a subsequent observation of the resulting
kilonova LIGOScientific:2017ync  signaled the beginning of the era of multi-
messenger astronomy. As a direct result, for the first time, it became
possible to determine limits on the neutron star tidal deformability, and due
to its connection with the equation of state (EoS) of dense neutron star
matter Hinderer:2009ca  also to obtain limits on the masses and radii of the
merging stars Annala:2017llu ; Bauswein:2017vtn . In addition, this event has
reignited the discussion on signals of quark- matter in the cores of neutron
stars Bauswein:2018bma ; Bauswein:2020aag .

The possibility of finding this rare state of matter inside neutron stars
provides a unique opportunity of accessing the cold and dense sector of the
Quantum Chromodynamics (QCD) phase diagram, which remains out of reach for
first–principle lattice QCD calculations. A large body of work exists on the
possible states of cold, strongly interacting matter and their relation to
neutron star properties (for recent reviews, see Baym:2017whm ;
Blaschke:2018mqw  and references therein), which were triggered by the
observation of pulsars with masses around $2\leavevmode\nobreak\ M_{\odot}$
Demorest:2010bx ; Antoniadis:2013pzd ; NANOGrav:2019jur ; Fonseca:2021wxt .
But without a radius measurement at this high mass, there are still too many
EoS scenarios possible, with and without a deconfinement phase transition.

Efforts are being made to solve the problem of a reliable radius measurement
by employing the X-ray timing method (cf. Bogdanov:2006zd  and references
therein). For that purpose, the NASA NICER mission was deployed to the
International Space Station in June 2017 and has since then gathered
luminosity data from a few millisecond pulsars. Results have been published
for two objects,: the heaviest known pulsar with a precisely determined mass,
PSR J0740+6620 Miller:2021qha ; Riley:2021pdl , and an intermediate mass
pulsar, PSR J0030+0451 Miller:2019cac ; Riley:2019yda . The latter only
marginally agrees with the analysis of GW170817 LIGOScientific:2018cki  in the
same mass range, resulting in a narrow region of possible radii for
intermediate-mass neutron stars Capano:2019eae , or perhaps hint at a signal
of the third family of compact stars Gerlach:1968zz  which would give rise to
the phenomenon of mass twin stars Glendenning:1998ag  that could occur in
different mass ranges Christian:2017jni , also in the mass range of GW170817,
see Blaschke:2018mqw  and references therein.

The high-mass pulsar PSR J0740+6620 is observed to have a rather large radius,
see figure 1. This creates tension with the idea of purely hadronic neutron
stars. Hadronic matter tends to soften at higher densities due to the onset of
hyperons Chatterjee:2015pua  which creates the "hyperon puzzle" problem for
nonrelativistic EoS models that are based on realistic nucleon-nucleon
interactions like the APR EoS Akmal:1998cf . Adding a repulsive few-nucleon
interaction in the form of a multi-pomeron potential solves the hyperon puzzle
for such models Yamamoto:2015lwa ; Yamamoto:2017wre  but this cannot explain
the large radius for PSR J0740+6620.

While not completely excluded, the scenario of purely hadronic neutron star
interiors is being increasingly disfavored by observation. A possible
alternative are multi–phase models, predicting a phase transition from purely
hadronic neutron stars at low mass to intermediate and high mass hybrid stars
with a core presenting exotic states of matter, for instance a quark–gluon
plasma (QGP). Such constructions are not completely free of the shortcomings
of hadronic models, as the common convention is to construct a Maxwell–type
first order phase transition from a stiffer hadronic equation of state (EoS)
to a softer one, represented by the QGP. Recently, a number of alternative
approaches were proposed Masuda:2012ed ; Baym:2017whm ; Ayriyan:2021prr  which
join a soft low-density hadronic EoS with stiff QGP models at high densities,
thus easily fulfilling all observational constraints.

In this manuscript, we will restrict ourselves to the standard Maxwell
construction of a phase transition and make use of the novel insights due to
the special point property of hybrid neutron star sequences Yudin:2014mla ;
Cierniak:2020eyh ; Blaschke:2020vuy ; Cierniak:2021knt  in order to determine
the impact of the recent observations on the available parameter space of a
class of constant–speed–of–sound (CSS) QGP models Alford:2013aca , which were
shown to provide an accurate description of color–superconducting quark matter
Zdunik:2012dj ; Antic:2021zbn .

The manuscript is organized as follows. After introducing the two-phase hybrid
EoS model for dense matter in the next section, we present results of an
extensive scan of the possible locations of the special point in the mass-
radius plane in section 3. In section 4, we introduce a fit formula that
relates the coordinates of the special point to the CSS model parameters $A$
and $c_{s}^{2}$ and obtain as the main result of this contribution the mapping
of the new NICER mass-radius constraint for PSR J0740+6620 onto the
$A-c_{s}^{2}$ plane. In section 5 we draw our Conclusions.

##  2 The two phase dense matter model

In the absence of a unified description of cold hadronic and quark matter with
a phase transition at high densities, the hybrid star matter EoS must be
constructed by matching separate EoS, comparing pressures derived using
hadrons, $P_{h}(\mu)$, or deconfined quarks, $P_{q}(\mu)$, as degrees of
freedom. The EoS with the higher pressure is the one that is preferred by
nature and describes the bulk thermodynamics. A crossing of both alternative
EoS at the critical chemical potential $\mu_{c}$ marks a first-order phase
transition fulfilling the condition $P_{h}(\mu_{c})=P_{q}(\mu_{c})$ for the
Maxwell construction.

The hadronic EoS discussed in this study is chosen from the class of
relativistic density-functional (RDF) models based on the DD2 parametrization
Typel:2009sy  with excluded volume effects modelled according to Typel:2016srf
. Its inputs are fitted to the properties of finite nuclei and it predicts the
EoS for nuclear matter at low densities up to and at nuclear saturation in
accordance with the phenomenological data. Therefore, it should provide a
reliable basis for extrapolations to supersaturation densities.

For the deconfined quark matter phase, the class of CSS quark matter models
will be used. The EoS in the form of pressure as a function of the
baryochemical potential reads (see also Cierniak:2020eyh  or the Appendix of
(Alford:2013aca, )),

$P(\mu)=A(\mu/\mu_{0})^{1+c_{s}^{-2}}-B,$ (1)

where the model parameters $A$, $B$ and $c_{s}^{2}$ are constant and
$\mu_{0}=1$ GeV. The baryon density $n$ follows from the canonical relation

$\frac{dP(\mu)}{d\mu}=n(\mu)=(1+c_{s}^{-2})\frac{A}{\mu_{0}}\left(\frac{\mu}{\mu_{0}}\right)^{c_{s}^{-2}}.$
(2)

Using the above, we arrive at the energy density

$\varepsilon=\mu n-P=B+c_{s}^{-2}A(\mu/\mu_{0})^{1+c_{s}^{-2}}.$ (3)

The relation between pressure and energy density takes the form

$P=c_{s}^{2}\varepsilon-(1+c_{s}^{2})B.$ (4)

It has been shown in Zdunik:2012dj  that the above CSS model fits well the EoS
of color superconducting quark matter in both, the 2SC and the CFL phases
which were obtained from a self-consistent solution of the three-flavor NJL
model in the mean field approximation (Blaschke:2005uj, ; Klahn:2013kga, ).

##  3 The special point

The special point of hybrid neutron stars is a property first described in
Yudin:2014mla  and recently studied in Cierniak:2020eyh ; Blaschke:2020vuy ;
Cierniak:2021knt . It is a characteristic feature of a broad range of
two–phase models (see, e.g., Blaschke:2010vd ; Kaltenborn:2017hus ;
Blaschke:2020qqj ), presenting itself as a stationary point in the mass–radius
diagram for hybrid star sequences. Any parametrization of the hybrid model,
which varies only by the onset density of the second phase, will cross this
point. This is illustrated in figure 1.

Figure 1: Special point as the intersection of hybrid star sequences (red
solid lines) on the neutron star mass radius diagram. The black line
represents the hadronic EoS with excluded volume DD2p40, naming convention
analogous to Typel:2016srf . The intersection of the red lines with the black
one mark the onset of CSS quark matter in the core, see also Cierniak:2021knt
. Observational data taken from Miller:2021qha ; Miller:2019cac ;
Riley:2021pdl ; Riley:2019yda ; LIGOScientific:2020zkf ; Bauswein:2017vtn ;
Annala:2017llu  Figure 2:  Location of the special point (green area) for
$0.25<c_{s}^{2}<1.0$ and $80<A[{\rm MeV/fm}^{3}]<300$, lines of constant
$c_{s}^{2}$ and $A$ are shown for comparison. Figure adopted from
Cierniak:2021knt . Observational data taken from Miller:2021qha ;
Miller:2019cac ; Riley:2021pdl ; Riley:2019yda ; LIGOScientific:2020zkf ;
Bauswein:2017vtn ; Annala:2017llu

As was shown in Cierniak:2020eyh ; Blaschke:2020vuy ; Cierniak:2021knt , this
feature can be used to characterize a broad family of EoS, without the need to
fully scan the parameter space of the model, and to easily compare model
predictions with observational data. Most recently, this has been demonstrated
in Cierniak:2021knt , where a prediction on the radius range of a hybrid star
fulfilling PSR J0740+6620 mass–radius measurement was derived.

A detailed map of the special point locations and their dependence on the CSS
model parameters has been made in Cierniak:2021knt , shown here as figure 2,
and will be used in the following chapter. It has been shown in
Cierniak:2020eyh ; Cierniak:2021knt  that this map is independent of the
underlying hadronic EoS, the phase transition density and the type of the
phase transition (first order, crossover etc.). Therefore, it can be used to
isolate the hybrid neutron star branch and analyze observational data in the
context of constraints on the parameters of the microscopic model for
deconfined quark matter.

##  4 Parameter fit

The coordinates of the special point are to a good approximation described by
the following relationship

$\begin{cases}R_{SP}[{\rm
km}]=a_{R}\cdot\left(c^{2}_{s}\right)^{b_{R}}\cdot\left(A[{\rm
MeV/fm}^{3}]\right)^{c_{R}}+d_{R},\\\
M_{SP}[M_{\odot}]=a_{M}\cdot\left(c^{2}_{s}\right)^{b_{M}}\cdot\left(A[{\rm
MeV/fm}^{3}]\right)^{c_{M}}+d_{M}.\end{cases}$ (5)

The accuracy of this approximation can be judged from figures 3 and 4, where
the data points were taken from Cierniak:2021knt  (see also figure 2) and the
black solid lines are obtained using the fit formulas of Eq. (5) with the
parameters listed in table 1.

Figure 3: Masses of the special point location (colored points) as function of
the CSS parameters $c^{2}_{s}$ (left panel) and $A$ (right panel) compared to
the fit formula (5) (black line). See text for details.

Figure 4: Radii of the special point location (colored points) as function of the CSS parameters $c^{2}_{s}$ (left panel) and $A$ (right panel) compared to the fit formula (5) (black line). See text for details. Table 1: Parameters of the fit formulas Eq. (5) for the special point location in the $M-R$ diagram. X | $a_{X}$ | $b_{X}$ | $c_{X}$ | $d_{X}$
---|---|---|---|---
R | 145.938 | 0.420 | -0.47 | -0.4
M | 35.332 | 0.648 | -0.47 | -0.4

By inverting the fit (5) we were able to translate existing observational
constraints on neutron star masses and radii to constraints on the CSS model
parameter space. This is the main result of this work, shown in Fig. 5.

Figure 5: Left panel: Neutron star mass–radius diagram with the 1$\sigma$
results of the NICER measurement of PSR J0740+6620 Miller:2021qha ;
Riley:2021pdl . The light green band is the mass range for the lighter object
in the binary merger GW190814. The grey area approximates the region of
special point locations for which the hybrid star sequence would fulfill the
radius constraint from PSR J0740+6620. For orientation, we show limiting
values of constant $A$ as well as $c_{s}^{2}=0.5$. In the small red area lie
all physical parametrizations of the color superconducting nonlocal NJL model
of quark matter Antic:2021zbn  Right panel: The same areas and lines as in the
left panel, translated to the coordinate system spanned by the CSS model
parameters $c^{2}_{s}$ and $A$.

##  5 Conclusions

The special point typically corresponds to a mass slightly below $M_{\rm
max}$, no less than $M_{\rm max}-M_{\rm SP}=0.2\leavevmode\nobreak\ M_{\odot}$
Cierniak:2020eyh . If we assume $M_{\rm SP}\approx M_{\rm max}$ and that PSR
J0740+6620 represents the heaviest possible neutron star, we arrive at a
narrow band of CSS model parameters that remain in agreement with this
observation. Further demanding the $A$ parameter not to fall below a value of
80 MeV/fm3, a value which ensures a phase transition not below the saturation
density (cf. Cierniak:2021knt ), we can determine a narrow band of possible
values for the speed of sound parameter $c^{2}_{s}$. For $A=80$ MeV/fm3, the
possible range is $0.36\leq c^{2}_{s}\leq 0.43$. Higher values of $A$ would
subsequently drive the speed of sound to greater values with a limit in the
vicinity of $c^{2}_{s}\approx 0.6$.

Translating back the requirement of a transition at supersaturation density,
we arrive at a limit represented by the black dashed line in the left panel of
Fig. 5. Only the left side of this limit can be achieved with the CSS model
assuming $M_{\rm max}\approx M_{J0740}$. A potential revision of the
measurement, that would suggest a higher radius for the star would signal,
that this is not the highest possible mass for a neutron star, or that the CSS
model does not accurately approximate the cold and dense matter EoS.

Finally, the grey band of the right panel of figure 5 shows the available
parameter space of the CSS model, assuming $M_{\rm max}>M_{J0740}$, for which
the model would remain in agreement with the NICER observation. For $A=80$
MeV/fm3, this results in an upper limit on $c^{2}_{s,max}=0.8$, along with the
previous lower limit of $c^{2}_{s,min}=0.36$.

We have demonstrated how the special point feature of hybrid neutron stars can
be applied to provide direct limits on the chosen quark matter model
parameters by translating the novel NICER measurement of PSR J0740+6620 to the
parameter space of the CSS model. Further constraints can be derived from
existing measurements, a task which we leave for subsequent studies. Due to
the link between the phenomenological CSS model and the color–superconducting
nlNJL model illustrated in Antic:2021zbn , such limits can provide valuable
insight into the phenomenon of color–superconductivity in quark matter at high
densities.

## Acknowledgements

M.C. and D.B. acknowledge support from the Polish National Science Centre
(NCN) under grant number 2019/33/B/ST9/03059. The work of D.B. was supported
by the Russian Foundation for Basic Research under grant No. 18-02-40137 and
by the Federal Program "Priority-2030".

## References

  * (1) B.P. Abbott et al., Phys. Rev. Lett. 119, 161101 (2017), 1710.05832
  * (2) B.P. Abbott et al., Astrophys. J. Lett. 848, L12 (2017), 1710.05833
  * (3) T. Hinderer, B.D. Lackey, R.N. Lang, J.S. Read, Phys. Rev. D 81, 123016 (2010), 0911.3535
  * (4) E. Annala, T. Gorda, A. Kurkela, A. Vuorinen, Phys. Rev. Lett. 120, 172703 (2018), 1711.02644
  * (5) A. Bauswein, O. Just, H.T. Janka, N. Stergioulas, Astrophys. J. Lett. 850, L34 (2017), 1710.06843
  * (6) A. Bauswein, N.U.F. Bastian, D.B. Blaschke, K. Chatziioannou, J.A. Clark, T. Fischer, M. Oertel, Phys. Rev. Lett. 122, 061102 (2019), 1809.01116
  * (7) A. Bauswein, S. Blacker, V. Vijayan, N. Stergioulas, K. Chatziioannou, J.A. Clark, N.U.F. Bastian, D.B. Blaschke, M. Cierniak, T. Fischer, Phys. Rev. Lett. 125, 141103 (2020), 2004.00846
  * (8) G. Baym, T. Hatsuda, T. Kojo, P.D. Powell, Y. Song, T. Takatsuka, Rept. Prog. Phys. 81, 056902 (2018), 1707.04966
  * (9) D. Blaschke, N. Chamel, Astrophys. Space Sci. Libr. 457, 337 (2018), 1803.01836
  * (10) P. Demorest, T. Pennucci, S. Ransom, M. Roberts, J. Hessels, Nature 467, 1081 (2010), 1010.5788
  * (11) J. Antoniadis et al., Science 340, 6131 (2013), 1304.6875
  * (12) H.T. Cromartie et al. (NANOGrav), Nature Astron. 4, 72 (2019), 1904.06759
  * (13) E. Fonseca et al., Astrophys. J. Lett. 915, L12 (2021), 2104.00880
  * (14) S. Bogdanov, G.B. Rybicki, J.E. Grindlay, Astrophys. J. 670, 668 (2007), astro-ph/0612791
  * (15) M.C. Miller et al., Astrophys. J. Lett. 918, L28 (2021), 2105.06979
  * (16) T.E. Riley et al., Astrophys. J. Lett. 918, L27 (2021), 2105.06980
  * (17) M.C. Miller et al., Astrophys. J. Lett. 887, L24 (2019), 1912.05705
  * (18) T.E. Riley et al., Astrophys. J. Lett. 887, L21 (2019), 1912.05702
  * (19) B.P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 121, 161101 (2018), 1805.11581
  * (20) C.D. Capano, I. Tews, S.M. Brown, B. Margalit, S. De, S. Kumar, D.A. Brown, B. Krishnan, S. Reddy, Nature Astron. 4, 625 (2020), 1908.10352
  * (21) U.H. Gerlach, Phys. Rev. 172, 1325 (1968)
  * (22) N.K. Glendenning, C. Kettner, Astron. Astrophys. 353, L9 (2000), astro-ph/9807155
  * (23) J.E. Christian, A. Zacchi, J. Schaffner-Bielich, Eur. Phys. J. A 54, 28 (2018), 1707.07524
  * (24) D. Chatterjee, I. Vidaña, Eur. Phys. J. A 52, 29 (2016), 1510.06306
  * (25) A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998), nucl-th/9804027
  * (26) Y. Yamamoto, T. Furumoto, N. Yasutake, T.A. Rijken, Eur. Phys. J. A 52, 19 (2016), 1510.06099
  * (27) Y. Yamamoto, H. Togashi, T. Tamagawa, T. Furumoto, N. Yasutake, T.A. Rijken, Phys. Rev. C 96, 065804 (2017), 1708.06163
  * (28) K. Masuda, T. Hatsuda, T. Takatsuka, PTEP 2013, 073D01 (2013), 1212.6803
  * (29) A. Ayriyan, D. Blaschke, A.G. Grunfeld, D. Alvarez-Castillo, H. Grigorian, V. Abgaryan, Eur. Phys. J. A 57, 318 (2021), 2102.13485
  * (30) A.V. Yudin, T.L. Razinkova, D.K. Nadyozhin, A.D. Dolgov, Astron. Lett. 40, 201 (2014), 1404.0865
  * (31) M. Cierniak, D. Blaschke, Eur. Phys. J. ST 229, 3663 (2020), 2009.12353
  * (32) D. Blaschke, M. Cierniak, Astron. Nachr. 342, 227 (2021), 2012.15785
  * (33) M. Cierniak, D. Blaschke, Astron. Nachr. 342, 819 (2021), 2106.06986
  * (34) M.G. Alford, S. Han, M. Prakash, Phys. Rev. D 88, 083013 (2013), 1302.4732
  * (35) J.L. Zdunik, P. Haensel, Astron. Astrophys. 551, A61 (2013), 1211.1231
  * (36) S. Antić, M. Shahrbaf, D. Blaschke, A.G. Grunfeld (2021), 2105.00029
  * (37) S. Typel, G. Ropke, T. Klahn, D. Blaschke, H.H. Wolter, Phys. Rev. C 81, 015803 (2010), 0908.2344
  * (38) S. Typel, Eur. Phys. J. A 52, 16 (2016)
  * (39) D. Blaschke, S. Fredriksson, H. Grigorian, A. Öztas, F. Sandin, Phys. Rev. D 72, 065020 (2005)
  * (40) T. Klähn, R. Lastowiecki, D. Blaschke, Phys. Rev. D 88 (8), 085001 (2013)
  * (41) D. Blaschke, T. Klahn, R. Lastowiecki, F. Sandin, J. Phys. G 37, 094063 (2010), 1002.1299
  * (42) M.A.R. Kaltenborn, N.U.F. Bastian, D.B. Blaschke, Phys. Rev. D 96, 056024 (2017), 1701.04400
  * (43) D. Blaschke, A. Ayriyan, D.E. Alvarez-Castillo, H. Grigorian, Universe 6, 81 (2020), 2005.02759
  * (44) R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 896, L44 (2020), 2006.12611


# Isospin diffusion measurement
from the direct detection of a Quasi-Projectile remnant

A. Camaiani<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Firenze, Italy  INFN, Sezione di Firenze, Italy  G. Casini  INFN, Sezione di
Firenze, Italy  S. Piantelli  INFN, Sezione di Firenze, Italy  A. Ono
Department of Physics, Tohoku University, Sendai 980-8578, Japan  E. Bonnet
SUBATECH, Université de Nantes, IMT Atlantique, IN2P3/CNRS, 4 Rue Alfred
Kastler, 44307 Nantes Cedex 3, France  R. Alba  INFN Laboratori Nazionali del
Sud, Via S. Sofia 62, 95125 Catania, Italy  S. Barlini  Dipartimento di
Fisica, Università di Firenze, Italy  INFN, Sezione di Firenze, Italy  B.
Borderie  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  R.
Bougault  Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000
Caen, France  C. Ciampi  Dipartimento di Fisica, Università di Firenze, Italy
A. Chbihi  Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DRF -
CNRS/IN2P3, Boulevard Henri Becquerel, F-14076 Caen, France  M. Cicerchia
INFN Laboratori Nazionali di Legnaro, 35020 Legnaro, Italy  M. Cinausero  INFN
Laboratori Nazionali di Legnaro, 35020 Legnaro, Italy  J.A. Dueñas  Depto. de
Ingeniería Eléctrica y Centro de Estudios Avanzados en Física, Matemáticas y
Computación, Universidad de Huelva, 21007 Huelva, Spain  D. Dell’Aquila
Dipartimento di Chimica e Farmacia, Università degli Studi di Sassari,
Sassari, Italy  INFN - Laboratori Nazionali del Sud, Catania, Italy  Q. Fable
Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen,
France  D. Fabris  INFN Sezione di Padova, 35131 Padova, Italy  C. Frosin
Dipartimento di Fisica, Università di Firenze, Italy  INFN, Sezione di
Firenze, Italy  J. D. Frankland  Grand Accélérateur National d’Ions Lourds
(GANIL), CEA/DRF - CNRS/IN2P3, Boulevard Henri Becquerel, F-14076 Caen, France
F. Gramegna  INFN Laboratori Nazionali di Legnaro, 35020 Legnaro, Italy  D.
Gruyer  Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000
Caen, France  K. I. Hahn  Department of Science Education, Ewha Womans
University, Seoul 03760, Republic of Korea  M. Henri  Grand Accélérateur
National d’Ions Lourds (GANIL), CEA/DRF - CNRS/IN2P3, Boulevard Henri
Becquerel, F-14076 Caen, France  B. Hong  Center for Extreme Nuclear Matters
(CENuM), Korea University, Seoul 02841, Republic of Korea  Department of
Physics, Korea University, Seoul 02841, Republic of Korea  S. Kim  Department
of Science Education, Ewha Womans University, Seoul 03760, Republic of Korea
A. Kordyasz  Heavy Ion Laboratory, University of Warsaw, 02-093 Warszawa,
Poland  M. J. Kweon  Center for Extreme Nuclear Matters (CENuM), Korea
University, Seoul 02841, Republic of Korea  Department of Physics, Inha
University, Incheon 22212, Republic of Korea  H. J. Lee  Department of
Physics, Inha University, Incheon 22212, Republic of Korea  J. Lemarié  Grand
Accélérateur National d’Ions Lourds (GANIL), CEA/DRF - CNRS/IN2P3, Boulevard
Henri Becquerel, F-14076 Caen, France  N. LeNeindre  Normandie Université,
ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France  I. Lombardo  INFN
Sezione di Catania, 95123 Catania, Italy  O. Lopez  Normandie Université,
ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France  T. Marchi  INFN
Laboratori Nazionali di Legnaro, 35020 Legnaro, Italy  S. H. Nam  Center for
Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Republic of
Korea  Department of Physics, Korea University, Seoul 02841, Republic of Korea
P. Ottanelli  Dipartimento di Fisica, Università di Firenze, Italy  INFN,
Sezione di Firenze, Italy  M. Parlog  Normandie Université, ENSICAEN, UNICAEN,
CNRS/IN2P3, LPC Caen, 14000 Caen, France  ”Horia Hulubei” National Institute
of Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest Magurele,
Romania  G. Pasquali  Dipartimento di Fisica, Università di Firenze, Italy
INFN, Sezione di Firenze, Italy  G. Poggi  Dipartimento di Fisica, Università
di Firenze, Italy  INFN, Sezione di Firenze, Italy  J. Quicray  Normandie
Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France  A. A.
Stefanini  Dipartimento di Fisica, Università di Firenze, Italy  INFN, Sezione
di Firenze, Italy  S. Upadhyaya  Faculty of Physics, Astronomy and Applied
Computer Science, Jagiellonian University, 30-348 Kracow, Poland  S. Valdré
INFN, Sezione di Firenze, Italy  E. Vient  Normandie Université, ENSICAEN,
UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France

###### Abstract

The neutron-proton equilibration process in 48Ca+40Ca at 35 MeV/nucleon
bombarding energy has been experimentally estimated by means of the isospin
transport ratio. Experimental data have been collected with a subset of the
FAZIA telescope array, which permitted to determine $Z$ and $N$ of detected
fragments. For the first time, the QP evaporative channel has been compared
with the QP break-up one in a homogeneous and consistent way, pointing out to
a comparable n-p equilibration which suggests close interaction time between
projectile and target independently of the exit channel. Moreover, in the QP
evaporative channel n-p equilibration has been compared with the prediction of
the Antisymmetrized Molecular Dynamics (AMD) model coupled to the GEMINI
statistical model as an afterburner, showing a larger probability of proton
and neutron transfers in the simulation with respect to the experimental data.

##  I Introduction

Since the end of the ’80s some experiments, mostly focused on dissipative
collisions below 20 MeV/nucleon, investigated how a colliding system with
projectile and target with different “chemical” composition, evolves towards
the charge equilibration Planeta  _et al._ (1988); Gippner  _et al._ (1988);
Madani  _et al._ (1995); Marchetti  _et al._ (1993). Later on, the so-called
isospin dynamics, namely the neutron-proton (n-p) exchange between two
interacting nuclei, gained much attention at Fermi energies (20-100
MeV/nucleon), where nuclear subsystems relatively far from the saturation
value of the baryon density can be explored; this, in turn, allows to
investigate how the nuclear Equation of State (nEoS) rules the dynamics Tsang
_et al._ (2004, 2009). In the Fermi energy domain, interesting signals have
been found mainly in binary semi-peripheral collisions, mosrly the clear
evidence of a neutron enrichment of the fragments emitted from the phase-space
region between the two main reaction products (also labeled mid-velocity or
neck region) Łukasik  _et al._ (1997); Plagnol  _et al._ (1999); Thériault
_et al._ (2005, 2006). A theoretical interpretation was proposed and timely
developed in the framework of nuclear reaction models, in order to describe
the isotopic composition of the emerging excited Quasi-Projectile (QP) and
Quasi-Target (QT) after the collision: the n-p equilibration is largely due to
the initial different concentration of neutrons and protons between projectile
and target (isospin diffusion) while the neutron enrichment of the mid-
velocity zone is ascribed to the density gradient, which arises between the
different regions of the colliding systems (isospin drift) Baran  _et al._
(2005a); Lionti  _et al._ (2005); Napolitani  _et al._ (2010). In this paper,
we discuss about isospin diffusion and how it guides the system towards the
n-p equilibration.

The degree of charge equilibration is strictly related both to the driving
force which rules the n-p exchange and to the interaction time. In particular,
the isospin diffusion is sensitive to the symmetry energy term
$E_{\mathrm{sym}}$ of the nEoS Napolitani  _et al._ (2010); Baran  _et al._
(2005a), and it has been used, in the past, to put some constraints on that
and on the whole parametrization Tsang  _et al._ (2004, 2009); Sun  _et al._
(2010). However, to date, a clear knowledge of the symmetry energy is still
lacking, namely the Taylor expansion coefficients are known with large
uncertainties (first order term, $L_{\mathrm{sym}}$) or not at all (second
order, $K_{\mathrm{sym}}$, and higher order coefficients) Margueron  _et al._
(2018). Concerning the interaction time, for a given restoring potential, the
longer the interaction time the more equilibrated in isospin the system Tsang
_et al._ (2004). In this sense, different effects contribute to the
equilibration, such as in-medium effects which significantly reduce the
nucleon-nucleon cross section with respect to the nucleon-nucleon value Lopez
_et al._ (2014), or cluster correlations that arise during the collision
Coupland  _et al._ (2011). Therefore, a characterization of the collision as a
function of the reaction centrality is mandatory in order to explore different
interaction times.

During the years the experimental investigations followed two main paths. The
first one exploited detection arrays covering a large part of the solid angle
in order to globally characterize the acquired events, although with
limitations in terms of isotopic separation (typically below $Z\approx 8$)
Souza  _et al._ (1990); Pouthas  _et al._ (1995); Pagano (2012). As a
consequence, in such studies Tsang  _et al._ (2004, 2009); Liu  _et al._
(2007); Galichet  _et al._ (2009); Sun  _et al._ (2010); Bougault  _et al._
(2018) only the lightest QP decay products could be used to extract
information on the isospin equilibration. The second one adopted mass
spectrometers, in order to directly access to the neutron-proton ratio ($N/Z$)
of the QP remnants, at the expense of covering a small part of the solid angle
and detecting only the main fragment of the event. Consequently, no
information on break-up events or Intermediate Mass Fragments (IMFs) and/or
Light Charged Particles (LCPs) accompanying the QP could be obtained in
typical configurations Souliotis  _et al._ (2006, 2014). On the other hand,
according to the literature Baran  _et al._ (2005b); Napolitani  _et al._
(2010), the experimental determination of the $N/Z$ content of the QP remnant
could be a good probe to put constraints on the symmetry energy. In such a
scenario, it could be useful to directly detect the isospin content of the QP
remnant, together with the accompanying particles or fragments. An example in
this direction is the recent paper of the NIMROD collaboration where the
authors reconstructed the isospin of the QP remnant May  _et al._ (2018).

The present work fits with this panorama, aiming at the investigation of the
isospin diffusion in peripheral and semi-peripheral reactions and trying to
overcome the limitation of previous detectors. In fact we investigated the
asymmetric reaction 48Ca+40Ca at 35 MeV/nucleon by means of the FAZIA multi-
telescope array, mainly for two reasons. Firstly, Ca isotopes allow to stress
the isospin unbalance of the entrance channel, moving from
$(\frac{N}{Z})_{{}^{48}Ca}=1.4$ to $(\frac{N}{Z})_{{}^{40}Ca}=1$. Secondly,
for such reactions the FAZIA array allows a mass resolution comparable with
that of a spectrometer The FAZIA Collaboration  _et al._ (2014), allowing to
fully access the isotopic content of the QP remnant. Moreover, thanks to the
good granularity of the detector, we can investigate also the break-up channel
in order to isotopically reconstruct the QP from the detected pair Camaiani
_et al._ (2018a). In light of this, we measured the n-p equilibration in the
QP evaporative channel, directly accessing the QP remnant; this will be
compared for the first time, in a homogeneous and coherent way, with the QP
break-up channel, where the QP can be reconstructed from the daughter
fragments.

In order to extract the equilibration degree in 48Ca+40Ca system, referred in
the following as the mixed one, we adopted the isospin transport ratio (also
known as imbalance ratio) Rami  _et al._ (2000), which normalizes an isospin
related observable measured in the asymmetric system to that measured for two
symmetric reactions, where the isospin diffusion is absent by definition. For
this reason, 48Ca+48Ca and 40Ca+40Ca reactions, both at 35 MeV/nucleon, have
been also measured and used as reference. The isospin transport ratio is
defined as follows Rami  _et al._ (2000):

$R(X)=\frac{2X-X^{4848}-X^{4040}}{X^{4848}-X^{4040}}$ (1)

where $X$ is an isospin sensitive observable evaluated for the three systems.
For the two symmetric systems 48Ca+48Ca and 40Ca+40Ca, $R(X)$ assumes the
value of +1 and -1, respectively. Such a method allows to enhance the
equilibration signal due to the isospin diffusion Tsang  _et al._ (2004); Liu
_et al._ (2007); May  _et al._ (2018), reducing the effects of any unwanted
overlapping process, and effectively cancelling those introducing a linear
transformation of $X$ Camaiani  _et al._ (2020). Moreover, we note that if the
chosen variable linearly depends on the isospin of the system, $R(X)=\pm 1$
represents the “No Equilibration” limit, while $R(X)=0$ the “Full
Equilibration” value Rami  _et al._ (2000). As done in the past Planeta  _et
al._ (1988); Gippner  _et al._ (1988); Madani  _et al._ (1995); Marchetti  _et
al._ (1993), in this paper, the n-p equilibration is followed as a function of
the reaction dissipation. Since the impact parameter is not directly
accessible as an experimental observable, as usual, we used a reaction
centrality estimator whose effectiveness to follow the impact parameter order
has been tested by means of the Antisymmetrized Molecular Dynamics (AMD) Ono
_et al._ (1992) model coupled with GEMINI++ Charity (2010) as an afterburner.

This paper is organized as follows. In Section II the experimental apparatus
and the adopted theoretical models are presented. Section III describes the
event selection criteria; also the gross properties of the studied systems are
presented. The adopted method to estimate the reaction centrality is presented
in Section IV. The n-p equilibration in both the QP evaporative and QP break-
up channels is presented in Section V, while the comparison of the QP
evaporative channel with the AMD+GEMINI++ prediction is reported in Section
VI. Summary and conclusions are given in Section VII.

##  II Investigation approach

We performed the experiment using beams of 40,48Ca at 35 MeV/nucleon,
delivered by the Superconducting Cyclotron of INFN-LNS with an average current
of 0.1 pnA, impinging on 40,48Ca targets with a thickness of 500$\,\mu$g/cm2.
Approximately 110, 70 and 15 millions of events have been collected for the
48Ca+48Ca, 48Ca+40Ca and 40Ca+40Ca, respectively. The vacuum inside the
scattering chamber was 2$\times$10${}^{-5}\,$mbar during the whole experiment.

In order to avoid Ca oxidation during the mounting of the targets, the Ca
layers were sandwiched between two Carbon foils of about 10$\,\mu$g/cm2 on
both sides of each target was used. Data of both 40,48Ca beams impinging on
12C (300$\,\mu$g/cm2 thick) have been collected in order to estimate the
carbon reaction background in the main reaction data. As observed in a
previous analysis where the same Ca targets have been used Piantelli  _et al._
(2020), no significant contribution of reactions on Carbon target has been
found, thus concluding that the background due to reaction on Carbon
negligibly affects the present results Camaiani (2019).

Data have been collected with four FAZIA blocks The FAZIA Collaboration  _et
al._ (2014); Valdre  _et al._ (2019) arranged in a wall configuration around
the beam axis covering polar angles from 2∘ up to 8∘ approximately, 80 cm far
from the target. A schematic representation of the apparatus geometry is shown
in Fig. 1.

Figure 1: Schematic polar representation of the apparatus geometry. The beam
axis passes through the symmetry center. View from the target.

The main features and performances of the FAZIA multi-telescope array are
fully described elsewhere The FAZIA Collaboration  _et al._ (2014); Valdre
_et al._ (2019); Pastore  _et al._ (2017); Frosin  _et al._ (2020). Here, we
remind that each block consists of 16 2$\times$2 cm2 Si-Si-CsI(Tl) telescopes,
where the thickness of different layers is 300$\,\mu$m, 500$\,\mu$m, and 10
cm, respectively. The telescopes are directly coupled to “custom” FEE cards,
featuring the preamplifiers and the fast digital sampling stages, also
allowing the on-line extraction of the energy parameters from the signals
Valdre  _et al._ (2019). Each FAZIA telescope allows to identify iostopes in
charge and mass up to $Z\approx 25$ with the $\Delta$E-E technique Carboni
_et al._ (2012) and up to $Z\approx 20$ via Pulse Shape Analysis in Silicon
detectors Pastore  _et al._ (2017) for fragments stopped in the first Silicon
layer with identification energy threshold depending on the ion charge Pastore
_et al._ (2017). The data presented in this paper refer to the QP phase-space;
as in most other experiments, energy thresholds do not allow to access the QT
phase-space, which results almost undetected.

Figure 2: (Color online) Experimental data for the 48Ca+48Ca reaction. Panel
a-b) Charge vs. parallel velocity correlation in laboratory frame of $BF$
ejectiles. a): events with $M_{BF}=1$, the rectangle shows the QPR selection;
b) events with $M_{BF}=2$. Beam ($v_{beam}$) and c.m. system ($v_{cm}$)
velocity are pointed out by the arrows. Panel c) $\theta_{rel}$ vs. $v_{rel}$
correlation between the two $BF$s of the same events as in panel b); the
rectangle points out the QPB selection. Each correlation is normalized to
unitary integral.

As anticipated, from the theoretical side, data are compared with the
predictions of the AMD model, belonging to the Quantum Molecular Dynamics
family Aichelin and Stöcker (1986); Aichelin (1991), due to its well assessed
capability to describe nuclear collision characteristics in a various range of
energy and impact parameters Ono (2019). In brief, this model describes a
many-body nuclear system by means of a Slater determinant of Gaussian wave
packets and the equation of motion is obtained via time dependent variational
principle Ikeno  _et al._ (2016). The version of the AMD code used in this
work implements the mean-field via the effective interaction Skyrme SLy4
Chabanat  _et al._ (1997), using $K_{sat}=230\,$MeV for the incompressibility
modulus of the nuclear-matter and $\rho_{0}=0.16\,$fm-3 for the saturation
density. Two parametrizations of the symmetry energy can be tested within AMD:
an asym-soft one with $E_{sym}=32\,$MeV and $L_{sym}=46\,$MeV, and an asym-
stiff one with $L_{sym}=108\,$MeV and the same value for $E_{sym}$, obtained
by changing the density dependent term in the SLy4 force Ikeno  _et al._
(2016). Such recipes are compatible with the reported values for realistic
parametrizations Margueron  _et al._ (2018). Nucleon-nucleon collisions are
taken into account by implementing test particles which are randomly generated
at every time step Ono (2019); Piantelli  _et al._ (2019). The transition
probability depends on the in-medium nucleon-nucleon cross section, which can
be considered, within some limits, as a free parameter of the model. In this
used code version, the parametrization proposed in Ref. Coupland  _et al._
(2011) has been used, i.e.
$\sigma=\sigma_{0}\tanh\left(\sigma_{\mathrm{free}}/\sigma_{0}\right)$, with
$\sigma_{0}=y\rho^{-2/3}$ , where $y$ is a screening parameter, set at
$y=0.85$ (according to Coupland  _et al._ (2011)). In order to take into
account cluster correlations arising during the dynamics, cluster states are
included among the possible achievable final states Tian  _et al._ (2017,
2018); Ono (2019); Piantelli  _et al._ (2019).

We produced about 40000 events for each system and symmetry energy
parametrization, stopping the dynamical calculation at 500 fm/c, a time when
the dynamical phase is safely concluded and the Coulomb interaction among QP
and QT can be considered negligible Piantelli  _et al._ (2019). Impact
parameters up to the grazing values $b_{gr}$ (10.4, 10.1 and 9.7 fm for the
n-rich, mixed and n-deficient system, respectively) have been randomly sorted,
with a triangular distribution. For each primary event, 2000 secondary events
have been generated by means of the GEMINI++ Charity (2010) statistical Monte
Carlo code. The simulated data have then been filtered through a software
replica of the apparatus, that takes into account the geometrical efficiency
and the identification thresholds, in order to consistently compare the
simulation output with the experimental results.

##  III Event selection and reaction characterization

In order to show the criteria adopted for selecting events we focus on the
48Ca+48Ca reaction for the sake of brevity. The same selection criteria have
been applied to the other systems. First of all, due to pile-up events, events
with the total detected charge $Z_{TOT}$ greater than the total system charge
are rejected, as well as events with a total parallel momentum greater than
the beam momentum (less than 2%). Only events with isotopically identified
ejectiles have been considered in the present work, which represents more than
80% of the total events.

The event selection is based on a detected multiplicity ($M$) condition. We
define as Big Fragments ($BF$s), any ejectile with $Z\geq 5$, and as IMFs only
Lithium and Beryllium ions. This choice is motivated by the fact that most
particles with $Z<5$ come from statistical emission according to the
AMD+GEMINI++ predictions. According to our goal, we want to select two main
channels, i.e. the evaporative channel, and the break-up one. In the
evaporative channel the primary QP de-excites emitting IMFs and LCPs, thus
only a BF is expected. Differently, in the break-up channel, the primary QP
splits in two BFs, possibly excited above the energy threshold for particle
decay and thus undergoing subsequent evaporation. Consequently, the first
class is identified by the presence of one $BF$ ($M_{BF}=1$), while the second
one includes two $BF$ ($M_{BF}=2$). It is worth mentioning that these classes
correspond to 65% and 2% of the total number of acquired events, respectively;
the remaining part, due to the limited solid angle coverage, contains events
with only LCP and/or IMF detected and it is discarded.

Figure 3: (Color online) Experimental (symbols) and simulated (lines)
properties of the QP in the QPE (black) and QPB (red) channels, for the
48Ca+48Ca (a-c) and 40Ca+40Ca (d-f) reactions. Panel a,d): charge
distributions. Panel b,e): parallel velocity in the laboratory frame. Panel
c,f): polar angle in the c.m. system. Each distribution is normalized to
unitary integral. QPB distributions are scaled by a factor 0.5 for sake of
clarity. Statistical errors are smaller than the marker size (line width).

Fig. 2(a,b) shows the $BF$ charge vs. the parallel velocity (along the beam
axis, $v_{par}$) correlation in the laboratory frame for events with
$M_{BF}=1$ and $M_{BF}=2$, respectively. Beam ($v_{beam}$) and center of mass
($v_{cm}$) velocities are pointed out by the arrows. Panel a) shows a quite
intense spot in the charge region $12\leq Z\leq 22$, with parallel velocity
between 60 and 80 mm/ns (i.e. $BF$s that preserve down to the 75% of the
projectile velocity). The $BF$s whose charge is greater than the projectile
charge are ascribable to a charge transfer from the target to the projectile
during the interaction phase. Both charge and velocity are compatible with a
$BF$ that is the QP remnant after the de-excitation through the emission of
LCP and/or IMF. The observed spot corresponds to a projectile that retains
down to 60% of its initial charge: such charge range complies with analogous
selections adopted in literature Galichet  _et al._ (2009); Galanopoulos  _et
al._ (2010). As a consequence, we select as QP evaporative channel (QPE) those
events containing a QP remnant (labeled as QPR), i.e. a $BF$ forward emitted
with $Z=12\div 22$), as pointed out by the red contour in Fig. 2(a). QPE
events represent 52% of the total collected data.

Fig. 2(b) shows the $Z-v_{par}$ correlation for events that we mostly ascribe
to QP break-up. Indeed four loci are mainly filled: according to the quadrants
defined by the dashed lines, we verified that $BF$s with $Z>10$ emitted at
$v_{par}>70\,$mm/ns “Heavy-Fast”) are mainly correlated with lighter $BF$s
with $v_{par}<70\,$mm/ns “Light-Slow”); $BF$s with $Z>10$ emitted at
$v_{par}<70\,$mm/ns (“Heavy-Slow”) are correlated with lighter $BF$s at
$v_{par}>70\,$mm/ns (“Light-Fast”). Such observation is compatible with the
well known QP break-up scenario Casini  _et al._ (1993); Stefanini  _et al._
(1995); De Filippo, E.  _et al._ (2012); Jedele  _et al._ (2017). We can
strenghten this selection by means of the correlation between the relative
angle of the two detected fragments $\theta_{rel}$ (in the system center of
mass) and their relative velocity $v_{rel}$. Indeed, in such a correlation QP
break-up events settle at low $\theta_{rel}$ and at a $v_{rel}$ compatible
with that of a Coulomb-driven split Piantelli  _et al._ (2020). On the
contrary, coincidence between QP and QT lies at $\theta_{rel}$ values close to
180∘. Results are shown in Fig. 2(c). Consequently, the QP break-up (QPB)
channel events are selected requiring $M_{BF}=2$ and the two $BF$s in the
phase-space region within the red contour of Fig. 2(c). In addition, we
require that the total charge of the two $BF$s is within the aforementioned
defined QP charge range (i.e. $12-22$). Events selected as described are the
1.5% of the total events (corresponding to the 75% of the $M_{BF}=2$ sample).

###  III.1 Evaporative and break-up channel characterization

Since both selected channels could contain partially detected events of higher
multiplicity, the study of their gross properties is mandatory in order to
validate the selections. For such purpose, we exploited the AMD+GEMINI++
model, which has shown to be able to reproduce the gross properties of heavy-
ion collisions in a large range of ions and bombarding energies Tian  _et al._
(2017, 2018); Piantelli  _et al._ (2019, 2020).

Preliminary, the percentages predicted by the simulation for QPE and QPB
events are 65% and 1.5%, i.e. in agreement with the values observed in the
experimental dataset. Moreover, the amount of QPB events within the QPE
selection is below 2% (due to the limited geometrical acceptance), thus
allowing to go further in the event characterization.

The measured distributions of the QPR charge, parallel velocity in the
laboratory frame, and diffusion angle in the system center of mass are
reported in Fig. 3(a, b, c), for the 48Ca+48Ca reaction, respectively; results
for the 40Ca+40Ca reaction are shown in fig Fig. 3(d, e, f). Both QPE and QPB
channels are shown. Each distribution is normalized to unity for a better
shape comparison with the model prediction; QPB distributions are further
scaled by a factor 0.5 for sake of clarity. We underline that in the QPB
channel, the QP is reconstructed from the two detected $BF$s.

For the experimental case, we observe that both the parallel velocity
($v_{par}$) and the diffusion angle ($\theta_{cm}$) show typical features of
binary dissipative collisions. Indeed, for QPE events extends downwards
starting from beam velocity, while the $\theta_{cm}$ is peaked at angles
slightly larger than the grazing angle Piantelli  _et al._ (2019). Similar
characteristics are also found in the QPB distributions. However, some
differences arise. The larger widths of the three distributions observed for
QPB are consistent with the expected broader phase-space region for QPB, and
the laboratory velocity tends to be on average smaller than for QPE events.
The AMD+GEMINI++ simulation is in global agreement with the observed
distributions, as also shown in a recent investigation on Kr+Ca reactions at
35 MeV/nucleon with four FAZIA blocks Piantelli  _et al._ (2020, ). We remind
that the simulation was subjected to the same constraints as the experimental
data. For the QPE channel, the simulation follows the experimental trend,
especially in the 40Ca+40Ca reaction, while some slight discrepancies appear
for the 48Ca+48Ca reactions. Such differences could be related to a different
dissipation degree between the experimental and the simulated data. Indeed,
the model seems to favor more dissipative events, i.e. lighter QPR (panel a),
lower parallel velocity (panel b), and with larger diffusion angle. Similar
findings have been found also in the Kr+Ca comparison with the AMD+GEMINI++
predictions Piantelli  _et al._ (2020, ).

As a final note of this section we observe that the QP distributions for the
asymmetric 48Ca+40Ca system are very similar to those of the symmetric
48Ca+48Ca case (fig. 3(a,b,c)). This is reasonable since we are observing very
similar QPR and none of the characteristics shown so far take into account the
detailed isotopic composition of the ejectiles. In conclusion, as also in the
recently investigated Kr+Ca reactions with four FAZIA blocksPiantelli  _et
al._ (2020, ), the AMD+GEMINI++ simulation offers a reasonable description of
both the QPE and QPB channels, thus confirming the validity of the adopted
selection criteria.

##  IV Reaction dissipation and centrality

Figure 4: (Color online) Experimental data for the 48Ca+40Ca system: panel a)
proton kinetic energy spectra in the QPR frame for different bins of
$p_{red}$, normalized to unitary area; panel b) average kinetic temperature
$T_{1}$ as a function of $p_{red}$ extracted from the Maxwellian fit (shown in
panel a)) for the proton kinetic energy spectra; the results for the three
systems are presented with symbols according to the legend; only statistical
errors of the fit are shown. Simulated data (filtered AMD+GEMINI++
simulation): panel c) reduced impact parameter $b_{red}$ vs. reduced momentum
$p_{red}$; panel d) average reduced impact parameter $\langle b_{red}\rangle$
vs. $p_{red}$ for each system. Comparison between experimental and simulated
data for the 48Ca+40Ca system: panel e) average QPR charge and sigma of the
charge distribution as a function of $p_{red}$; panel f) same as e) for the
neutron number distribution. Symbols according to the legend.

In this section, we aim at extracting an experimental observable which can be
used to order the events as a function of the reaction dissipation, to
quantify the isospin diffusion from peripheral to more central events. The
chosen observable is based on the momentum of the detected (or reconstructed)
QPR. We define the reduced momentum ($p_{red}$), defined as
$p_{red}=\left(\frac{p^{QP}_{par}}{p^{beam}}\right)_{cm}$, i.e. the QP remnant
(or reconstructed) parallel momentum ($p^{QP}_{par}$) normalized to the beam
momentum ($p^{beam}$), both of them in the c.m. frame.

We first verify, for the experimental data, that the reduced momentum scales
as a function of the reaction dissipation. We report the results from the
48Ca+40Ca reaction as a representative case. We focus on the QPE channels
since no significant amount of LCPs are detected in the QPB channel due to the
limited angular setup. For such purpose we exploited the LCPs forward emitted
with respect to the QPR, that more reliably can be attributed to the QP decay,
being less affected by other contributions. However, in this phase-space other
contributions could be present, as LCPs associate to pre-equilibrium
emissions. One expects that the LCP coming from the statistical decay of the
QP present a Maxwellian-like kinetic energy spectra: the apparent temperature
increases with the reaction dissipation. Fig. 4(a) shows the experimental
invariant proton kinetic energy spectra, in the QPR frame, for the 48Ca+40Ca
system: each distribution refers to a different bin of $p_{red}$, according to
the legend, and is normalized to unitary area for better shape comparison. We
observe that each distribution presents two slopes, corresponding to two
apparent temperatures $T_{1}$ and $T_{2}$, and this deserves some comments.
The QPR is the matching source only for protons that contribute to the low
energy tail ($T_{1}$), i.e. the thermal-part of the distributions Vient  _et
al._ (2018); the high energy tail ($T_{2}$) could be due to different
mechanism, such as pre-equilibrium emission from the neck Vient  _et al._
(2018) or from the deformed QP Piantelli  _et al._ (2002, 2007); Rudolf  _et
al._ (1993), i.e. due to protons emitted from different sources. For what is
relevant to the present discussion, a two-temperature fit can be used in order
to disentangle the thermal part from the non-thermal one, thus obtaining a
crude indication on the excitation scale of the QP source.

The results of the fitting procedure using two Maxwellian contributions are
depicted in Fig. 4(a), superimposed to the experimental spectra. The values of
the fitted parameter $T_{1}$ are shown in Fig. 4(b) as a function of $p_{red}$
for all the systems. The obtained $T_{1}$ scaling as a function of the reduced
momentum confirms that, on average, we are indeed selecting collisions with
increasing dissipation when $p_{red}$ decreases from 1 to 0.3.

Within the AMD+GEMINI++ model, on the other hand, we can directly verify the
relationship between $p_{red}$ and the reduced impact parameter $b_{red}$
($b/b_{gr}$). Fig. 4(c) shows the $b_{red}$ vs. $p_{red}$ correlation
predicted by the AMD+GEMINI++ simulation, filtered with the detector response:
the correlation is narrow for peripheral collisions and tends to broaden for
low $b_{red}$. For this reason, we restrict the following analysis only to the
upper-right region indicated by the dashed lines in the figure. Here, the
correlation is relatively narrow and permits to reliably explore the range
$b_{red}\approx 0.5-1$. These findings are quite the same for the three
studied Ca reactions as evidenced in Fig. 4(d) by the evolution of the average
reduced impact parameter ($\langle b_{red}\rangle$) as a function of
$p_{red}$.

Finally, the QPR average charge $\langle Z\rangle$ and the rms width $\sigma$
of the charge distribution are reported as a function of $p_{red}$ in Fig.
4(e). Panel f) is for the average QPR neutron number distribution $\langle
N\rangle$. In particular, the experimental data are shown in black, with the
bars indicating the $\pm$1 $\sigma$ values. The model results are in magenta
and the $\pm$1 $\sigma$ values are drawn as a contour. As $p_{red}$ decreases,
$\langle Z\rangle$ and $\langle N\rangle$ decrease starting from values very
close to the projectile ones. The average trends as a function of $p_{red}$
are well reproduced by the simulation and, to a lower extent, also the
$\sigma$ of both distributions. The global agreement between the experimental
results and simulation strengthens the use of $p_{red}$ as an order variable,
in order to explore neutron-proton equilibration as a function of the reaction
centrality.

##  V Neutron-proton equilibration: evaporative and break-up channels

Figure 5: (Color online) Average neutron-proton ratio as a function of
$p_{red}$. Panel a) QPE channel; panel b) QPB channel. Magenta and cyan dashed
arrow point out the EAL Charity (1998) value for Ca and Mg nuclei. Statistical
errors are smaller than the marker size. Lines are drawn to guide the eyes.
Figure 6: Isospin transport ratio for the QPE and QPB channels as a function
of $p_{red}$. Statistical errors are smaller than the marker size. Symbols
according to the legend. Lines are drawn to guide the eyes.

The n-p equilibration can be now explored using the average neutron-proton
ratio ($\langle N/Z\rangle$) of the various sources as a function of the
reduced momentum. Fig. 5(a,b) shows the evolution of $\langle N/Z\rangle$ vs.
$p_{red}$ for the three systems for both the QPE and the QPB channel,
respectively. In particular, in the QPE channel, the values refer to the QPR,
while in the QPB channel to the reconstructed (from the two $BF$s) QP. For
sake of clarity, we remind that the accompanying LCPs and/or IMFs are not
taken into account.

As suggested in Sec. III, we observe that the break-up channel is detectable
at lower $p_{red}$. Apart from this, we observe comparable trends in the two
channels. Namely, the bound neutron abundances of the 48Ca and 40Ca detected
(or recontructed) ejectiles are very different as expected, with much larger
values for the n-rich case. Such effects are in agreement with studies at
lower bombarding energies, mainly dedicated to the investigation of the
initial neutron-proton unbalance effects in fusion reactions Bonnet  _et al._
(2008); Ademard  _et al._ (2011); Pirrone  _et al._ (2019). Moreover, the
$\langle N/Z\rangle$ ratios evolve with dissipation in a different way
depending on the initial neutron abundance. 48Ca projectiles we observe a
sizable decrease of $\langle N/Z\rangle$ with centrality, while for the 40Ca
case the n-p ratio is essentially constant after a slight increase in
peripheral events. These different trends can be interpreted in the light of a
dominating statistical decay process for n-rich or n-deficient excited nuclei.
Indeed, the steep decrease of the average $\langle N/Z\rangle$ with respect to
the projectile values (1.4 and 1 for the n-rich and n-deficient system,
respectively) is mainly due to the statistical decay Camaiani  _et al._
(2020). As explained in Charity (1998), excited nuclei follow an average path
in the $N-Z$ plane during the decay and, with increasing initial excitation,
tend to approach a specific region of that plane, called Evaporation Actractor
Line (EAL) Charity (1998), described by a $N/Z$ ratio, depending on the
nuclear size. In Fig. 5, just for reference, the EAL $N/Z$ ratios indicated
with dashed arrows for ion charges $Z=12,20$ representing relevant values for
our QP remnant selection. We see that, with increasing dissipation, QPR from
48Ca and from 40Ca have $\langle N/Z\rangle$ values that move towards the EAL
predictions, although coming from different sides.

The comparison between the $\langle N/Z\rangle$ of QPR from 48Ca of the
symmetric and asymmetric reactions reveals the trend to isospin equilibration.
Focusing on the QPE case (Fig. 5(a)), a clear hierarchy is observed: a reduced
neutron content is detected for the asymmetric case (black solid circles in
fig. 5(a)) with a gap with respect to the symmetric reference (green solid
triangles in fig. 5(a)) increasing towards central collisions, as the result
of the interaction with a n-deficient partner so that the two colliding nuclei
tend to equilibrate their $N/Z$ ratios De Filippo, E.  _et al._ (2012);
Barlini  _et al._ (2013); Piantelli  _et al._ (2017). Remarkably, very similar
observations can be repeated for the QPB channel, where the same hierarchy and
evolution are evident.

In order to more quantitatively establish the isospin equilibration process we
show in Fig. 6 the isospin transport ratio $R(X)$ built with $X=\langle
N/Z\rangle$ (Eq.1) as a function of the reaction dissipation represented by
$p_{red}$. Concerning the evaporative channel, we observe the expected trend.
The equilibration degree smoothly and monotonically evolves from $R\approx 1$
for $p_{red}\approx 1$ to $R\approx 0.6$ for $p_{red}\approx 0.3$, which,
according to the AMD average prediction (Fig. 4(d)), corresponds to a range of
centrality $\langle b_{red}\rangle\in[1,0.5]$. Also the experimental result
for the same Ca+Ca collision Fable (2018); Boisjoli, M.  _et al._ (2012);
Wigg, P.C.  _et al._ (2012) obtained with the INDRA+VAMOS experimental
apparatus points out in this direction : the n-p equilibration for such
experiment is compatible, as discussed in Camaiani (2019), with that here
reported. The isospin diffusion sets in for the asymmetric reactions and makes
the QP and QT to approach a common $N/Z$ values. Since the QP size selection
is somewhat arbitrary (Sec. 3), we tested the result by changing the adopted
QP charge range. In particular, we increased and decreased the lower limit of
two units with respect to our previous “standard” ($Z=12$) value (as done in
Galichet  _et al._ (2009); Galanopoulos  _et al._ (2010)), taking into account
other reasonable choices reported in the literature. For instance Ref.
Thériault  _et al._ (2005) fixes as a lower limit of the QP charge the 36% of
the projectile charge. By using the ranges $Z^{QP}\in[14,22]$ or
$Z^{QP}\in[10,22]$, we found that the trend of $R$ is negligibly affected in
the studied range of $p_{red}$ Camaiani (2019).

An important point of this work, as anticipated, is the access to the isospin
diffusion process looking at the QPB channel, in a manner that - to our
knowledge - has not been yet attempted before. In Fig. 6 the open dots show
the $\langle N/Z\rangle$ for QP reconstructed from the break-up fragments. As
a first comment we can say that the general trend is the same, with slight
differences: for the QPB we find a weak process at least for the less
dissipative accessible bins. It is very difficult to judge and conclude about
these small differences which, in any case, are out of statistical errors.
Such an observation suggests a heavier primary source in the QPB channel,
which can lead the system to a lower n-p equilibration for the most explored
peripheral events. For instance, the average charge and neutron number of the
reconstructed QP in the QPB channel are on average 2 units larger than the
values of the QPR in the evaporative channel. On the other hand, the
differences can be also related to subtle effects associated to the different
evaporation paths followed by the excited break-up fragments (before and after
the split) with respect to the case without break-up.

Such a topic will be further investigate in the INDRA+FAZIA experimental
campaign at GANIL, thus combining with the isotopic capabilities of the FAZIA
multi-telescope array the large angular coverage of the INDRA detector, in
order to more precisely select the reaction centrality. Here, we can only
conclude that this roughly common trend of the two geometrical loci in fig.6
suggests that, irrespective of the final state channel, the isospin diffusion
acts in a similar way. In other words, it appears that the isospin
equilibration process acts before any de-excitation process. This observation
is rather in line with some old results Planeta  _et al._ (1988) for lower
energy collisions. There, a general conclusion was suggested that the n-p
degree of freedom tends to relax rather quickly during the interaction. The
complete equilibrium could be reached only for rather central impacts, not
accessible here according to the AMD centrality estimation of Fig. 4(c,d),
associated with relatively long interaction times.

##  VI Neutron-proton equilibration: comparison with the simulation

In this section, we aim at comparing the isospin evolution extracted from
experimental data with that predicted by the transport model AMD, coupled with
GEMINI++ as an afterburner. We will focus on the evaporative channel, as it
corresponds to 65% of the collected data. The break-up channel is
experimentally around 35 times less abundant and since also the model predicts
a similar event partition, the simulation statistics results to be to low for
a reliable comparison. For sake of clarity, we remind that the simulated data
have been treated as the experimental ones.

Fig.7(a,b) shows the simulated $\langle N/Z\rangle$ vs. $p_{red}$ trend
(lines), compared with that obtained experimentally (same points of Fig. 5(a))
for the asym-stiff and asym-soft parametrization of the symmetry energy,
respectively. As for the experimental data, we observe the clear hierarchy
among the three systems, and the tendency to approach $\langle N/Z\rangle$
values around the EAL loci (magenta and cyan arrows for $Z=20$ and $Z=12$,
respectively) with increasing dissipation. The agreement with the $\langle
N/Z\rangle$ of the 40Ca data is excellent while, as noticed for the gross
properties of the QPR (see. Sec. 3), there are some differences for the 48Ca
case. Weak differences between the two calculations can be seen, in
particular, the asym-stiff choice predicts a more neutron-rich QPR with
respect to the asym-soft one, as expected Baran  _et al._ (2005a, b).

The corresponding isospin transport ratio are shown in Fig. 8 as a function of
$p_{red}$, with dot-dashed and dotted line for the asym-stiff and asym-soft
parametrizations, respectively. We first underline that the $R$ variable
depends on the gap between the asymmetric and the symmetric references. The
way how the gap evolves vs. $p_{red}$ dictates the shape of the R as a
function of the dissipation, thus a precise reproduction of the $\langle
N/Z\rangle$ values is not mandatory. However, Fig 8 shows a sizable
disagreement between experiment and model predictions concerning the isospin
diffusion process. In particular, the model predicts an initial fast
relaxation followed by a slower trend, whereas the experiment suggests a
smoother evolution. As for the asym-stiffness, we can see that the very small
differences in the two model results for $\langle N/Z\rangle$ give a quite
small gap in the equilibration degree; however, as expected, the asym-soft
assumption slightly favors the isospin relaxation.

Figure 7: Comparison of the measured average neutron-proton ratio for the QPE
channel as a function of $p_{red}$ with the AMD+GEMINI++ simulation. Panel a)
AMD asym-stiff parametrization; panel b) AMD asym-soft parametrization.
Magenta and cyan dashed arrow point out the EAL Charity (1998) for relevant
nuclei. Symbols according to the legend. Statistical errors are smaller than
the marker size (line width). Figure 8: Comparison of the isospin transport
ratio for the QPE channel as a function of $p_{red}$ between the experimental
results and the AMD+GEMINI++ simulation, using asym-stiff and asym-soft
parametrizations. Statistical errors are smaller than the marker size (line
width).

Some comments and arguments on the possible origin of the observed
disagreement are in order. A first comment deals with the role of the
emissions from the primary QP, i.e. the fragment emerging just at the end of
the interaction which we would like to access in order to measure the isospin
diffusion. Indeed, any particle or fragment emission before the detection
perturbs the final isotopic distribution. One can thus wonder if the found
disagreement is related to a partially wrong description of the dynamics
(reaction times and/or nuclear potential terms ruling the isospin transfer) or
to a somehow wrong evaporation scheme. In this respect, we must stress that
isospin transport ratio has been introduced Rami  _et al._ (2000); Tsang  _et
al._ (2004) just to bypass any perturbation which introduces a linear
transformation of the isospin variable in use (Eq. 1). Such behavior has been
recently investigated in a specific work Camaiani  _et al._ (2020), in a full
model framework, for the systems here discussed. In this paper one
demonstrates, by means of the AMD simulation coupled with statistical models,
that the charge equilibration process measured via isospin transport ratio is
indeed affected by perturbations introduced by the dynamical and statistical
emissions from the fragments after their separation. In particular, the
statistical emission (described by Gemini++ code) tends to introduce non-
linear spurious distortions at low excitation energies (where structure
effects are well known to affect the particle emission Camaiani  _et al._
(2018b); Morelli  _et al._ (2019); Bruno  _et al._ (2019)), i.e. for large
impact parameters, while the distortion becomes smoother and linear with
increasing excitation. Instead, at least for the considered systems, the
contribution of emissions occurring during the interaction phases and
predicted by the AMD model, increases with centrality but remains relatively
scarce and negligibly affects the $R$ variable. As a consequence, we checked
that despite some residual distortions related to emissions, the variable $R$
is robust and keeps memory of the primary isospin history; this suggests that
the observed discrepancy between measured and predicted $R$ can be safely
ascribed to the dynamical modelization.

By analyzing the evolving output of the model, we can access to the end of the
projectile-target interaction phase (labeled as $t_{DIC}$), by means of the
procedure described in refs. Camaiani  _et al._ (2020); Piantelli  _et al._
(2020). In order to pin down the mechanism responsible for the observe
discrepancies with experiment, we applied some special conditions on the
analyzed events, as follows.

Figure 9: (Color online) Comparison of the experimental isospin transport
ratio with the equilibration obtained at the projectile-target separation time
($t_{DIC}$), for the asym-stiff parametrization; the isospin transport ratios
only due to a net charge (red line) and neutron (green line) number change are
shown. The equilibration obtained after a re-scaling of the proton and neutron
transfer probabilities is shown with magenta line. See text for details.
Errors are statistical.

The n-p equilibration obtained at $t_{DIC}$, for the asym-stiff simulation, is
shown in Fig. 9 as a black line; for sake of comparison also the experimental
trend of Fig. 8 is here reported. For each system (i.e. the asymmetric and the
symmetric references), we start allowing only the net neutron transfers (green
line): this corresponds to retain only the reaction channels where the QP
emerges as a Ca isotope. Vice versa, we allow only the net proton exchanges
(red line), i.e. events where the QP retains the neutron number of the
projectile. As expected, limiting the n-p exchange produces a lower
equilibration. More interesting, we observe that the equilibration obtained
via only charge change lies close to the total one, pointing out to an
important role of the p transfers in the isospin equilibration mechanism. This
can be quantitatively understood keeping into account that, in order to
restore the $N/Z$ unbalance, a p transfer is more effective than a n transfer,
since the former counts as 1/20 whereas the latter as 1/28.

Starting from the indication that the nucleon transfer in AMD may be too
frequent, we now aim at quantifying the degree of the overestimation of the
transfer probability. We introduce a multiplying factor ($f$), depending on
the net number of transferred neutrons and protons, $\Delta_{n}$ and
$\Delta_{p}$ respectively. Assuming that nucleon transfers in the same event
are independent of each other, we modelled a parametrization as:
$f=\alpha^{|\Delta_{n}|}\beta^{|\Delta_{p}|}$, where $\alpha$ and $\beta$ are
parameters to suppress (or enhance) the net transfer probability of single
neutrons and single protons, respectively. The probability of the non-transfer
channel (at $t_{dic}$) is adjusted for the total probability conservation. For
each system, we then proceed to classify the various channels as function on
the net p/n changes at $t_{DIC}$: we modify these initial populations via a
change of the $(\alpha,\beta)$ pair and thus obtain different average isospin
values. The isospin transport ratio is then computed via eq.(1), adopting the
$\langle N/Z\rangle(\alpha,\beta)$ as $X$ variable ($R_{AMD}(\alpha,\beta)$).
The parameters $\alpha$ and $\beta$ have been selected by means of fit
procedure on the experimental data $R_{exp}$. Specifically, we looked for the
minimum of a $\mathbf{M}^{2}$ variable defines as follows:

$\mathbf{M}^{2}=\sum_{i=0}^{N}\frac{\left[R_{exp}^{i}-R_{AMD}^{i}(\alpha,\beta)\right]^{2}}{\sigma_{exp}^{2}(i)+\sigma_{AMD}^{2}(i)},$
(2)

where $R_{exp}^{i}$ and $R_{AMD}^{i}$ are the values of the experimental and
simulated $R$ at the $i$th point along the $p_{red}$ axis;
$\sigma_{exp}^{2}(i)$ and $\sigma_{AMD}^{2}(i)$ the statistical errors of each
point. The fitted values of the parameters are: $\alpha=0.60\pm 0.05$,
$\beta=0.3\pm 0.1$. The equilibration degree obtained for such values is shown
in Fig. 9 with magenta line (Modified AMD), which follows the experimental
trend proving the satisfactory quality of the fit. This show that the nucleon
transfer is overestimated in AMD by about a factor of two. Moreover, it is
likely that proton transfer is more overestimated than neutron transfer.

In conclusion, this first attempt to compare the n-p equilibration measured
via the isospin transport ratio built from the $\langle N/Z\rangle$ of the QPR
has shown a faster equilibration of the model prediction with respect to that
observed in the experimental sample. Such discrepancy can be recovered acting
on the transfer probability, reducing it approximately of a factor two. It is
not easy to identify a reason behind this problem, as many factors could
contribute to it, e.g. the nucleon-nucleon cross section or the nucleon
effective masses or their interplay. For instance, a simple variation of the
screening paramenter $y$ of the nucleon-nucleon cross section from $y=0.42$ up
to the free nucleon-nucleon cross section did not produce significant
variations of the isospin transport ratio. Such topics will be investigated in
future works.

##  VII Summary and Conclusion

In this paper, we have presented the experimental results of an experiment
dedicated to the investigation of the n-p equilibration in 48Ca+40Ca semi-
peripheral reactions at 35 MeV/nucleon, performed with four blocks of the
FAZIA multi-telescope array at the INFN-LNS. For the first time, thanks to the
FAZIA identification performances coupled to its good granularity, we could
study the isospin relaxation for the two main QP decay channels, the
evaporative and the break-up one.

The equilibration trend has been investigated by means of the isospin
transport ratio, which which improves the sensitivity to the effect sought
after and normalizes the mixed system evolution with the limiting values of
the symmetric reactions 48Ca+48Ca and 40Ca+40Ca, investigated under the same
experimental conditions. Despite the relatively small coverage of the setup
($2-8^{\circ}$ in the laboratory frame), the main achievements have been
proved not to be strongly affected by the apparatus response: indeed we focus
on the QP phase-space for which we have reasonable acceptance. We have
introduced a reaction dissipation estimator ($p_{red}$), which has been linked
with the reaction centrality by means of the model.

The results reported in this paper are the following. As expected, the
relaxation of the isospin degree of freedom has been observed in the
48Ca+40Ca, via the use of the isospin transport ratio of the average neutron-
proton ratio ($\langle N/Z\rangle$) of QP remnants.

The comparative analysis of the QP evaporative and break-up channels has shown
the typical signature of the isospin diffusion: as the reaction centrality
increases, the system evolves to restore the charge equilibrium. The
similarity of the behavior for the two channels suggests a comparable
dynamical evolution before the decay, whatever it is. Specifically, this is
consistent with an isospin exchange mechanism that acts on a similar timescale
(that of the interaction phase) shorter than the evaporation cascade or the QP
split phase.

Concerning the comparison with the AMD model coupled with the GEMINI++
statistical code, we observed that the model globally reproduces the main
features of the QP in both the evaporative and break-up channels; the
agreement is better for the QP evaporation channel than for the break-up one,
where the model produces lighter and slower fragments than the measured ones.
Also, the agreement is quite good for the 40Ca system while for the 48Ca
reactions it less nicely reproduces the QP data. The detailed isospin
distributions of the final (post-evaporative) fragments are, again, less well
reproduced for the n-rich systems; for the 40Ca reaction the comparison is
excellent.

The main difference between measured and model data is observed in the
evolution towards the charge equilibration for the evaporative exit channel.
The model predicts a faster relaxation of the initial neutron-proton unbalance
with respect to the experiment. This discrepancy seems to be associated with
an overestimated probability of nucleon transfers, mainly and more
specifically for the protons: in particular a reduction of about a factor two
accounts for the experimental path. However, a deeper investigation on this
point is in program. In this respect we plan to extend the analysis of this
paper to the data obtained by the first recent INDRA-FAZIA experiment on Ni+Ni
reactions at comparable energies. Here, we have the almost full isotopic
identification of QP ejectiles coupled with a much larger acceptance, allowing
to adopt and cross-check several variables, several variables, to extend the
analysis to the full panel of exit channels, and to more precisely select the
reaction centrality.

###### Acknowledgements.

This work required the use of a lot of computation time for the production of
the simulated data. We would like to thank the GARR Consortium for the kind
use of the cloud computing infrastructure on the platform cloud.garr.it. We
would like to thank also the INFN-CNAF for the use of its cloud computing
infrastructure. A. Ono was supported by JSPS KAK-ENHI Grant No. JP17K05432.
This work was also supported by the National Research Foundation of Korea
(NRF) (Grant No. 2018R1A5A1025563).

## References

  * Planeta  _et al._ (1988) R. Planeta, S. H. Zhou, K. Kwiatkowski, W. G. Wilson, V. E. Viola, H. Breuer, D. Benton, F. Khazaie, R. J. McDonald, A. C. Mignerey, A. Weston-Dawkes, R. T. de Souza, J. R. Huizenga, and W. U. Schröder, Phys. Rev. C 38, 195 (1988).
  * Gippner  _et al._ (1988) P. Gippner _et al._ , Zeit. fur Phys. A 330 (1988).
  * Madani  _et al._ (1995) H. Madani, A. C. Mignerey, A. A. Marchetti, A. P. Weston-Dawkes, W. L. Kehoe, and F. Obenshain, Phys. Rev. C 51, 2562 (1995).
  * Marchetti  _et al._ (1993) A. A. Marchetti, A. C. Mignerey, H. Madani, A. Gökmen, W. L. Kehoe, B. Libby, K. Morley, H. Breuer, K. Wolf, and F. Obenshain, Phys. Rev. C 48, 266 (1993).
  * Tsang  _et al._ (2004) M. B. Tsang, T. X. Liu, L. Shi, P. Danielewicz, C. K. Gelbke, X. D. Liu, W. G. Lynch, W. P. Tan, G. Verde, A. Wagner, H. S. Xu, W. A. Friedman, L. Beaulieu, B. Davin, R. T. de Souza, Y. Larochelle, T. Lefort, R. Yanez, V. E. Viola, R. J. Charity, and L. G. Sobotka, Phys. Rev. Lett. 92, 062701 (2004).
  * Tsang  _et al._ (2009) M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch, and A. W. Steiner, Phys. Rev. Lett. 102, 122701 (2009).
  * Łukasik  _et al._ (1997) J. Łukasik, J. Benlliure, V. Métivier, E. Plagnol, B. Tamain, M. Assenard, G. Auger, C. O. Bacri, E. Bisquer, B. Borderie, R. Bougault, R. Brou, P. Buchet, J. L. Charvet, A. Chbihi, J. Colin, D. Cussol, R. Dayras, A. Demeyer, D. Doré, D. Durand, E. Gerlic, S. Germain, D. Gourio, D. Guinet, P. Lautesse, J. L. Laville, J. F. Lecolley, A. Le Fèvre, T. Lefort, R. Legrain, O. Lopez, M. Louvel, N. Marie, L. Nalpas, M. Parlog, J. Péter, O. Politi, A. Rahmani, T. Reposeur, M. F. Rivet, E. Rosato, F. Saint-Laurent, M. Squalli, J. C. Steckmeyer, M. Stern, L. Tassan-Got, E. Vient, C. Volant, J. P. Wieleczko, M. Colonna, F. Haddad, P. Eudes, T. Sami, and F. Sebille, Phys. Rev. C 55, 1906 (1997).
  * Plagnol  _et al._ (1999) E. Plagnol, J. Łukasik, G. Auger, C. O. Bacri, N. Bellaize, F. Bocage, B. Borderie, R. Bougault, R. Brou, P. Buchet, J. L. Charvet, A. Chbihi, J. Colin, D. Cussol, R. Dayras, A. Demeyer, D. Doré, D. Durand, J. D. Frankland, E. Galichet, E. Genouin-Duhamel, E. Gerlic, D. Guinet, P. Lautesse, J. L. Laville, J. F. Lecolley, R. Legrain, N. Le Neindre, O. Lopez, M. Louvel, A. M. Maskay, L. Nalpas, A. D. Nguyen, M. Pârlog, J. Péter, M. F. Rivet, E. Rosato, F. Saint-Laurent, S. Salou, J. C. Steckmeyer, M. Stern, G. Tăbăcaru, B. Tamain, L. Tassan-Got, O. Tirel, E. Vient, C. Volant, and J. P. Wieleczko (The INDRA Collaboration), Phys. Rev. C 61, 014606 (1999).
  * Thériault  _et al._ (2005) D. Thériault, A. Vallée, L. Gingras, Y. Larochelle, R. Roy, A. April, L. Beaulieu, F. Grenier, F. Lemieux, J. Moisan, M. Samri, C. St-Pierre, S. Turbide, B. Borderie, R. Bougault, P. Buchet, J. L. Charvet, A. Chbihi, J. Colin, D. Cussol, R. Dayras, D. Durand, J. D. Frankland, E. Galichet, D. Guinet, B. Guiot, P. Lautesse, J. F. Lecolley, N. L. Neindre, O. Lopez, A. M. Maskay, L. Nalpas, M. Parlog, P. Pawlowski, M. F. Rivet, E. Rosato, J. C. Steckmeyer, B. Tamain, E. Vient, C. Volant, J. P. Wieleczko, I. Collaboration, S. J. Yennello, E. Martin, and E. Winchester, Phys. Rev. C 71, 014610 (2005).
  * Thériault  _et al._ (2006) D. Thériault, J. Gauthier, F. Grenier, F. Moisan, C. St-Pierre, R. Roy, B. Davin, S. Hudan, T. Paduszynski, R. T. d. Souza, E. Bell, J. Garey, J. Iglio, A. L. Keksis, S. Parketon, C. Richers, D. V. Shetty, S. N. Soisson, G. A. Souliotis, B. C. Stein, and S. J. Yennello, Phys. Rev. C 74, 051602 (2006).
  * Baran  _et al._ (2005a) V. Baran, M. Colonna, V. Greco, and M. D. Toro, Physics Reports 410, 335 (2005a).
  * Lionti  _et al._ (2005) R. Lionti, V. Baran, M. Colonna, and M. D. Toro, Physics Letters B 625, 33 (2005).
  * Napolitani  _et al._ (2010) P. Napolitani, M. Colonna, F. Gulminelli, E. Galichet, S. Piantelli, G. Verde, and E. Vient, Phys. Rev. C 81, 044619 (2010).
  * Sun  _et al._ (2010) Z. Y. Sun, M. B. Tsang, W. G. Lynch, G. Verde, F. Amorini, L. Andronenko, M. Andronenko, G. Cardella, M. Chatterje, P. Danielewicz, E. De Filippo, P. Dinh, E. Galichet, E. Geraci, H. Hua, E. La Guidara, G. Lanzalone, H. Liu, F. Lu, S. Lukyanov, C. Maiolino, A. Pagano, S. Piantelli, M. Papa, S. Pirrone, G. Politi, F. Porto, F. Rizzo, P. Russotto, D. Santonocito, and Y. X. Zhang, Phys. Rev. C 82, 051603 (2010).
  * Margueron  _et al._ (2018) J. Margueron, R. Hoffmann Casali, and F. Gulminelli, Phys. Rev. C 97, 025805 (2018).
  * Lopez  _et al._ (2014) O. Lopez, D. Durand, G. Lehaut, B. Borderie, J. D. Frankland, M. F. Rivet, R. Bougault, A. Chbihi, E. Galichet, D. Guinet, M. La Commara, N. Le Neindre, I. Lombardo, L. Manduci, P. Marini, P. Napolitani, M. Pârlog, E. Rosato, G. Spadaccini, E. Vient, and M. Vigilante (INDRA Collaboration), Phys. Rev. C 90, 064602 (2014).
  * Coupland  _et al._ (2011) D. D. S. Coupland, W. G. Lynch, M. B. Tsang, P. Danielewicz, and Y. Zhang, Phys. Rev. C 84, 054603 (2011).
  * Souza  _et al._ (1990) R. D. Souza, N. Carlin, Y. Kim, J. Ottarson, L. Phair, D. Bowman, C. Gelbke, W. Gong, W. Lynch, R. Pelak, T. Peterson, G. Poggi, M. Tsang, and H. Xu, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 295, 109 (1990).
  * Pouthas  _et al._ (1995) J. Pouthas, B. Borderie, R. Dayras, E. Plagnol, M. Rivet, F. Saint-Laurent, J. Steckmeyer, G. Auger, C. Bacri, S. Barbey, _et al._ , Nucl. Instr. and Methods A 357, 418 (1995).
  * Pagano (2012) A. Pagano, Nuclear Physics News 22, 25 (2012).
  * Liu  _et al._ (2007) T. X. Liu, W. G. Lynch, M. B. Tsang, X. D. Liu, R. Shomin, W. P. Tan, G. Verde, A. Wagner, H. F. Xi, H. S. Xu, B. Davin, Y. Larochelle, R. T. d. Souza, R. J. Charity, and L. G. Sobotka, Phys. Rev. C 76, 034603 (2007).
  * Galichet  _et al._ (2009) E. Galichet, M. F. Rivet, B. Borderie, M. Colonna, R. Bougault, A. Chbihi, R. Dayras, D. Durand, J. D. Frankland, D. C. R. Guinet, P. Lautesse, N. L. Neindre, O. Lopez, L. Manduci, M. Pârlog, E. Rosato, B. Tamain, E. Vient, C. Volant, and J. P. Wieleczko (INDRA Collaboration), Phys. Rev. C 79, 064614 (2009).
  * Bougault  _et al._ (2018) R. Bougault, E. Bonnet, B. Borderie, A. Chbihi, D. Dell’Aquila, Q. Fable, L. Francalanza, J. D. Frankland, E. Galichet, D. Gruyer, D. Guinet, M. Henri, M. La Commara, N. Le Neindre, I. Lombardo, O. Lopez, L. Manduci, P. Marini, M. Pârlog, R. Roy, P. Saint-Onge, G. Verde, E. Vient, and M. Vigilante (INDRA Collaboration), Phys. Rev. C 97, 024612 (2018).
  * Souliotis  _et al._ (2006) G. A. Souliotis, D. V. Shetty, A. Keksis, E. Bell, M. Jandel, M. Veselsky, and S. J. Yennello, Phys. Rev. C 73, 024606 (2006).
  * Souliotis  _et al._ (2014) G. A. Souliotis, P. N. Fountas, M. Veselsky, S. Galanopoulos, Z. Kohley, A. McIntosh, S. J. Yennello, and A. Bonasera, Phys. Rev. C 90, 064612 (2014).
  * Baran  _et al._ (2005b) V. Baran, M. Colonna, M. D. Toro, M. Zielinska-Pfabé, and H. H. Wolter, Phys. Rev. C 72, 064620 (2005b).
  * May  _et al._ (2018) L. W. May, A. Wakhle, A. B. McIntosh, Z. Kohley, S. Behling, A. Bonasera, G. Bonasera, P. Cammarata, K. Hagel, L. Heilborn, A. Jedele, A. Raphelt, A. R. Manso, G. Souliotis, R. Tripathi, M. D. Youngs, A. Zarrella, and S. J. Yennello, Phys. Rev. C 98, 044602 (2018).
  * The FAZIA Collaboration  _et al._ (2014) The FAZIA Collaboration, Bougault, R., Poggi, G., Barlini, S., Borderie, B., Casini, G., Chbihi, A., Le Neindre, N., Pârlog, M., Pasquali, G., Piantelli, S., Sosin, Z., Ademard, G., Alba, R., Anastasio, A., Barbey, S., Bardelli, L., Bini, M., Boiano, A., Boisjoli, M., Bonnet, E., Borcea, R., Bougard, B., Brulin, G., Bruno, M., Carboni, S., Cassese, C., Cassese, F., Cinausero, M., Ciolacu, L., Cruceru, I., Cruceru, M., D´Aquino, B., De Fazio, B., Degerlier, M., Desrues, P., Di Meo, P., Dueñas, J. A., Edelbruck, P., Energico, S., Falorsi, M., Frankland, J. D., Galichet, E., Gasior, K., Gramegna, F., Giordano, R., Gruyer, D., Grzeszczuk, A., Guerzoni, M., Hamrita, H., Huss, C., Kajetanowicz, M., Korcyl, K., Kordyasz, A., Kozik, T., Kulig, P., Lavergne, L., Legouée, E., Lopez, O., Lukasik, J., Maiolino, C., Marchi, T., Marini, P., Martel, I., Masone, V., Meoli, A., Merrer, Y., Morelli, L., Negoita, F., Olmi, A., Ordine, A., Paduano, G., Pain, C., Palka, M., Passeggio, G., Pastore, G., Pawlowski, P., Petcu, M., Petrascu, H., Piasecki, E., Pontoriere, G., Rauly, E., Rivet, M. F., Rocco, R., Rosato, E., Roscilli, L., Scarlini, E., Salomon, F., Santonocito, D., Seredov, V., Serra, S., Sierpowski, D., Spadaccini, G., Spitaels, C., Stefanini, A. A., Tobia, G., Tortone, G., Twaróg, T., Valdré, S., Vanzanella, A., Vanzanella, E., Vient, E., Vigilante, M., Vitiello, G., Wanlin, E., Wieloch, A., and Zipper, W., Eur. Phys. J. A 50, 47 (2014).
  * Camaiani  _et al._ (2018a) A. Camaiani _et al._ , Il Nuovo Cimento, in Proceedings of the International Workshop on Multifragmentation IWM-EC 2018 041 (2018a).
  * Rami  _et al._ (2000) F. Rami, Y. Leifels, B. de Schauenburg, A. Gobbi, B. Hong, J. P. Alard, A. Andronic, R. Averbeck, V. Barret, Z. Basrak, N. Bastid, I. Belyaev, A. Bendarag, G. Berek, R. Čaplar, N. Cindro, P. Crochet, A. Devismes, P. Dupieux, M. Dželalija, M. Eskef, C. Finck, Z. Fodor, H. Folger, L. Fraysse, A. Genoux-Lubain, Y. Grigorian, Y. Grishkin, N. Herrmann, K. D. Hildenbrand, J. Kecskemeti, Y. J. Kim, P. Koczon, M. Kirejczyk, M. Korolija, R. Kotte, M. Kowalczyk, T. Kress, R. Kutsche, A. Lebedev, K. S. Lee, V. Manko, H. Merlitz, S. Mohren, D. Moisa, J. Mösner, W. Neubert, A. Nianine, D. Pelte, M. Petrovici, C. Pinkenburg, C. Plettner, W. Reisdorf, J. Ritman, D. Schüll, Z. Seres, B. Sikora, K. S. Sim, V. Simion, K. Siwek-Wilczyńska, A. Somov, M. R. Stockmeier, G. Stoicea, M. Vasiliev, P. Wagner, K. Wiśniewski, D. Wohlfarth, J. T. Yang, I. Yushmanov, and A. Zhilin (FOPI Collaboration), Phys. Rev. Lett. 84, 1120 (2000).
  * Camaiani  _et al._ (2020) A. Camaiani, S. Piantelli, A. Ono, G. Casini, B. Borderie, R. Bougault, C. Ciampi, J. A. Dueñas, C. Frosin, J. D. Frankland, D. Gruyer, N. LeNeindre, I. Lombardo, G. Mantovani, P. Ottanelli, M. Parlog, G. Pasquali, S. Upadhyaya, S. Valdré, G. Verde, and E. Vient, Phys. Rev. C 102, 044607 (2020).
  * Ono  _et al._ (1992) A. Ono, H. Horiuchi, T. Maruyama, and A. Ohnishi, Progress of Theoretical Physics 87, 1185 (1992), http://oup.prod.sis.lan/ptp/article-pdf/87/5/1185/5272175/87-5-1185.pdf .
  * Charity (2010) R. J. Charity, Phys. Rev. C 82, 014610 (2010).
  * Piantelli  _et al._ (2020) S. Piantelli, G. Casini, A. Ono, G. Poggi, G. Pastore, S. Barlini, A. Boiano, E. Bonnet, B. Borderie, R. Bougault, M. Bruno, A. Buccola, A. Camaiani, A. Chbihi, M. Cicerchia, M. Cinausero, M. D’Agostino, M. Degerlier, J. A. Dueñas, Q. Fable, D. Fabris, J. D. Frankland, C. Frosin, F. Gramegna, D. Gruyer, M. Henri, A. Kordyasz, T. Kozik, N. Le Neindre, I. Lombardo, O. Lopez, G. Mantovani, T. Marchi, L. Morelli, A. Olmi, P. Ottanelli, M. Pârlog, G. Pasquali, A. A. Stefanini, G. Tortone, S. Upadhyaya, S. Valdré, G. Verde, E. Vient, M. Vigilante, R. Alba, and C. Maiolino, Phys. Rev. C 101, 034613 (2020).
  * Camaiani (2019) A. Camaiani,  _Complete isotopic characterization of projectile fragments in Ca+Ca reactions at Fermi energies with the FAZIA array_ , Ph.D. thesis, Università degli Studi di Firenze (2019).
  * Valdre  _et al._ (2019) S. Valdre, G. Casini, N. L. Neindre, M. Bini, A. Boiano, B. Borderie, P. Edelbruck, G. Poggi, F. Salomon, G. Tortone, R. Alba, S. Barlini, E. Bonnet, B. Bougard, R. Bougault, G. Brulin, M. Bruno, A. Buccola, A. Camaiani, A. Chbihi, C. Ciampi, M. Cicerchia, M. Cinausero, D. Dell’Aquila, P. Desrues, J. Dueñas, D. Fabris, M. Falorsi, J. Frankland, C. Frosin, E. Galichet, R. Giordano, F. Gramegna, L. Grassi, D. Gruyer, M. Guerzoni, M. Henri, M. Kajetanowicz, K. Korcyl, A. Kordyasz, T. Kozik, P. Lecomte, I. Lombardo, O. Lopez, C. Maiolino, G. Mantovani, T. Marchi, A. Margotti, Y. Merrer, L. Morelli, A. Olmi, A. Ordine, P. Ottanelli, C. Pain, M. Pałka, M. Pârlog, G. Pasquali, G. Pastore, S. Piantelli, H. de Préaumont, R. Revenko, A. Richard, M. Rivet, J. Ropert, E. Rosato, F. Saillant, D. Santonocito, E. Scarlini, S. Serra, C. Soulet, G. Spadaccini, A. Stefanini, G. Tobia, S. Upadhyaya, A. Vanzanella, G. Verde, E. Vient, M. Vigilante, E. Wanlin, G. Wittwer, and A. Zucchini, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 930, 27 (2019).
  * Pastore  _et al._ (2017) G. Pastore, D. Gruyer, P. Ottanelli, N. L. Neindre, G. Pasquali, R. Alba, S. Barlini, M. Bini, E. Bonnet, B. Borderie, _et al._ , Nucl. Instr. and Methods A 860, 42 (2017).
  * Frosin  _et al._ (2020) C. Frosin, S. Barlini, G. Poggi, G. Casini, M. Bini, A. Stefanini, S. Valdré, D. Gruyer, M. Ciemała, A. Maj, M. Ziebliński, B. Sowicki, K. Mazurek, N. Cieplicka-Oryńczak, M. Matejska-Minda, E. Bonnet, B. Borderie, R. Bougault, M. Bruno, A. Buccola, A. Camaiani, A. Chibhi, M. Cinausero, M. Cicerchia, J. Dueñas, D. Fabris, J. Frankland, F. Gramegna, M. Henri, A. Kordyasz, T. Kozik, N. Le Neindre, I. Lombardo, O. Lopez, G. Mantovani, T. Marchi, A. Olmi, P. Ottanelli, M. Parlog, S. Piantelli, G. Pasquali, S. Upadhyahya, G. Verde, and E. Vient, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 951, 163018 (2020).
  * Carboni  _et al._ (2012) S. Carboni, S. Barlini, L. Bardelli, N. L. Neindre, M. Bini, B. Borderie, R. Bougault, G. Casini, P. Edelbruck, A. Olmi, _et al._ , Nucl. Instr. and Methods A 664, 251 (2012).
  * Aichelin and Stöcker (1986) J. Aichelin and H. Stöcker, Physics Letters B 176, 14 (1986).
  * Aichelin (1991) J. Aichelin, Physics Reports 202, 233 (1991).
  * Ono (2019) A. Ono, Progress in Particle and Nuclear Physics 105, 139 (2019).
  * Ikeno  _et al._ (2016) N. Ikeno, A. Ono, Y. Nara, and A. Ohnishi, Phys. Rev. C 93, 044612 (2016).
  * Chabanat  _et al._ (1997) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nuclear Physics A 627, 710 (1997).
  * Piantelli  _et al._ (2019) S. Piantelli, A. Olmi, P. R. Maurenzig, A. Ono, M. Bini, G. Casini, G. Pasquali, A. Mangiarotti, G. Poggi, A. A. Stefanini, S. Barlini, A. Camaiani, C. Ciampi, C. Frosin, P. Ottanelli, and S. Valdré, Phys. Rev. C 99, 064616 (2019).
  * Tian  _et al._ (2017) G. Tian, R. Wada, Z. Chen, R. Han, W. Lin, X. Liu, P. Ren, F. Shi, F. Luo, Q. Sun, L. Song, and G. Q. Xiao, Phys. Rev. C 95, 044613 (2017).
  * Tian  _et al._ (2018) G. Tian, Z. Chen, R. Han, F. Shi, F. Luo, Q. Sun, L. Song, X. Zhang, G. Q. Xiao, R. Wada, and A. Ono, Phys. Rev. C 97, 034610 (2018).
  * Galanopoulos  _et al._ (2010) S. Galanopoulos, G. Souliotis, A. Keksis, M. Veselsky, Z. Kohley, L. May, D. Shetty, S. Soisson, B. Stein, S. Wuenschel, and S. Yennello, Nuclear Physics A 837, 145 (2010).
  * Casini  _et al._ (1993) G. Casini, P. G. Bizzeti, P. R. Maurenzig, A. Olmi, A. A. Stefanini, J. P. Wessels, R. J. Charity, R. Freifelder, A. Gobbi, N. Herrmann, _et al._ , Phys. Rev. Lett. 71, 2567 (1993).
  * Stefanini  _et al._ (1995) A. A. Stefanini, G. Casini, P. R. Maurenzig, A. Olmi, R. J. Charity, R. Freifelder, A. Gobbi, N. Herrmann, K. D. Hildenbrand, M. Petrovici, _et al._ , Zeitschrift für Physik A Hadrons and Nuclei 351, 167 (1995).
  * De Filippo, E.  _et al._ (2012) De Filippo, E., A. Pagano, P. Russotto, F. Amorini, A. Anzalone, L. Auditore, V. Baran, I. Berceanu, B. Borderie, Bougault, _et al._ , Phys. Rev. C 86, 014610 (2012).
  * Jedele  _et al._ (2017) A. Jedele, A. B. McIntosh, K. Hagel, M. Huang, L. Heilborn, Z. Kohley, L. W. May, E. McCleskey, M. Youngs, A. Zarrella, and S. J. Yennello, Phys. Rev. Lett. 118, 062501 (2017).
  * (53) S. Piantelli _et al._ , Phys. Rev. C accepted.
  * Vient  _et al._ (2018) E. Vient, L. Augey, B. Borderie, A. Chbihi, D. Dell’Aquila, Q. Fable, L. Francalanza, J. D. Frankland, E. Galichet, D. Gruyer, _et al._ , The European Physical Journal A 54, 96 (2018).
  * Piantelli  _et al._ (2002) 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).
  * Piantelli  _et al._ (2007) S. Piantelli, P. R. Maurenzig, A. Olmi, L. Bardelli, M. Bini, G. Casini, A. Mangiarotti, G. Pasquali, G. Poggi, and A. A. Stefanini, Phys. Rev. C 76, 061601 (2007).
  * Rudolf  _et al._ (1993) G. Rudolf, S. Tomasevic, M. Aboufirassi, J. Adloff, B. Bilwes, R. Bilwes, G. Bizard, R. Bougault, R. Brou, Y. Cassagnou, J. Colin, F. Cosmo, F. Delaunay, D. Durand, J. Ferrero, A. Genoux-Lubain, M. Glaser, F. Guilbault, G. Jin, J. Laville, C. Le Brun, C. Lebrun, J. Lecolley, F. Lefèbvres, R. Legrain, J. Lemière, O. Lopez, M. Louvel, M. Mahi, A. Péghaire, J. Péter, B. Rastegar, E. Rosato, F. Scheibling, J. Steckmeyer, L. Stuttgé, and B. Tamain, Physics Letters B 307, 287 (1993).
  * Charity (1998) R. J. Charity, Phys. Rev. C 58, 1073 (1998).
  * Bonnet  _et al._ (2008) E. Bonnet, J. P. Wieleczko, J. G. Del Campo, M. La Commara, S. Barlini, C. Beck, B. Borderie, R. Bougault, A. Chbihi, R. Dayras, G. De Angelis, J. D. Frankland, A. Galindo-uribarri, T. Glodariou, V. Kravchuk, P. Lautesse, J. Moisan, N. Le Neindre, B. Martin, L. Nalpas, A. D. Onofrio, M. Parlog, D. Pierroutsakou, F. Rejmund, M. F. Rivet, M. Romoli, E. Rosato, R. Roy, D. Shapira, G. Spadaccini, B. Tamain, and M. Vigilante, International Journal of Modern Physics E 17, 2359 (2008).
  * Ademard  _et al._ (2011) G. Ademard, J. P. Wieleczko, J. Gomez del Campo, M. La Commara, E. Bonnet, M. Vigilante, A. Chbihi, J. D. Frankland, E. Rosato, G. Spadaccini, S. A. Kalandarov, C. Beck, S. Barlini, B. Borderie, R. Bougault, R. Dayras, G. De Angelis, J. De Sanctis, V. L. Kravchuk, P. Lautesse, N. Le Neindre, J. Moisan, A. D’Onofrio, M. Parlog, D. Pierroutsakou, M. F. Rivet, M. Romoli, R. Roy, G. G. Adamian, and N. V. Antonenko, Phys. Rev. C 83, 054619 (2011).
  * Pirrone  _et al._ (2019) S. Pirrone, G. Politi, B. Gnoffo, M. La Commara, E. De Filippo, P. Russotto, M. Trimarchi, M. Vigilante, M. Colonna, S. A. Kalandarov, F. Amorini, L. Auditore, C. Beck, G. Cardella, A. D’Onofrio, E. Geraci, D. Lacroix, E. La Guidara, G. Lanzalone, A. Pagano, E. V. Pagano, M. Papa, E. Piasecki, L. Quattrocchi, F. Rizzo, E. Rosato, G. Spadaccini, and A. Trifirò, The European Physical Journal A 55, 22 (2019).
  * Barlini  _et al._ (2013) S. Barlini, S. Piantelli, G. Casini, P. R. Maurenzig, A. Olmi, M. Bini, S. Carboni, G. Pasquali, G. Poggi, Stefanini, _et al._ (FAZIA Collaboration), Phys. Rev. C 87, 054607 (2013).
  * Piantelli  _et al._ (2017) S. Piantelli, S. Valdré, S. Barlini, G. Casini, M. Colonna, G. Baiocco, M. Bini, M. Bruno, A. Camaiani, _et al._ , Phys. Rev. C 96, 034622 (2017).
  * Fable (2018) Q. Fable, Ph.D. thesis, Université de Caen Normandie (2018).
  * Boisjoli, M.  _et al._ (2012) Boisjoli, M., Chbihi, A., and Wigg, P. C., EPJ Web of Conferences 31, 00040 (2012).
  * Wigg, P.C.  _et al._ (2012) Wigg, P.C., Boisjoli, M., Chartier, M., Chbihi, A., Lemmon, R., Frankland, J.D., Le Neindre, N., and Marini, P., EPJ Web of Conferences 31, 00015 (2012).
  * Camaiani  _et al._ (2018b) A. Camaiani, G. Casini, L. Morelli, S. Barlini, S. Piantelli, G. Baiocco, M. Bini, M. Bruno, A. Buccola, M. Cinausero, M. Cicerchia, M. D’Agostino, M. Degelier, D. Fabris, C. Frosin, F. Gramegna, F. Gulminelli, G. Mantovani, T. Marchi, A. Olmi, P. Ottanelli, G. Pasquali, G. Pastore, S. Valdré, and G. Verde, Phys. Rev. C 97, 044607 (2018b).
  * Morelli  _et al._ (2019) L. Morelli, M. Bruno, M. D’Agostino, G. Baiocco, F. Gulminelli, S. Barlini, A. Buccola, A. Camaiani, G. Casini, C. Ciampi, C. Frosin, N. Gelli, A. Olmi, P. Ottanelli, G. Pasquali, S. Piantelli, S. Valdré, M. Cicerchia, M. Cinausero, F. Gramegna, G. Mantovani, T. Marchi, M. Degerlier, D. Fabris, and V. L. Kravchuk, Phys. Rev. C 99, 054610 (2019).
  * Bruno  _et al._ (2019) M. Bruno, M. D’Agostino, M. V. Managlia, L. Morelli, G. Baiocco, F. Gulminelli, C. Frosin, S. Barlini, A. Buccola, A. Camaiani, G. Casini, M. Cicerchia, M. Cinausero, M. Degerlier, D. Fabris, F. Gramegna, G. Mantovani, T. Marchi, P. Ottanelli, G. Pasquali, S. Piantelli, and S. Valdré, Journal of Physics G: Nuclear and Particle Physics 46, 125101 (2019).