Q Fable

L Baldesi

S Barlini

Eric Bonnet

Bernard Borderie

Remi Bougault

A Camaiani

G Casini

A Chbihi

Caterina Ciampi

J A Dueñas

J D Frankland

T Génard

D Gruyer

M Henri

B Hong

S Kim

A J Kordyasz

T Kozik

A Le Fèvre

N Le Neindre

I Lombardo

O Lopez

T Marchi

P Marini

S H Nam

A Ono

J Park

M Pârlog

S Piantelli

A Rebillard-Soulié

G Verde

E Vient

Study of isospin diffusion from 40,48 Ca+ 40,48 Ca experimental data at Fermi energies: direct comparisons with transport model calculations

Keywords: numbers: 21.65.Ef, 25.70.-z, 25.70.Lm, 25.70.Mn, 25.70.Pq

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The nuclear equation of state (EoS) is a major issue in modern astrophysics and nuclear physics, as it is a central it is usually defined as the second derivative of the energy per nucleon in nuclear matter with respect to the neutron-to-proton asymmetry (isospin). Better knowledge of the symmetry energy is necessary in the context of astrophysical simulations which aim to describe neutronrich systems over a wide range of densities. At suprasaturation densities, constraints can be extracted by relativistic HIC [7] and multi-messenger measurements [8], whose interest has been renewed by the recent breakthrough in gravitational waves measurements. At subsaturation, HIC offer a unique opportunity to study the modification of the chemical composition induced by the formation and dissolution of clusters in dilute nuclear matter [9]. The study of these reactions allows to probe the thermodynamical properties of the expanding nuclear system, thus the effect of cluster formation on the symmetry energy. Various HIC observables have been used to study the symmetry energy density dependence of neutron-rich matter. Experimental probes such as nuclear masses [10], isobaric analog states [11], collective flows [12], pion yield ratios [13] and isospin transport [14] have been used to constrain the symmetry energy functional. More recently, it has been shown that such measurements are not only consistent with astrophysical observations but they also provide important constraints on the nuclear EoS at supra-saturation densities [5,6]. It was furthermore pointed out that improvements in the interpretation of experimental observables are needed if one wants to refine our knowledge of the EoS. In this context, isospin transport is particularly interesting as it led to one of the first experimental constraints on the symmetry energy [15]. This process, expected to occur in binary dissipative reactions, corresponds to a stochastic and differential exchange of nucleons between projectile and target. In particular, in collisions with different projectile and target N/Z asymmetries, a balancing in the neutron-richness of different isospin regions is expected to take place, leading to a rearrangement of the neutronto-proton ratio of the colliding nuclei, called isospin diffusion. The degree of isospin equilibration, directly related to the magnitude of the symmetry energy (at a given local density), can be experimentally estimated from the isospin transport ratio [16], widely used over the years [17][18][19][20][21].

Constraints on the EoS are obtained by comparing nuclear experimental data to transport model calculation, where it is possible to test the interplay between the mean-field effects and nucleon-nucleon collisions. In addition to the inherent difficulty of implementing various algorithms for solving the transport equations, uncertainties also arise from the comparison protocols between the experimental data and the transport models. Indeed, as most of the relevant observables are not directly measurable experimentally, surrogate variables are usually used, complicating the interpretations. Suitable comparisons should therefore focus on an identical set of observables, with calculations ideally run over the same domain of impact parameter probed by the experiment and taking into account the intrinsic limits of the experimental setup. To this aim, this work presents an investigation of isospin diffusion in 40,48 Ca+ 40,48 Ca reactions at 35 MeV/nucleon, from both experimental and transport model simulation data. The experimental setup is described in Sec.II. The model framework is discussed in Sec.III along with global comparisons with the experimental data. The impact parameter estimation method is presented in Sec.IV, while the results on isospin diffusion are presented in Sec.V. A summary and conclusions are finally drawn in Sec.VI.

II. EXPERIMENTAL SETUP AND EVENT SELECTION

A. Experimental setup

The experiment was performed at the GANIL facility, where beams of 40,48 Ca at 35 MeV/nucleon impinged on self-supporting 1.0 mg/cm 2 40 Ca or 1.5 mg/cm 2 48 Ca targets placed inside the INDRA vacuum chamber, for a typical beam intensity around 5.10 7 pps. The detection apparatus consisted of the coupling of the 4π charged particle array INDRA and the VAMOS high-acceptance spectrometer. INDRA is composed of detection telescopes arranged in rings and centered around the beam axis. A detailed description of the INDRA detector and its electronics can be found in [22,23]. For this experiment, INDRA covered polar angles from 7 • to 176 • , as rings 1 to 3 were removed to allow the mechanical coupling with VAMOS in the forward direction. Rings 4 to 9 (7 • -45 • ) consisted each of 24 three-layer detection telescopes: a gas-ionization chamber operated with C 3 F 8 gas at low pressure, a 300 or 150 µm silicon wafer, and a CsI(Tl) scintillator (14 to 10 cm thick) read by a photomultiplier tube. Rings 10 to 17 (45 • -176 • ) included 24, 16, or 8 two-layer telescopes: a gas-ionization chamber and a CsI(Tl) scintillator of 8, 6, or 5 cm thickness. Fragment identification thresholds were about 0.5 and 1.5 MeV/nucleon for the lightest (Z 10) and the heaviest fragments, respectively. INDRA allowed charge and isotope identification up to Be-B and charge identification for heavier fragments.

VAMOS is composed of two large magnetic quadrupoles focusing the incoming ions in the vertical and horizontal planes, followed by a large magnetic dipole [24,25]. In the used configuration, the spectrometer was rotated at 4.5 • with respect to the beam axis, to cover the forward polar angles from 2.56 • to 6.50 • , favoring the detection of a fragment emitted slightly above the grazing angles of the studied reactions. The momentum acceptance was about ±5% and the focal plane was located 9 m downstream the target, giving a large enough time of flight (ToF) base to obtain a mass resolution of about ∆A/A ≈ 1/165 for the isotopes produced in the collisions. VAMOS detection setup included two position-sensitive drift chambers used to determine the trajectories of the reaction products at the focal plane, followed by a seven-module ionization chamber, a 500 µm thick Si wall (18 independent modules), and a 1 cm thick CsI(Tl) wall (80 independent modules), allowing the measurements of the ToF, energy loss (∆E), and energy (E) parameters. Around 12 magnetic rigidity settings, from 0.661 to 2.220 Tm, were used for each system to cover the full velocity range of the fragments. At least one hit on the VAMOS silicon wall was required for each event to be acquired.

An overview of the acquired data has been presented in [26], with a detailed description of the identification, reconstruction and event normalization procedures. In a second paper, a study dedicated to isospin diffusion and migration was presented, demonstrating the potential of the INDRA-VAMOS coupling to provide further experimental constraints on the nuclear symmetry energy [21].

B. Data selection

The following preliminary selections have been applied for the present analysis : i) Only multiplicity "1" events in the VAMOS Si wall were considered. This offline selection was applied to make sure that the positions measured in the drift chambers are correct and to remove events with ambiguous trajectory reconstruction.

ii) Elastic-like events, defined as events with no hit in INDRA and a fragment identical to the projectile in VAMOS, were also removed offline.

iii) The most incomplete events were discarded by requesting a measured total charge Z tot ≥ 10 and a total parallel momentum (along the beam axis) p tot z ≥ 0.5 • p beam , where p beam is the beam initial momentum.

iv) Events with a potential QP residue measured in INDRA were discarded by requesting that Z P LF > Z max,f wd I , where Z P LF and Z max,f wd I are respectively the charge of the fragment detected in VAMOS and of the heaviest fragment forward-emitted in the reaction center of mass (c.m.) identified in charge with INDRA. Since we focus on the study of isospin-sensitive observables from the QP remnant, such events are indeed not relevant due to the limited mass identification (Z 5) with INDRA.

In summary, the applied data selections criteria represent a subset of about 90% and 87% of the total statistics of the experimental data for 40 Ca and 48 Ca projectile reactions, respectively.

The present analysis relies on the comparison of experimental data with the output of the antisymmetrized molecular dynamics (AMD) model [27]. AMD is a microscopic transport model widely used to describe various features of ground state nuclei [28] and out of equilibrium many-body dynamics [29], recently reviewed in [30].

A general issue encountered when comparing transport model simulations with experimental HIC data at Fermi energies is the underestimation of light cluster yields. For example, in a recent study Frosin et al. investigated the role of clustering at Fermi energies by comparing 32 S+ 12 C and 20 Ne+ 12 C reactions at 25 and 50 MeV/nucleon measured with four FAZIA blocks with the output of the AMD model [31]. It was shown that accounting for cluster formation helps to better reproduce experimental multiplicities of both light-charged particles (LCP, Z ≤ 2) and intermediate mass fragments (IMF, 3 ≤ Z ≤ 8) but also their kinetic energy distributions. Furthermore, to our knowledge, in the literature very few works adressed the specific issue of clustering in semiperipheral to peripheral collisions, where binary dissipative collisions exhaust a large part of the total reaction cross section [32].

In this perspective, we present in this section a comparison of the INDRA-VAMOS data with a basic version of AMD, namely AMD-NC, and a more recent version with cluster correlations, namely AMD-CC. More precisely, in AMD-CC cluster correlations are explicitly taken into account by allowing each of the scatterd nucleons to form light clusters (A ≤ 4) and also heavier nuclei (up to A 10) from the introduction of intercluster correlations as a stochastic process of intercluster binding, as described in [33].

Concerning AMD-NC, the same version as the one detailed in [34] was employed, proven to be suitable for the study of Ca+Ca collisions at 35 MeV/nucleon [35,36]. This version was used in our previous work on isoscaling [26] and isospin transport [21]. Collisions were followed up to t lim = 300 fm/c, for a total of approximately 100000 primary events per system. Clusters were defined at t lim according to a proximity criterion of the Gaussian centroids in spatial coordinates, two-nucleon pair separated by less than 5 fm contributing to the formation of a cluster. In this version the mean-field description is based on the momentum-dependent Gogny force, consistent with the incompressibility modulus of symmetric matter K sat = 228 MeV and ρ 0 = 0.16 fm -3 . Based on such effective interaction, two parametrizations of the density-dependence of the nuclear symmetry energy were exploited, namely the Gogny (soft dependence) and Gogny-AS (stiff dependence) forces [37].

Concerning AMD-CC, we have employed the same simulated data as those published in [17]. Collisions were followed up to t lim = 500 fm/c, for a total of about 40000 primary events per system. In this version, the meanfield description is based on the momentum-dependent SLy4 effective Skyrme interaction [38], with K sat = 230 MeV and ρ 0 = 0.16 fm -3 . Two parametrizations of the symmetry energy were tested: a soft symmetry energy dependence with E sym = 32 MeV and L = 46 MeV, as zero and first order terms, respectively, and a stiff one with L = 108 MeV. Finally, a reduction factor of γ = 0.85 was employed for the in-medium correction of nucleon-nucleon cross-section, following the prescription proposed in [39].

For both models, the input impact parameter followed a triangular distribution from 0 to a value b max , that is slightly above the geometrical grazing impact parameter b gr (about 9.7 fm for 40 Ca+ 40 Ca and 10.4 fm for 48 Ca+ 48 Ca). The statistical decay code GEMINI++ was employed [40] to de-excite the hot nuclei produced at t lim (afterburner step), using the default parameters of the code, for a total of 50 and 100 secondary events per primary event for AMD-NC and AMD-CC respectively. The number of decays per primary event was chosen to limit their effect on the uncertainties associated to the imbalance ratio.

We note an important discrepancy between the two versions of AMD, due to the difference in stopping time t lim . Indeed, in our former work with AMD-NC we have employed a standard t lim value widely used in the literature with this version of the code [36]. In comparison, in the case of AMD-CC the authors of ref. [18] ensured that the primary fragments have reached thermodynamical equilibrium (while Coulomb repulsion is negligible) at t lim , for the systems under study [41]. Given the previous comment, some caveats are to be considered when comparing the primary events of the two model versions.

B. Filter and event selection

For both aforementioned versions of the AMD model, simulated events were filtered with the same software replica of the experimental setup, to allow comparison between the predictions and the experimental data.

Event detection was simulated within the KaliVeda framework [START_REF] Frankland | Kaliveda: Heavy-ion collisions analysis toolkit[END_REF]. Concerning VAMOS, the experimental polar and azimuthal angular distribution in the laboratory frame were used to apply cuts on the simulated events, while the energy and identification thresholds of the Si and CsI detectors were considered. Concerning INDRA, KaliVeda allows a complete description of the detector configuration for the experiment, including geometrical coverage, dead-zones, detector resolutions and identification thresholds. It must be noted that the majority of the simulated events are discarded due to the spectrometer angular acceptance and trigger condition, for a total of about ≈ 75% of the whole statistics.

In addition to the numerical filtering, the same offline selections as the experiment, described in Sec.II A, were applied to the filtered events. In summary, once the preliminary selections are applied, the filtered simulated events represent a subset of about of 22% and 18% of the total statistics for the AMD-CC and AMD-NC models, respectively.

C. General features

In this section we present a comparison of several relevant global observables between the two versions of the AMD model, to better apprehend the results on isospin diffusion presented in the next section. For the sake of clarity and as we focus on the general features, only the results obtained from a soft symmetry energy are presented at this point of the analysis. Similarly to our previous work with the AMD-NC version [26], the unfiltered model exhibits two main components on both sides of the c.m. velocity (black dashed line), with a third region of LCP and IMF spreading over the whole velocity domain. By comparing the unfiltered and filtered simulations, we mainly observe the effect of VAMOS angular acceptance, which drastically reduces the measured yields while the right-most component becomes concentrated in a region of charge and velocity close to the projectile one. Such topology is representative of dissipative binary collisions, where the fragments detected in VAMOS are mostly the products of the quasiprojectile decay (namely the projectile-like fragment, PLF) resulting from peripheral to semi-peripheral collisions, while the products of the quasitarget (targetlike fragment, TLF) decay are occasionally identified in INDRA at backward angles (left-most component). Finally, by comparing the experimental data and the filtered model, a satisfactory agreement is observed. It must be noted that similar topologies are observed independently of the version of the AMD model and for all the systems, with a similar effect of the numerical filter and data selection [26].

We present in Fig. 2 the effect of the filter and event selection criteria on the reduced impact parameter distributions for the 48 Ca+ 48 Ca system. The input triangular distributions, represented as solid lines, present a small difference between the models as AMD-CC was ran up to slightly larger b/b max values than AMD-NC. Interestingly, we observe a strong difference between the two versions of the code once the filter and data selection are applied, AMD-NC (open circles) leading to the detection of more peripheral events compared to AMD-CC (full circles), even if the latter was run up to higher impact parameter values. Such remarkable difference is not trivial as it comes from the combined effect of the model dynamics, the detectors acceptance and the applied offline selections. In a sense, this plot highlights the inherent difficulty of comparing models with experimental data as the subset of simulated data can vary from a model to example expected that the AMD-NC version leads, in average, to more LCP emissions compared to AMD-CC, therefore to a different pattern of multiplicities and kinetic energy spectra. In fact, a more detailed study of the primary fragments from AMD-NC (not presented here) leads to the conclusion that this version of the code tends to overproduce inelastic collisions in the b/b max = 0.8 -1 region as compared to AMD-CC, while these fragments have still enough excitation energy to emit LCP in the afterburner phase.

Study of the QP residue

We present in this section a comparison of the global observables related to the fragment detected in VAMOS, expected to be the QP remnant (PLF). According to the simulations, more than 95% of the nuclei detected in VAMOS (after data selection criteria) correspond to the remnant of the QP, the latter defined as the biggest primary fragment forward emitted in the c.m. reference frame.

The parallel velocity (laboratory frame) and charge distributions of the filtered PLF are given in Figs. 3(a-d) and (e-h) respectively, for the four reactions under study. The main plots present the distributions obtained from the experiment (full circles) and filtered models (continuous lines) when the selection criteria are applied. Concerning the velocity distributions (top panels), we observe decreasing values of the yields with decreasing velocity of the PLF, reflecting the impact parameter distribution. As expected from Fig. 2, the differences between the two versions of AMD are more pronounced for the most peripheral collisions (b/b gr 0.8, V z 8 cm/ns) as AMD-NC tends to overproduce inelastic-like events compared to AMD-CC. It is interesting to note that the opposite effect is observed for the unfiltered models represented in the inner plots, showing that the experimental selection criteria (and indirectly the consideration of cluster formation) may have a strong impact on the filtered distributions. An overall better agreement with experimental data is obtained with AMD-CC, while it tends to produce slighlty slower QP compared to the experiment and AMD-NC. Similar results were obtained from various systems simulated with AMD-CC followed by GEMINI++ as afterburner [18][19][20], and it was concluded that the model is more dissipative than the experimental case, although the effect of the filter, event selection (event multiplicity) and clustering process were not detailed. Concerning the atomic number distributions (bottom panels), similar conclusions can be drawn: AMD-CC reproduces remarkably well the charge distributions for all the systems, even for the neutron-rich 48 Ca+ 48 Ca, while AMD-NC predicts more heavy products than experimentally observed. The same conclusion can also be drawn from the mass distributions of the PLF (not shown here). Finally, it is interesting to note that the observed experimental odd-even staggering [START_REF] Piantelli | [END_REF], stronger for neutron-deficient systems, is partially reproduced by both filtered simulations, as a consequence of the sequential decays (GEMINI++).

We present in Fig. 4 the evolution of the average neu- a-d) and charge (e-h) distributions of the PLF from the experiment (full circles) and filtered AMD-NC and AMD-CC models (blue and solid lines, respectively) . The inner plots represent the PLF from the models when the numerical filter is not applied. The distributions are normalized to their integral.

tron excess of the PLF detected in VAMOS as a function of its atomic number, for all the reactions. We first ob- serve that similarly to the experiment (full cirlces), both models (full lines) present an evolution according to the neutron content of the projectile, progressively driven by the evaporative attractor line (EAL, dashed-dotted line extracted from [44]) with decreasing charge [26]. Concerning the 40 Ca projectile reactions, we notice a satisfying agreement for both models with the experimental data, except for the most peripheral collisions (Z = 20) for the neutron-rich target with AMD-CC. Concerning the 48 Ca projectile reactions, we observe that both models tend to underestimate the neutron-richness of the PLF. This could be ascribed to a too high excitation energy of the QP in both AMD versions, leading to the emission of too many neutrons in the afterburner step. Since model predictions remain too close to the EAL compared to the data, their mutual difference increases with increasing charge. We also note more discrepancies between the models in the Z P LF = 15 -20 peripheral region, AMD-NC predicting values closer to the experiment compared to AMD-CC. Finally, the neutron excess of the associated QP, as predicted by the model (before GEM-INI++), is also presented in thin and thick dashed lines for AMD-NC and AMD-CC respectively. We remark that the AMD-NC model exhibits systematically more neutron-rich QP than AMD-CC. The neutron excess is partially preserved for the largest PLF (Z P LF > 15).

LCP and IMF multiplicities

In addition to the PLF properties, we present in this section the multiplicity of the nuclei measured in coincidence in INDRA. We aim at studying the evolution of cluster emission with the dynamical properties of the reactions, using the parallel velocity V P LF z of the fragment detected in VAMOS as a sorting parameter. Indeed, such a selection allows one to remove the trivial bias induced by the excess of peripheral collisions in AMD-NC, as seen in Fig. 3 in the V P LF z 7.5 -8 cm/ns region. For this reason, we focus our interpretations on the semiperipheral region (V P LF z 7.25 cm/ns). Figure 5 shows a comparison of the average total, LCP and IMF multiplicity distributions of the nuclei detected in INDRA as a function of V P LF z . We first observe that all multiplicities increase with decreasing V P LF z , reflecting an increase of fragment production with increasing dissipation as higher excitation energies are reached on average. Then, we notice that the total multiplicity is governed by the LCP emissions, which both models tend to overproduce for all V P LF z . This overproduction increases with decreasing V P LF z , reaching about ∆ M LCP 1 and ∆ M LCP 2 for neutrondeficient and neutron-rich projectile reactions, respectively, for AMD-NC, while the overproduction remains much smaller for AMD-CC. A noticeable difference is also observed in the IMF multiplicity distributions that present an underproduction in the case of AMD-NC and an overproduction for AMD-CC, more evident for the 48 Ca target reactions.

We present in Figs. 6 and 7 the average multiplicities of Z = 1 and Z = 2 nuclei isotopically identified in INDRA as a function of V P LF z . The unfiltered neutron distributions are also shown in Fig. 6. As a first general comment, we observe that the models exhibit a dominant emission of proton and α-particles, similarly to the data. Concerning the Z = 1 isotopes, we observe an overproduction of protons by AMD-NC, reaching about ∼ 40% for the most dissipative reactions, while AMD-CC present values close to the experimental one at low velocity. Interestingly, an overproduction of neutrons by AMD-NC compared to AMD-CC is also observed for all the systems except the neutron-deficient 40 Ca+ 40 Ca. This indicates that in the case of AMD-NC more free protons (and neutrons) are produced to the detriment of clusters, while the difference in filtered total multiplicity remains reasonable. This seems to be confirmed by the 2,3 H isotopes multiplicities which are systematically underproduced by AMD-NC while the inclusion of cluster formation in AMD-CC leads to a better agreement with the data. Nevertheless, one should exercise caution regarding the 2 H isotopes because the SLy4 effective force employed in AMD-CC leads to an overestimation of their binding energy for the soft parametrization, leading to an anomalous increase of their multiplicity [41].

Concerning now the Z = 2 isotopes, we first notice an underproduction of 3 He isotopes from the models, reach-ing a reduction of about 75% and 50% for AMD-NC and AMD-CC, respectively. The situation is even worse for 6 He isotope multiplicities, even if the average multiplicities are very small. Finally, a similar behaviour is observed in the case of α-particles for both models, without a clear evidence of what model performs the best.

Transverse kinetic energy of LCP

We present in Fig. 8 the distributions of the total transverse energy of the LCP identified in charge, namely E t12 , extracted from the filtered models and the experiment.

This global variable is of particular interest for centrality estimation in the analysis of isospin transport (see Sec.V) as it is well-suited to the performance of INDRA array for which LCP are detected with a 90% efficiency. It is defined as

E t12 = i:Zi≤2 E i sin 2 θ i (1)
where in the sum i runs over the detected (filtered) products of each event with Z i ≤ 2, laboratory kinetic energy E i and laboratory polar angle θ i . We first notice in Fig. 8 a good agreement between the experimental distributions (full circles) and the filtered AMD-CC simulations (thick red lines) that reproduce both the experimental trends and E t12 values, including the tail of the distributions. In the case of AMD-NC, we observe more narrow-down distributions with higher statistics for the less dissipative collisions (E t12 20 MeV), as already anticipated from Fig. 3, while the tails of the experimental data are not reproduced. We clearly observe in this figure the relevance of considering clustering to reproduce the experimental kinetic properties of the LCP, as recently evidenced in [31].

D. Discussion

As a general comment, we can conclude that both versions of AMD reproduce reasonably well the experimental topology (charge and parallel velocity correlations), highlighting the effect of VAMOS acceptance (and trigger) to favor the measurement of the QP remnant in binary dissipative collisions. A comparison of the filtered simulation impact parameter distributions shows that the data selection criteria have a strong impact on the subset of simulated data from a model to another.

We have shown that the AMD-CC version reproduces remarkably well the experimental velocity, charge and mass distributions of the PLF detected in VAMOS, while a sizable disagreement is obtained with AMD-NC, mostly for the less dissipative collisions. This was anticipated as the AMD-NC version leads to an overproduction of inelastic-like events compared to AMD-CC. Furthermore, (cm/ns

) PLF z V 0.5 1.5 IMF 〉 M 〈 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
Figure 5: Average total (a-d), LCP (e-h) and IMF (i-l) multiplicity measured with INDRA for the experiment (full circles) and the filterd AMD-NC (thin blue lines) and AMD-CC (thick red lines) simulations followed by GEMINI++. a focus on the evolution of the neutron-richness of the PLF shows that both models manage to reproduce, to some extent, the experimental trends. This point is particularly relevant for the study of isospin diffusion, as it is expected to be responsible of such an evolution according to the neutron content of the projectile.

By exploiting the parallel velocity of the PLF as a surrogate for the collision dissipation, we have shown that it is also possible to discuss several aspects of the dynamical emission of clusters. More specifically, comparisons of the average multiplicities have stressed that the inclusion of cluster correlations helps to better reproduce the experimental isotopic multiplicities. A remarkable agreement is obtained for 1,2,3 H isotopes in the case of AMD-CC while AMD-NC tends to overproduce free protons to the detriment of clusters. This is nonetheless less obvious for 3,4,6 He isotopes, as both models tend to underproduce 3,6 He but present a satisfying agreement for 4 He. It was also shown that AMD-CC tends to better reproduce the average IMF multiplicties.

To conclude, similarly to [32], this work demonstrates that AMD-CC is not only relevant in the study of central collisions, but also for semi-peripheral to peripheral collisions, where binary dissipative collisions exhaust a large part of the total reaction cross section. We also showed that the inclusion of cluster formation is a mandatory step to better reproduce the experimental features, like the multiplicity and transverse kinetic energy distributions. This is particularly interesting in order to improve the comparison protocols used in the study of HIC, as these global observables are used for experimental impact parameter sorting.

IV. IMPACT PARAMETER RECONSTRUCTION

In this section, we apply a method for impact parameter distribution estimation [45,46], recently adapted from relativistic HIC to the Fermi energy domain by Frankland et. al [47]. The goal of the method is to infer realistic impact parameters from the inclusive distribution of an experimentally measured observable. Such method is mandatory for suitable comparisons with dynamical models where, as shown in the previous section, we can expect a strong variation in the inclusive impact parameter distribution. More details of the implementation of the method for the INDRA-VAMOS data are given in Appendix A.

We would like to highlight that the software necessary to perform the present analysis is currently implemented in the KaliVeda heavy-ion analysis toolkit [START_REF] Frankland | Kaliveda: Heavy-ion collisions analysis toolkit[END_REF].

A. Validation of the method with AMD-CC

In the following, we present the results of applying the impact parameter reconstruction method presented in Appendix A to the filtered AMD-CC simulated data. Since, as discussed in Section III, AMD-CC reproduces several experimental findings much better than the AMD-NC version, we will now exclusively use and focus on the former. Figure 9 shows an example of the quality of the fits to the inclusive E t12 data histograms from AMD-CC stiff, achieved using Eq.A1 and the gammadistribution parametrization. The fit parameters and the reduced χ 2 values are given in Table I for all simulated data and both parametrizations of the symmetry energy. We observe that the shapes of the filtered distributions are globally well-reproduced by the fits, with a satisfactory goodness-of-fit parameter (χ 2 ∼ 1) and similar parameter values for both parametrizations (for a given system). Fitting the P (E t12 ) distributions allows to determine the conditional probability distribution P (E t12 |c b ), which is then used to extract the centrality and absolute impact parameter distributions for any sample S (see Eq.A3 and A4). For this analysis, we have adopted a sampling of 20 bins of 5% experimental centrality c Et12 . Results for the 40 Ca+ 40 Ca collisions are shown in Fig. 10 for the AMD-CC model followed by GEMINI++ as afterburner, for six centrality intervals. The distributions reconstructed from the deduced form of P (E t12 |c b ) (solid lines) are compared to the actual distributions (filled areas), directly obtained by applying the corresponding cuts in c Et12 to the model. Also, the average impact parameter expected to be probed in the most central collisions is about b 3 ± 0.5 fm, corresponding to semi-central collisions. This limit is mostly induced by VAMOS trigger condition, preventing to measure the most central collisions. Similar results are obtained from AMD-CC with a soft symmetry energy. We observe a reasonable agreement between the distributions for all the samplings, with similar results for all the simulated systems. Also, similarly to [47], the most central sampling (0.05 < c E12 ≤ 0.1) presents a shift that would lead to a slight underestimation of the mean impact parameter.

Model

Finally, we present in Fig. 11 the evolution of the mean centrality c b and impact parameter b as a function of the applied sampling in c Et12 , for all the considered systems. The associated standard deviation are shown as error bars. This plot allows to illustrate the performances of the method to quantitatively characterize the centrality of selected event samples, in a modelindependent way (at least for c b ). We observe a linear correlation between the reconstructed c b and c Et12 in the c b 0.05 -0.95 range, independently of the system. This evolution is not observed for the reconstructed b, where values present a system-dependence and increase non-linearly from b 3 fm to b 9 fm.

B. Application to the experimental data

Following the same procedure as in the previous section, we can estimate the impact parameter distributions for the experimental data. Figure 12 shows an example of the quality of the fits to the inclusive E t12 experimental data, achieved using Eq.A1 and the gamma-distribution parametrization, similarly to Fig. 9. The fit parameters and the reduced χ 2 values are given in Table II The centrality and absolute impact parameter distributions for all the samples in experimental centrality c Et12 are then estimated using Eqs. A3 and A4, respectively. The evolution of the average centrality c b and impact parameter b as a function of c Et12 are presented in Fig. 13(a) and (b), respectively, for the four reactions under study. For both observables we find, as expected, increasing values with increasing c Et12 , even if a stronger system-dependence than the model is observed for b . Concerning b , we have used the inclusive impact parameter of the filtered model (after data selection criteria) as a surrogate for P (b) in Eq.A4, similarly to Sec. IV A. The plots given in Fig. 13(b) are thus obtained with the inclusive impact parameter distributions extracted from the filtered AMD-CC with a stiff symmetry energy. It was verified that the same quantitative results are obtained with the soft symmetry energy (within the error bars), as the P (b) distributions are almost not sensitive to the employed parametrization.

V. ISOSPIN DIFFUSION

This section is dedicated to the study of isospin diffusion from the quasiprojectile (QP) and its remnant (PLF), defined as the fragment measured in VAMOS. The isospin transport ratio, described in the following, is used to quantify the degree of isospin equilibration, while its evolution is followed as a function of the centrality or impact parameter, estimated using the method described in Sec.IV. In the first part, we focus on the AMD-CC model, in order to characterize the effect of the impact parameter reconstruction on the isospin equilibration measured from the QP along with the effect of sequential decays. In the continuity with our previous works [21,26], the experimental method applied to reconstruct the primary QP fragments, including the estimation of evaporated neutrons, is also tested and compared to the result obtained from the actual QP predicted by the model. Details about the method are given in Appendix B. In the second part, the same protocol will be applied to the experimental data, allowing direct comparisons with the model.

A. Isospin transport ratio

The isospin transport ratio (also called imbalance ratio) was introduced by Rami et. al to deduce quantitative signals of isospin diffusion in experimental data [16]. It consists of combining an isospin-sensitive observable measured, under the same experimental conditions, with four systems differing in their initial neutron-to-proton ratio. It is defined as

R x = 2x M -x N R -x N D x N R -x N D (2)
where x is an isospin-sensitive observable expected to be univocally related to the N/Z of the systems under study, measured for the symmetric neutron-rich (NR) and neutron-deficient (ND) reactions, while the two mixed reactions (M) reach a neutron content in between these two references. The evolution of R x towards N/Z equilibration is thus followed as a function of a centrality parameter correlated to the dissipation of the collision. By construction, R x = ±1, in the limit of fully non-equilibrated conditions (isospin transparency), while equilibration is generally signaled by both mixed reactions reaching the same value of the ratio.

B. Isospin diffusion in AMD-CC

We present in Fig. 14(a) and (b) the evolution of the asymmetry δ = (N -Z)/A as a function of the centrality c Et12 from AMD-CC (stiff) followed by GEMINI++ for the PLF and the reconstructed QP, respectively. The same sampling as the one used for the impact parameter estimation has been used (20 bins of 5% in c Et12 ).

We first observe that both the PLF and the reconstructed QP exhibit the same hierarchy according to the neutron-richness of the projectile and, to a lesser extent, of the target, as observed in the experimental data [21]. The latter is also observed for the QP predicted by the model, represented in dashed line for each reaction. Nonetheless, the model exhibits for all systems a significant difference between the asymmetry values obtained for the QP and its residue, resulting from the effect of sequential decays [48], that seems to be restored by the QP reconstruction method for 48 Ca projectile reactions but not for 40 Ca. Using the estimated c b and asymmetry δ for each sampling in c Et12 , presented in Figs. 11(a) and 14 respectively, the corresponding isospin transport ratio can be computed from Eq.2 and plotted as a function of c b . The results are presented in Fig. 15 (a) and(b) for the PLF and the QP, respectively, for the stiff and soft parametrization. First, we observe converging values of the isospin transport ratio when moving from peripheral collisions (high c b values) to more central collisions (low c b values), indicating that an evolution towards isospin equilibration is predicted by the AMD-CC model. Second, we notice that for the most central collisions probed ( c b < 0.1) the full equilibration condition is not reached. As observed in Fig. 11(b), the average impact parameter expected to be probed in this reagion of centrality is about b 3 ± 0.5 fm, corresponding to semi-central collisions. Third, the results indicate a weak sensitivity of the isospin transport ratio to the stiffness of the employed symmetry energy within the model, for both the PLF and the reconstructed QP. Indeed, we no- tice that for the stiff asymmetry term (close squares), R δ exhibits systematically less isospin equilibration compared to the soft asymmetry term (open circles), for both the PLF and the reconstructed QP. This is also observed from the QP predicted by the filtered model (not reconstructed), superimposed in thin and thick dashed lines for the soft and stiff parametrization, respectively. Such behaviour has already been mentioned in [49] and [50] with BUU and SMF calculations, respectively.

0 0.2 0.4 0.6 0.8 1 〉 b c 〈 1 - 0.5 - 0 0.5 1 PLF δ R PLF -Stiff EOS PLF -Soft EOS PLF -Stiff EOS PLF -Soft EOS (a) 0 0.2 0.4 0.6 0.8 1 〉 b c 〈 1 - 0.5 - 0 0.5 1 QP δ R rec. QP -Stiff EOS rec. QP -Soft EOS QP -Stiff EOS QP -Soft EOS rec. QP -Stiff EOS rec. QP -Soft EOS QP -Stiff EOS QP -Soft EOS (b)
In a similar way, the evolution of the isospin transport ratio as a function of the average reduced impact parameter, namely b /b gr , can be extracted from Figs. 11(b) and 14. The results are presented in Fig. 16 (a) and(b) for the PLF and the QP, respectively, for the stiff and soft parametrizations. We observe that the same conclusion as the one from Fig. 15 can be drawn, at least in the b /b gr > 0.3 region. It is worth noting that the sensitivity to the stiffness of the employed symmetry energy remains weak, independently of the centrality or the QP reconstruction.

C. Isospin diffusion in experimental data

In this Section, we apply the same protocol used with the AMD-CC model to the INDRA-VAMOS experimental data.

Concerning the evaporated neutron estimation, the mean neutron multiplicities extracted from filtered AMD-CC followed by GEMINI++ calculations were used as a surrogate. More precisely, a constant scaling factor k = 0.8 is applied to the model neutron distribution in order to take into account the overstimation of light particles observed with AMD-CC (see Sec.III C 3). More details are given in Appendix B.

We present in Figs. 17(a) and (b) the evolution of the experimental asymmetry δ as a function of the centrality c Et12 (with a sampling of 5%) for the PLF and the reconstructed QP, respectively.

Concerning the PLF, we notice that the experimental data (symbols) exhibit systematically higher asymmetry values than AMD-CC (dashed lines), while the same hier-archy is obtained. This effect is even more remarkable for the 48 Ca projectile reactions, leading to the conclusion that AMD-CC (followed by GEMINI++) fails to reproduce the experimental QP remnant neutron-enrichment over the full c Et12 domain. Our results are consistent with the analysis of 40,48 Ca+ 40 Ca reactions performed with four blocks of FAZIA, reported in [18]. Concerning the reconstructed QP, we observe that the applied reconstruction method leads to a better agreement between the experimental data and the model, except for the 40 Ca+ 48 Ca asymmetric reaction. Based on the results presented in Figs. 13(b) and 17, the experimental isospin transport ratio was computed from Eq.2 as a function of the estimated reduced impact parameter b /b gr . For more consistency, the scaling factor k used for the evaporated neutron estimation was varied from k = 0.7 -0.9, corresponding to a variation of one standard deviation of the average neutron multiplicity obtained for k = 0.8, while the results presented thereafter are averaged over this domain in k. The difference in average neutron multiplicity induced by the vari-ation of k is larger than the one induced by the stiffness of the symmetry energy. Thus, this average also allows to remove the dependence of the average neutron multiplicity on the employed stiffness for the experimental QP reconstruction.

The results are presented in Figs. 18 (a) and (b) for the experimental PLF and reconstructed QP (full circles), respectively. It must also be noted that similar results can be drawn from the evolution of the experimental R δ with c b . For both PLF and reconstructed QP, we first observe an evolution of the isospin transport ratio towards isospin equilibration, with decreasing values from about R δ ±0.75 towards R δ -0.1 and R δ 0.3 for the neutron-deficient and neutron-rich mixed reactions, respectively, when moving from the most peripheral ( b /b gr 0.9) to the most central ( b /b gr 0.3) measured collisions. Furthermore, we remark a noticable difference between the slopes of the ratios obtained from the two mixed reactions, leading to experimental values closer to the AMD-CC stiff distributions for the 48 Ca+ 40 Ca as mixed system, while the PLF presents more equilibration than the QP at small impact parameters, consistently with the model (open symbols). It is worth noting that a similar experimental trend is obtained for the PLF with 48 Ca+ 40 Ca mixed reaction in [18], where it was concluded that the observed discrepancy can be ascribed to an overestimated probability of nucleon transfer in AMD. In our understanding, the larger model-experiment discrepancy of R δ in 40 Ca+ 48 Ca reaction originates from the significant difference in the evolution of the asymmetry δ with centrality c Et12 , as observed in Fig. 17.

VI. SUMMARY AND CONCLUSIONS

In this work, we have presented an investigation of isospin diffusion in 40,48 Ca+ 40,48 Ca reactions at 35 MeV/nucleon, by comparing INDRA-VAMOS experimental data to the output of the filtered AMD model followed by GEMINI++ as afterburner.

Two versions of AMD, with and without cluster correlations, were employed in order to study the role of clustering in the global observables in the measured reactions. Generally, both versions reproduce the experimental topology (charge and parallel velocity correlation). Nonetheless, it was shown that the inclusion of clusters allows to better reproduce the characteristics (velocity, mass and charge distributions) of the QP remnant detected in VAMOS (PLF). Furthermore, comparisons of the average multiplicities have evidenced that clustering allows to better reproduce the experimental multiplicities while the latter is mandatory to reproduce the LCP transverse kinetic energy distributions. It is also important to note that, aside from clustering, the treatment of dissipation (collision term), nuclear stopping and excitation energy within the models plays an important role in reproducing the experimental data. We believe that reproducing the global observables outlined in this study sets a fundamental standard for the trustworthiness of dynamic models. Such standard ensures that, when comparing them to experimental data, the comparison becomes meaningful and enhances the precision in constraining the equation of state of nuclear matter.

We have applied a method for impact parameter reconstruction specifically adapted to the Fermi energy domain, allowing to infer information on the centrality (and the impact parameter) from the inclusive distribution of an experimentally measured observable. The method was adapted to INDRA-VAMOS, using the inclusive distribution of total transverse kinetic energy, and tested within the AMD-CC model, proving its relevance over the whole impact parameter domain.

The isospin diffusion phenomenon was investigated by means of the isospin transport ratio computed from the asymmetry δ = (N -Z)/A of both PLF and reconstructed QP. For the first time, we have highlighted that the employed impact parameter reconstruction method allows a more direct comparison to the experimental data for the study of isospin diffusion. Furthermore, our results show that the weak sensitivity of the isospin transport ratio to the stiffness of the employed symmetry energy, expected for the QP predicted by the model (before sequential decays), holds for both the PLF and the reconstructed QP. Nonetheless, a noticable disagreement is observed in the centrality dependence of the ratios between the experimental data and the model, more specifically for the 40 Ca+ 48 Ca mixed system. Such difference can be anticipated from the individual evolution of δ with centrality obtained from the PLF, that the AMD model followed by GEMINI++ fails to reproduce for all reactions. Interestingly, the reconstruction of the QP indicates a better agreement between the model and the experimental data for the neutron rich 48 Ca reactions. The evolution of the average neutron excess obtained from the filtered QP also indicates that, even if the clusters are considered, the 48 Ca reactions are mostly driven by the Evaporative Attractor Line as compared to the 40 Ca projectile reactions.

The results presented in this work constitute a further step to improve the comparison protocol employed in the study of isospin transport and of the nuclear EoS. The subsequent phase involves implementing the suggested protocol across a variety of dynamical models, including both QMD-like and BUU-like formalisms, for which we can anticipate that variations in the Mean-Field implementation will result in distinct behaviors in the isospin transport ratio. This work is currently in progress.

The right hand side of Eq.A1 is obtained by introducing the centrality c b , defined as the cumulative distribution function of P (b):

c b ≡ b 0 P (b )db (A2)
with P (c b ) = 1 ∀c b . As described in [46,47], Eq.A1 can be used to determine P (X|c b ) by fitting the experimentally measured inclusive distribution P (X), with a suitable probability density function (p.d.f) which encodes both the centrality dependence of the mean value X(c b ) and the fluctuation of X about this mean value.

Once P (X|c b ) is obtained by fitting the experimental P (X) distribution, the centrality distribution of any experimental generic sample S, P (c b |S), can be deduced from the Bayes' theorem:

P (c b |S) = P (X|c b ) P (X|S) P (X) dX P (X|S)dX (A3)
where P (X|S) is an histogram of X filled with the events of the sample while the integrals are performed over the full domain of X.

Finally, the absolute impact parameter distribution can be deduced from the previously calculated centrality distribution, such as

P (b|S) = P (b)P (c b (b)|S) (A4)
The method described above is model-independent and allows to take into account the fluctuations inherent to the relationship between any experimental observable and the impact parameter. It should nonetheless be noted that it is necessary to assume a specific form for P (b) in order to use Eq.A3 and calculate c b (b), i.e. the relation between c b and b.

Implementation for INDRA-VAMOS data

In the present analysis the impact parameter distributions are reconstructed by using the inclusive distributions in total transverse energy of the LCP, namely P (E t12 ), where E t12 is defined in Eq.1. It is worth noting that the PLF properties are largely independent of this quantity, avoiding possible trivial bias due to autocorrelation with the event sorting for the study of the isospin transport ratio [16]. Also, as it can be seen in Fig. 8, this quantity is relatively well reproduced by the AMD-CC filtered model while we have verified that it presents a monotonic relationship with centrality c b within the model.

Secondly, the data are sampled according to the experimental centrality c Et12 , defined as the complementary cumulative distribution function of the P (E t12 ) distribution:

c Et12 ≡ +∞ Et12 P ( Ẽt12 ) d Ẽt12 (A5)
By construction, c Et12 decreases from 1 to 0 as E t12 goes from its minimum (∼ 0) to its (system-dependent) maximum value. Therefore, large (∼ 1) c Et12 values are associated with the most peripheral collisions while smaller values (c Et12 → 0) indicate smaller average impact parameters. This choice is based on our previous work and aims to remove the system-dependence of the E t12 distributions in the data sampling [21]. Indeed, since bins of fixed width of c Et12 contain the same number of events, we expect them to have the same statistical significance, whatever the centrality. Based on the aforementioned considerations, a fixed step of 5% experimental centrality c Et12 have been used for the data sampling, for a total of 20 bin samples.

Thirdly, we have adopted the same specific implementation as Frankland et. al for the fluctuation kernel and the centrality dependence of E t12 . Concerning the fluctuation kernel, since E t12 is positive the gammadistribution is an ideal choice [46]. Accordingly, the p.d.f used to fit the inclusive distribution reads:

P (X|c b ) = 1 Γ(k)θ k X k-1 e -X/θ (A6)
where k and θ are two positive parameters related to the mean value X and its variance, respectively. As pointed out in [46], these parameters generally depend on the centrality, while the fit to the inclusive distribution P (X) using Eq.A6 to extract k(c b ) and θ(c b ) is underconstrained. In the framework of the gammadistribution of Eq.A6, the centrality dependence of k must be parametrized, and it is generally assumed that the variance parameter θ is constant for all centralities. Therefore, similarly to [47], in this work k is parametrized with a monotonically decreasing function of centrality, while θ is a free parameter of the fit constrained by the tail of P (X). In summary, 5 free parameters (α, γ, θ, X min , X max ) are required for the fit, which are explicitly defined in [47].

Finally, we would like to stress that if the model calculations were run with a geometric impact parameter distribution, the filtered impact parameter distribution P (b) can not be well-fitted by the usual approximation observed for most of the INDRA dataset (see Eq.12 in [47]). This can be seen in Fig. 2, where we clearly observe the effect of the VAMOS angular cuts that favor the detection of semi-peripheral and peripheral events, even for AMD-CC, in the b/b gr = 0.6 -0.9 region. For this reason, and at the cost of losing the model independence for P (X|S), we have used the inclusive impact parameter distribution directly obtained from the filtered model as a surrogate for P (b) in Eq.A4. Therefore, for more consistency the results on isospin diffusion will be presented using both Eqs.A3 and A4, keeping in mind that the former is model-independent.

Z QP = Z P LF + M I i Z i (B1)
where Z P LF is the charge of the fragment measured (filtered) in VAMOS and the sum runs over the charge Z i of the M I selected particles.

Similarly, the quasiprojectile mass number (without evaporated neutron contribution) ÃQP reads

ÃQP = A P LF + M I i A i (B2)
where A P LF is the mass of the fragment in VAMOS and the sum runs over the mass A i of the M I selected particles. As the neutrons were not measured in the INDRA-VAMOS experiment, the distributions of the neutrons evaporated by the reconstructed QP from the AMD filtered model calculations were used as a substitute. More precisely, for each event with reconstructed charge Z QP and mass without the neutron ÃQP , the evaporated neutron multiplicity was estimated from a random number generator following the filtered model neutron multiplicity distribution (histogram).

The estimated evaporated neutron multiplicity is thus the following:

M n (Z QP , ÃQP ) = M rdm n (Z QP , ÃQP ) • k (B3)
where M rdm n is the random neutron multiplicity extracted from the model histogram, k is a correction factor and is the ceiling function.

The scaling factor k is intended to correct from the fact that both AMD models tends to overestimate the proton multiplicities compared to the experiment, as observed in Sec.III C 3. Thus, assuming that the experimental and filtered model average neutron-to-proton multiplicity ratios are equivalent, we have:

M n exp = M n mod • M p exp M p mod = M n mod • k (B4)
where M n,p exp,mod are the neutron and proton average multiplicities of the experiment and model, respectively. Therefore, k = 1 when applying the evaporated neutron correction within the model, while a constant value of k = 0.8 is applied in the case of the experiment, which corresponds to the average scaling factor over all systems and charges. The resulting experimental average neutron distributions are presented in Fig. 19, for the four systems under study. In this figure, the symbols correspond to the values obtained for k = 0.8, while the dashed areas represent a variation from k = 0.7-0.9, corresponding to a variation of one standard deviation.

1 .

 1 Topology of the reaction products

Figure 1

 1 Figure1depicts the correlations between the charge and the parallel velocity (laboratory frame) for all particles identified in charge in the48 Ca+48 Ca reactions, for the experiment (a) and the AMD-CC model without (b) and with (c) filter and event selection criteria.Similarly to our previous work with the AMD-NC version[26], the unfiltered model exhibits two main components on both sides of the c.m. velocity (black dashed line), with a third region of LCP and IMF spreading over the whole velocity domain. By comparing the unfiltered and filtered simulations, we mainly observe the effect of VAMOS angular acceptance, which drastically reduces the measured yields while the right-most component becomes concentrated in a region of charge and velocity close to the projectile one. Such topology is representative of dissipative binary collisions, where the fragments detected in VAMOS are mostly the products of the quasiprojectile decay (namely the projectile-like fragment, PLF) resulting from peripheral to semi-peripheral collisions, while the products of the quasitarget (targetlike fragment, TLF) decay are occasionally identified in INDRA at backward angles (left-most component). Finally, by comparing the experimental data and the filtered model, a satisfactory agreement is observed. It must be noted that similar topologies are observed independently of the version of the AMD model and for all the systems, with a similar effect of the numerical filter and data selection[26].We present in Fig.2the effect of the filter and event selection criteria on the reduced impact parameter distributions for the 48 Ca+ 48 Ca system. The input triangular distributions, represented as solid lines, present a small difference between the models as AMD-CC was ran up to slightly larger b/b max values than AMD-NC. Interestingly, we observe a strong difference between the two versions of the code once the filter and data selection are applied, AMD-NC (open circles) leading to the detection of more peripheral events compared to AMD-CC (full circles), even if the latter was run up to higher impact parameter values. Such remarkable difference is not trivial as it comes from the combined effect of the model dynamics, the detectors acceptance and the applied offline selections. In a sense, this plot highlights the inherent difficulty of comparing models with experimental data as the subset of simulated data can vary from a model to

Figure 1 :

 1 Figure 1: Atomic number Z as a function of the parallel velocity Vz (in the laboratory frame) for the 48 Ca+ 48 Ca system. Experimental data (a). Secondary fragments from AMD-CC (after GEMINI++ de-excitation) without (b) or with (c) the experimental filter. The black and red dashed vertical lines indicate the reaction c.m. and the projectile velocities, respectively.

Figure 2 :

 2 Figure 2: Reduced impact parameter distributions from filterd AMD-NC (open circles) and AMD-CC (full circles) models with data selection. Solid lines correspond to the input triangular distributions. The plots are normalized to the integral of their respective input distributions.

Figure 3 :

 3 Figure3: Parallel velocity (a-d) and charge (e-h) distributions of the PLF from the experiment (full circles) and filtered AMD-NC and AMD-CC models (blue and solid lines, respectively) . The inner plots represent the PLF from the models when the numerical filter is not applied. The distributions are normalized to their integral.

  filt. AMD-NC pr. AMD-CC filt.AMD-CC pr.

48 Figure 4 :

 484 Figure4: Average neutron excess of the PLF for the experimental data (full circles), filtered AMD-NC and AMD-CC (blue and red continuous curves, respectively). The dasheddotted line is the EAL from[44]. Results from the QP predicted by the model (before GEMINI++) are plotted in thick red and thin blue dashed lines, respectively.

Figure 6 :

 6 Figure6: Average multiplicities of neutrons and Z = 1 nuclei (isotopically identified) for the experiment (full circles) and the filterd AMD-NC (thin blue lines) and AMD-CC (thick red lines) simulations followed by GEMINI++. The neutrons distributions are not filtered.

Figure 7 : 48 Figure 8 :

 7488 Figure 7: Same as Fig. 6 for Z = 2 nuclei.

Figure 9 :

 9 Figure 9: Model calculations: fits (dashed curves) to the inclusive distribution of Et12 from AMD-CC (stiff) filtered model predictions (followed by GEMINI++), for the 40 Ca+ 40 Ca and 48 Ca+ 48 Ca reactions. Each distribution is presented in both (a) linear and (b) logarithmic y-axes. Statistical uncertainties are not shown when smaller than the symbols.

Figure 10 :

 10 Figure 10: Model calculations: impact parameter probability distributions for various sampling of experimental centrality cE t12 , returned by the fit (solid lines) and calculated directly (shaded area), for the same 40 Ca+ 40 Ca AMD-CC (stiff) filtered events as in Fig 9.

Figure 11 :

 11 Figure 11: Model calculations: evolution of the average bcentrality (a) and impact parameter (b) with the applied sampling of 5% in cE t12 , for the AMD-CC (stiff) filtered simulations (followed by GEMINI++). The error bars correspond to the standard deviation.

Figure 12 :

 12 Figure 12: Experimental data: fits (dashed curves) to the inclusive distribution of Et12 for the 40 Ca+ 40 Ca and 48 Ca+ 48 Ca reactions. Each distribution is presented in both (a) linear and (b) logarithmic y-axes. Statistical uncertainties are not shown when smaller than the symbols.

Figure 13 :

 13 Figure 13: Experimental data: evolution of the average bcentrality (a) and impact parameter (b) with the applied sampling of 5% in cE t12 . The error bars correspond to the standard deviation.

Figure 14 :

 14 Figure 14: Model calculations: distribution of the average asymmetry δ as a function of the centrality cE t12 for (a) the PLF and (b) the reconstructed QP, for AMD-CC (stiff) filtered simulations (followed by GEMINI++). The dashed lines correspond to the QP predicted by the filtered model before GEMINI++. Statistical error bars are smaller than the symbols.

Figure 15 :

 15 Figure 15: Model calculations: isospin transport ratio as a function of the estimated average c b for the PLF (a) and the reconstructed QP (b), respectively, from the AMD-CC model with a stiff (full squares) and a soft (open circles) parametrization. Results with the QP predicted by the model are superimposed in dashed lines.

  -Stiff EOS rec. QP -Soft EOS QP -Stiff EOS QP -Soft EOS rec. QP -Stiff EOS rec. QP -Soft EOS QP -Stiff EOS QP -Soft EOS (b)

Figure 16 :

 16 Figure 16: Model calculations: isospin transport ratio as a function of the estimated average reduced impact parameter b /bgr for the PLF (a) and the reconstructed QP (b), respectively, from the AMD-CC model with a stiff (squares) and a soft (circles) parametrization. Error bars are not shown when smaller than the symbols. Results with the QP predicted by the model are shown in dashed lines.

Figure 17 :

 17 Figure 17: Experimental data : distribution of the average asymmetry δ as a function of the centrality cE t12 for (a) the PLF and (b) the reconstructed QP. The dashed lines correspond to (a) the PLF and (b) the reconstructed QP obtained from AMD-CC (stiff) filtered simulations (followed by GEM-INI++). Statistical error bars are smaller than the symbols.

Figure 18 :

 18 Figure 18: Experimental isospin transport ratio computed from the asymmetry δ for the average reduced impact parameter b /bgr estimated from the method described in Sec.IV, for (a) the PLF and (b) the reconstructed QP. Open symbols represent the AMD-CC filtered simulations (followed by GEMINI++) for a stiff (open squares) and a soft (open circles) symmetry energy.

48 Figure 19 :

 4819 Figure 19: Average neutron multiplicities distributions used for the mass reconstruction of the experimental quasiprojectile as a function of its charge. The symbols correspond to the values obtained for k = 0.8 while the dashed areas represent a variation of k = 0.7 -0.9 (roughly one standard deviation).

Table I :

 I Model calculations: results of the fits to the total transverse energy. χ 2 is the reduced chi-square value of each fit.

		System	α	γ	θ	Xmin Xmax χ 2
					[MeV] [MeV] [MeV]
		40 Ca+ 40 Ca 0.10 0.52 8.95	1	265 1.2
	AMD-CC	40 Ca+ 48 Ca 0.26 0.81 9.42	4	197 1.7
	stiff	48 Ca+ 40 Ca 0.12 0.59 8.68	5	251 1.4
		48 Ca+ 48 Ca 0.33 0.93 8.48	8	179 1.1
		40 Ca+ 40 Ca 0.15 0.58 7.12	1	246 1.5
	AMD-CC	40 Ca+ 48 Ca 0.31 0.84 9.65	5	187 1.2
	soft	48 Ca+ 40 Ca 0.15 0.60 8.71	5	241 1.3
		48 Ca+ 48 Ca 0.35 0.96 9.68	10	175 1.1

Table II :

 II for the four systems under study. Experimental data: results of the fits to the total transverse energy. χ 2 is the reduced chi-square value of each fit.

	System	α	γ	θ	Xmin Xmax χ 2
				[MeV] [MeV] [MeV]
	40 Ca+ 40 Ca 0.14 0.69 10.29	4	278 1.3
	40 Ca+ 48 Ca 0.37 0.82 12.90	7	167 1.3
	48 Ca+ 40 Ca 0.17 0.71 9.11	6	257 1.1
	48 Ca+ 48 Ca 0.10 0.55 13.01	1	233 1.4

Acknowledgments

The authors would like to thank: The technical staff of GANIL for their continued support for performing the experiments; The CNRS/IN2P3 Computing Center (Lyon -France) for providing data-processing resources needed for this work. This work was partly supported by the IN2P3-GSI agreement (Grant No. 03-45); The National Research Foundation of Korea (NRF) (Grant No. 2018R1A5A1025563); IBS (Grant No. IBS-R031-D1) in Korea. Q. F. gratefully acknowledges the support from CNRS-IN2P3.

Appendix A: Impact parameter reconstruction 1. Method By definition, the inclusive distribution of an observable X, P (X), resulting from all measured collisions with an unknown impact parameter distribution P (b), is related to the conditional probability distribution of X at fixed b, P (X|b), such as P (X) ≡ Similarly to our previous works, the relative velocities between the reactions products measured (filtered) in INDRA and (i) the PLF detected (filtered) in VA-MOS and (ii) the fragment with the largest identified charge at backward angles (supposed to be the TLF) are calculated. Numerically, cuts on the associated relative velocities, respectively V rel,P LF and V rel,T LF , were applied in order to include fragments whose velocities verify

The values of the cut-off thresholds were estimated from both AMD-NC and AMD-CC filtered simulations, to optimize the contribution of the actual QP to the reconstruction. Interestingly, the values of the cuts are similar for both versions of AMD, while their effect remains consistent for the models and the experiment.

The quasiprojectile atomic number Z QP reads


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Taking care of people, taking care of spaces: vulnerable actors and architectural renovation in the context of urban austerity Federica Gatta (PACTE-UGA), Cécile Léonardi (AE&CC-ENSAG), Alouna Nicolas (Otopo)

1. Austerity and care of public buildings This paper will draw on insights from a research project1 that aims to analyse the effects of "care for repair". Fitz and Krasny understand care for repair as something beyond mere restoration, whose quality is to bring new uses or new functions as reparative for buildings, humans and nonhumans alike (2019: 20). Following Denis and Pontille we argue that is even mostly the idea of maintenance that becomes central in our cities today: "This 'ecocentric' ethic can inspire an ambitious and desirable policy of maintenance. It invites us to develop ways of caring for things that are aware of the multiplicity of material interdependencies, and that cultivate an awareness of the fragility of environments and the relationships that develop within them." (Denis & Pontille, 2022: 219).

We would like, first of all, to replace the maintenance in the context of austerity urbanism. This is to be understood as a reduction in public budgets in favour of new cycles of innovation in the fields of outsourcing and privatisation of urban public services [START_REF] Peck | Austerity urbanism", City : Analysis of urban trends, culture[END_REF], which becomes a routine policy rather than a response to the urgency of the 2008 economic crisis. The spatial effects of austerity urbanism can vary due to the interaction between the redistributive capacities of state policies and the local policy capacities [START_REF] Addison | Four Types of Urban Austerity. Public Land Privatizations in French and Italian Cities[END_REF]. Among those effects, the alienation and optimization of municipal real estate is becoming more and more important in cities' policies and it is leading to new forms of regulatory documents and planning devices. The repair and maintenance of existing buildings is in fact a key issue of the budget reductions as it concerns not only the structural and internal security expenses but also, more and more, their energy efficiency. In an austerity context, municipalities are realising that the burden of hundreds or thousands of square metres not only need to be repaired, but ask for a more intelligent maintenance, when they do not simply become a source of income through selling. In this sense, the transformation of those municipal assets can lead to ambiguous policies: some, clearly neoliberal, oriented to alienation; others, more or less alternative, inventing new forms of regeneration and governance. Public buildings are then a good lens to see how the imposition of austerity measures in cities opens up a discussion on care for both people and spaces. In some cities, the management of public buildings is actually allowing new forms of partnership between institutions and citizens in the co-creation of public services. Through an ethnographic approach, our research attempts to understand which factors enable these alternative policies to facilitate non-capitalist economies (Gibson-Graham, 2016) and commoning practices. The hypothesis is that caring attitudes in political and architectural spheres are one of the main conditions for this emergence. The research is observing closely the tangled histories of buildings and communities in some French, Italian and Spanish cities with municipalist traditions (in particular Grenoble, Nantes, Bologna, Turin and Barcelona).

In this paper we will focus in particular on one case study in Grenoble. It concerns a project designed to accommodate care practices, combining housing and activities, in a 19th-century villa that first has to be refurbished. Like five other municipal buildings, this villa was the subject of a call for innovative urban projects in 2017 [START_REF] Gatta | De nouvelles formes de partenariat public-privé ? Un appel à projet innovant à Grenoble[END_REF]. This call was inspired by calls experimented in Paris since 2014 whose objective is to adapt "public land transfer procedures to the challenges of innovation. Sites are put out to tender [by a local authority], accompanied by a set of 'special conditions' [...] expressing expectations in terms of innovation -constructive, technical, but also in terms of uses. The definition of programs is thus at least partially transferred to the candidate groups" (Gomes, 2020: 5). In Grenoble the call has the ambition to "reawaken" certain under-used buildings in the municipal heritage "by helping Grenoblois rediscover them through new uses" (Gren' de Projets, 2017). While most of these buildings are of undeniable heritage interest, their main need is for maintenance and renovation. The call is linked to a Plan de sauvegarde des services publics, created by the ecologist municipality in 2014 in order to react to austerity conditions. One of its goals is to optimise the costs of managing the city's extensive property holdings and to make the existing stock more energy-efficient, by renovating the busiest facilities and thinking about pooling the occupancy of those that are less busy (city technician, November 2020). At the same time the municipality wants to be sure that the projects respect the general interest, the openness to the public and the patrimonial value of the buildings. As a result, winners or shortlisted, the five projects finally selected for the call are not backed by any private developer and just one of them (the smaller) has been alienated. Most of the project leaders are coalitions of small business owners or of local associations that propose cultural and/or social service activities.

Those groups have obtained long-term leases and their renovation projects are accompanied by the city which ensures that projects respect the criteria of the call. The building we are interested in here 2 , has been the subject of a project by a collective of thirdsector players associated with a social landlord. The project involves the cohabitation of both living and working spaces for precarious people and various shared spaces designed to facilitate encounters between precarious residents and people from the outside. We are dealing here with an atypical living space that will have to protect the privacy of its occupants, facilitate the work of the caretakers who will manage it, and at the same time lend itself to uses that are open and programmatically uncertain. The analysis of this case study will allow us to observe an ethic of care in the making. Our observations show how this ethic is the fruit of a process of negotiation and dialogue within the project process between different agents. By asking "how it all [will] work in everydaylife" [START_REF] Denis | Le soin des choses : politiques de la maintenance[END_REF], we are looking at what is often invisible: the process of designing the maintenance of a place and its commoning conditions.

Agents, caretakers and caresupports

In this second part we will outline the portrait of four agents involved in the project in order to underline the forms of interdependence and care which characterise them: the villa itself, the collective and users, the architects in charge of refurbishing the site and the contract signed with

2 The Villa Clément case study is the object of an ethnographic survey involving interviews with members of the collective and the architects in charge of the rehabilitation project, as well as participant observations at collective meetings over one year in 2020-2021.

the City. Afterwards we will see how these actors negotiate the project and its ethics through an example of a meeting about the accessibility of the space.

The villa

This 19th-century bourgeois residence has a distinctive architectural style, making it a key part of Grenoble's heritage especially for its monumental woodwork staircase, 19th-century cement tiles and stained glass windows on the veranda. Situated on the right bank of the Isère river, to the east of the Bastille foothills, this streetside building complex occupies the corner of a wooded plot of approximately 4 255 m2. The vast park is home to a number of remarkable tree species and has preserved its early 20th-century garden features, with pathways and rockery borders. Villa Clément was built at the end of the 19th century. At the time, it comprised eleven rooms and benefited from its current ornamental park. An extension was added to the west at the beginning of the 20th century. The villa is a building that has already undergone numerous transformations over the course of its history, and has suffered from twenty years of under-occupation. Previously used for residential purposes, it became, after the war, the site of municipal activities (a leisure centre) and then a centre for disabled children, before being abandoned and left unused, it is now partly occupied by a regional association for employment training. As mentioned by the architects, in the 70s the villa was already heavily transformed in order to fit the accessibility norms of the time. Today the building is not in a bad situation, but seepage into the walls of certain garden-floor rooms has weakened the floors in this part of the building (architect, October 2019). Within the call for innovative urban projects, the villa was one of the buildings with the most identified (but also damaged) heritage value. It is one of the first "vulnerables" in this project.

The collective and the users

The villa will host: 9 boarding houses, a flat for short stays for single parents in precarious situations, workshops for professional reintegration, a kennel for homeless and precarious people, social integration projects, a shop selling refurbished bicycles and a bicycle repair workshop, a beehive, some work and exhibition spaces. The beneficiaries of the project will be: single, ageing, homeless people; single parents for parental care; young people far from employment; dogs; integration employees and customers of the bicycle workshop; local associations and other city inhabitants. The project therefore aims to accommodate people who live, work and provide services, as well as more occasional users: customers and work space tenants. This raises issues of cohabitation: "Unlike most boarding houses, it proposes that the collective facilities be shared with other associations, the public and partners. Because of this particularity, which enables a very 'civic openness' but can also call into question the residents' appropriation of the common areas, from the outset of the project, we wanted to involve boarding house residents in the project process." (Response for the call, 2018) The Villa Clément collective is made up of a wide range of players: future users of the site (both permanent and temporary), local associations, members of the associations involved in the project, coordinators as well as the social landlord (Grenoble Habitat). The RSA38 association 3 , is a key element in the project, representing knowledge on the history of the associative sector in Grenoble guaranteeing also the history of the site. The aim of the associations involved in the project is to relay the needs of the precarious public, such as the person who is in charge of the kennel project and who seeks to respond to all types of needs encountered by people who own a pet, especially those who are homeless. The member of the RSA38 association points out that the project is aimed at all types of precariousness:

"We have social integration, RSA recipients, and the street. If we can demonstrate, on the scale of a single site, that user expertise can give shape to programs adapted to these groups and be sustainable over time, we'll have won!" (October 2019). The collective therefore includes associations specialised in housing of precarious populations (the Relais Ozanam and Oiseau Bleu), as well as in professional reintegration and peer workers (Mutualité française and Cycle and go). What's new about the project is the early collaboration of these several pioneering structures in the fields of social integration and care, enabling them to make the most of their expertise in bringing together activities that have their own specificities. Moreover, the idea for this collaboration originally came from Grenoble Habitat, the social landlord, who wanted to involve the members of the collective closely in the process of building the project, as well as the architects commissioned, who have specific expertise in the repair and preservation of heritage buildings.

The architects

PNG is an architecture workshop created in 2007 by three architects, all graduated from Grenoble architecture school. One of them has a specific diploma in architectural restoration. They work between two sites: in Paris and in Voiron, a rural area nearby Grenoble. They have been chosen by Grenoble Habitat for the heritage expertise necessary for this kind of historical renovation but also for their youth and open-mindedness. On one hand, they are called to defend the attention paid on the site and its fragility:

"We were approached by Grenoble Habitat because we're architects and we've been trained as heritage architects since 2015, and we were interested in 'what already exists' in the architectural process. We started out working on projects like this, more or less by chance, and we found a lot of interest in trying to do with what was already there, to bring things back to life." (November 2020) But they defend also the fact that the uses planned by the collective will be just a part of all the past and future uses that this space will have to support:

"[There are some requests] we were never able to translate into architecture because it had far too great an impact on the villa. And for a unique and particular use that is not even guaranteed to last. [...] So there's a real trade-off to be made here." (November 2020). On the other hand, they also showed interest in the participatory process, which was a novelty for them, and sometimes a challenge.

The contract

After the call, the projects have been followed by a city technician specialised in project management. This technician represents the presence of the institution that provides support, not financial, but in terms of management engineering, helping to find support and realise the project business plan. The technician also guarantees compliance with the innovation criteria required by the call, firstly the respect of patrimonial value of the site and the openness of the activities. This task takes shape both in regular meetings, but also, and most of all, on the realisation of a specific contract:

"We're doing it for the first time. We adapt the tools. [...] Through this partnership agreement it was a give-and-take. We each have rights and duties [...] We're bound by them, we've chosen them, we've pampered them, we've brought them to what they are today. [...] In the deliberations of the jury that selected the projects, it is written that the City would formalise the partnership through a partnership agreement that recognizes the fact that the City has the right to negotiate by saying, for example, as you have seen: [...] we're going to work to a certain timetable; we're going to optimise the architectural project; we're going to look at the costs. [...] In fact, there were some safeguard measures." (July 2020). Even if the technician is absent from the internal meetings of the collective, the contract with the City is the foundation of the duty of taking care of the space and also not privatising it, which can sometimes be a source of contradictory positions.

Public heritage and caresupports

In order to understand those contradictions, this section aims to show how these agents had to negotiate in order to reach agreement on the building project. The architectural planning of the site often faces disagreements or requires regular arbitration, given the multiplicity of planned uses and the villa's heritage constraints. We're going to focus in particular on a scene that played out during a meeting on 17th of July 2020 that brought together the collective, the architects and representatives of some of the associations involved (RSA38, the solidarity kennel, a neighbourhood association, the Mutualité française et the Relais Ozanam). The meeting deals with the most complex issue: managing access between permanent users of the villa and visitors. It starts in the morning only with the collective talking about the governance of the association. Following a meeting that took place one month before, they discuss again of the articulation of different levels of governance: the association's office, some thematic divisions linked to the different functions (living, farming, programming, sharing), a house council for everyday management (including users). Moreover, they agree on the fact that a householder is needed in order to take care of the everyday life of the place: managing stocks, taking out garbage, mediation between users, secretary and communication. At the same time the people representing the inhabitants of the district and the volunteers express their fear that this organisation is going to be too rigid and that it will exclude those who come from the outside. They agree then on the idea of "guaranteeing the void" i.e. leaving some uncertainty for making space for the unplanned. Then the architects join the meeting. The purpose is to discuss with them the architectural plans in order to fix the uses assigned to each space. Three issues appear to be problematic in particular: the shared kitchen, the ground floor office and the doors opening system. Each of them has to deal with the conflict between two kinds of necessities: on one side, letting the different users meet and protect them; and on the other respecting the rules of a public-access building (which is a quite ambitious choice of the collective) and the needs of an ancient building. The architects bring technical plans without furniture, only a A3 print per floor. They maybe do not have thought to this moment as needing some more comfortable mediums of discussion. This obliges everybody to get close and to take time to interpret the plan. Talking about the kitchen, one of the members representing the boarding house explains that, even if people have kitchens in their apartments, the collective kitchen is needed for regular common meals that allow the cohesion and prevent solitude of people. They are going to be 10 on those occasions. But one architect says that the kitchen is designed for 5 people. People try to find solutions to enlarge it, but the architects remember that the plans approved by the Municipality concern a "familial kitchen". This creates some tension, a woman says: "you should come to see one of our boarding houses to see how important the kitchen is!". This by implying the architects are not caring enough about the vulnerable people that are going to live there and their specific needs:

"The boarding house will accommodate people who will be living there for many years, some of whom will pass away within the walls of the villa. The collective kitchen is going to be their kitchen. I think we need to be careful about that, we need to take care of this place!" (collective member, July 2020) The architects then explain that, even if the kitchen space and furniture is not very big, the "A room"4 , just close, can be used to extend the kitchen space with an opening wall. This seems to be enough to satisfy and talk about other subjects. The collective proposes to talk about the workshops, where there is again a discussion on some big incompatibilities between the need of workers in terms of spaces, accessibility and comfort, and the constraint of the space, in particular the facade. No solution is found even if all the actors try to find alternatives from their point of view, so the meeting moves to the next point: the ground floor office. This office is needed for the boarding house, but it must also become the place of work of the householder which is now very important for the project. The architects are surprised, they designed only a small space of 5 square metres with a small window:

"In terms of fire regulations, C room must have a second access to the outside to provide an emergency exit that's why the space we had allocated to the office must remain throughaccess." (architect, July 2020) The collective explains the idea of the householder and the security needs of the boarding house office. It is actually something new for the architects, but they try to find a solution proposing that the "C room" becomes then the office as it was in any case though as a workspace. After some discussion everybody seems satisfied with this idea which seems to give some flexibility to the future uses. The atmosphere is now getting a bit more relaxed. But it becomes soon a bit tense again when it comes to the discussion of space accessibility. Through the debate it emerges that the fact of keeping the original doors for patrimonial reasons, plus the fire security norms for a public-access building, creates a series of difficulties.

"When you close off a room in the part of the villa that will be a public reception area ." (architect, July 2020) In fact, the ground floor is going to be at the same time the entrance hall for the boarding houses, a workspace, a collective kitchen, a space for events, the single parents house. The future boarding house responsible remembers the importance of not giving access to all the spaces to the residents that can sometimes be disruptive or even "paranoiac": "I had a case in a boarding house where a resident repainted an entire corridor!". She tries to propose programmable entry passes saying they work very well in other boarding houses she works in. But the architects react with some rigidity by underlying that the other places she knows are new buildings, while this is an ancient one. This is why they asked for an organisation chart in order to decide who is going to have keys of which door. Even if people are frowning eyebrows, in the end everybody agrees on the idea that the householder will help in getting this organisation more easily. We could ask ourselves if the householder will be able to "guarantee the void" within all these rules and limitations due to the complexity of mixed uses.

Conclusions

Apparently, this situation is showing a lack of care as it is reproducing an opposition between relational and spatial issues: "The point is not that contemporary architects and planners are all uncaring; the point is that they are caring wrongly. They are caring about things, and, often, about the wrong things. [...] The problem is that the pay check money itself is not care; it needs to be transformed into clean clothing, food, a safe and pleasant place to live. Doing so requires participating in the ongoing relations of those who are cared for. Buildings protect people from the elements. But by themselves, they do not provide care; what happens within the buildings, how the building fits within its location and context, how it was built, who it will house or displace, all of these aspects vitally affect the nature of the caring that the building does." (Tronto, 2019 : 27) But what we observed here is actually only a part of the construction of a dialogue which engages all the agents in the compatibility of future uses and the interdependencies of people relations. The architects are forced here to participate in the relations of those who are cared for and each participant becomes a care professional who emphasises his expertise in caring. Some are attentive to ensuring that the vulnerable members of the public in their care are well looked after, while others are attentive to ensuring that the doors they open and close all day long are well taken care of. By confronting these two gestures of attention, architects and social workers have taken the time to confront together what repair tends to underestimate, or even invisibilize: maintenance and its relational dimension. What [START_REF] Denis | Le soin des choses : politiques de la maintenance[END_REF] call:

"the role that humans allow themselves to play in the life of things" (21) and that obliges us to "concern ourselves with what is not an event, but on the contrary (...) the small, unimportant gestures that neither break nor restore the order of things, and yet play a full part in their existence and are even vital to the stability of the relationships that humans maintain with most of them." (52) Denis and Pontille, following Hilary Sample, point out that maintenance relies on the incessant and often invisibilised agitation of caretakers who take charge of a variety of minute operations. Placed end to end, these operations "weave the fabric of an alternative time which, although it is the condition of possibility, seems foreign to the time of the architects and that of the inhabitants" (Ibid: 53). In the case study of the Villa Clément, we can see how architects and social workers together have come to reconsider this alternative time and reintroduce it into the heart of a process that too often excludes it: that of the project that precedes the actual maintenance and putting into use of a building. This attention made to the maintenance brings to light the role of both spatial and relational caresupports in enabling an ethic of care to be discussed and negotiated. We argue that this very small case allows us to make the hypothesis that today regaining control of municipal assets is not simply a matter of space optimization but a matter of finding a way to ensure that the general interest corresponds to the increasingly transversal needs of people and their living environments. At the same time the austerity context in which these new needs emerge raises the question of the neoliberal logics which are surrounding the project: which responsibility is given to private actors to take in charge the general interest, especially in economic terms? Behind the contractualisation process, which is the role that the public administration is taking in the daily care of the place and its users?

  Taking care of people. Taking care of spaces: vulnerable actors and architectural renovation in the context of urban austerity Federica Gatta, Cécile Léonardi, Alouna Nicolas To cite this version: Federica Gatta, Cécile Léonardi, Alouna Nicolas. Taking care of people. Taking care of spaces: vulnerable actors and architectural renovation in the context of urban austerity. Housing is caring. Housing through gender studies -Habiter le care. Le logement et ses abords au défi des gender studies, UCLouvain/Saint-Louis (CTRL+H -Uses & Spaces -Loci/Lab); Angela.D asbl; ULB (La Cambre Horta -Hortence), Nov 2023, Bruxelles, Belgium. hal-04318736

  [which has specific standards], you have to empty it completely and close it off over its entire surface area. If you open up part of this space, you have to open it up completely. [...] That's the problem with the villa. Here, we decided to keep the original doors for heritage reasons and the doors do not open in the good direction [usually recommended for fire and public access standards]

CARE Commoning processes in Austerity urbanism: spatial REgeneration and governance (ANR-22-CE55-0002),

[2023][2024][2025][2026], founded by the National Research Agency (https://www.pacte-grenoble.fr/programmes/care), coordinated by Federica Gatta. This research has been preceded by the research Grinnurb Grenoble face à l'innovation urbaine, 2019-2021, founded by the Idex Grenoble Alpes and coordinated by Federica Gatta.

The Revenu de solidarité active (RSA) is a French social welfare benefit providing a minimum income to people with few resources. 38 is the code of the department Isère in which Grenoble is situated.

In the architect's plans, ground floor rooms are named A, B and C.