A Méry

V Kumar

X Fléchard

B Gervais

S Guillous

M Lalande

J Rangama

W Wolff

A Cassimi

Coulomb explosion imaging of carbon monoxide dimers

We report on experimental results obtained from collisions of slow highly charged Ar 9+ ions with a carbon monoxide dimer (CO)2 target. A COLTRIMS setup and a Coulomb explosion imaging approach are used to reconstruct the structure of the CO dimers. The three dimensional structure is deduced from the 2-body and 3-body dissociation channels from which both the intermolecular bond length and the relative orientation of the two molecules are determined. For the 3-body channels, the experimental data are interpreted with the help of a classical model in which the trajectories of the three emitted fragments are numerically integrated. We measured the equilibrium intermolecular distance to be Re = 4.2 Å. The orientation of both CO molecules with respect to the dimer axis is found to be quasi-isotropic due to the large vibrational temperature of the gas jet.

I. INTRODUCTION

Carbon monoxide dimers (CO) 2 have been studied since the 90 s using infrared and millimeter-wave spectroscopy [1,2,3]. However, the interpretation of the measured rovibrational spectra has been puzzling for many years because of the possible contribution of two distinct isomers of the dimer with very little binding energy difference. The lowest in energy has been assigned to a planar C-bonded geometry in which the carbon atoms of each CO molecule are located in the inner region of the dimer. In contrast, the other isomer would correspond to a planar O-bonded configuration with two inner oxygen atoms. Moreover, these two isomers have been found to have different equilibrium intermolecular distance R e . The corresponding three-dimensional structures of the dimer can be described using four independent parameters: the intermolecular bond length R e , the relative orientation θ 1 and θ 2 of each molecule with respect to the dimer axis and the torsional angle Φ (see Fig. 1). Table I shows a summarized comparison of the geometry of the C-bonded and O-bonded structures.

Conjointly, theoretical efforts were dedicated to the detailed investigation of the potential energy surface of the neutral CO dimer [4,5,6]. Han et al. [4] reported about the possible existence of T-shaped structures (corresponding to θ 1 ≈ 0 • and θ 2 ≈ 90 • ) as well as slipped anti-parallel structures. The likely presence of planar anti-parallel geometries has been confirmed more recently through extensive efforts in the determination of the potential energy surface (PES) of CO dimers [5,6]. The PES shows a global minimum corresponding to the Cbonded geometry and a local minimum corresponding to the O-bonded geometry (see table I). These two stable structures differ both by their relative orientation θ 1 and θ 2 but also by their intermolecular distance R e and thus may be identified thanks to two distinct experimental observables. The energies of these two minima have been calculated to be about -135 cm -1 and -120 cm -1 respectively. The two potential wells are separated FIG. 1. Definition of the dimer geometry using internal coordinates. The CO molecular bond length is fixed to RCO = 1.13 Å. Upper panel: complete three-dimensional geometry (θ1, θ2, Φ, Re). Lower panel: Partial geometry accessible through the 2-body and 3-body dissociation of the dimer (θ, Re). The non-dissociating CO + molecular ion is considered as a point-like particle located at its center of mass and the measurement is not sensitive to the Φ coordinate (see details in the text).

by low barriers following a dis-rotatory path along the X = θ1+θ2 2 coordinate while passing through some intermediate parallel conformations. The low barrier (about 18 cm -1 ≈ 2 meV) easily enables vibrational excitation along this dis-rotatory coordinate [6] even for relatively low excitation energies of the dimer.

The Coulomb explosion imaging (CEI) technique is known to be a powerful tool to investigate the fragmentation dynamics of covalent molecules [7,8,9,10]. The method has also recently allowed the experimental determination of the structure of van der Waals complexes such as rare gas trimers [11] but also mixed ArN 2 [12] or ArCO [13] dimers. The CEI technique is particularly well suited for van der Waals clusters where large intermolecular distances lead to dissociation following a pure Coulomb potential energy curve [14]. This experimental method also requires a rapid ionization of the target molecules in such a way that the nuclear motion can be considered as frozen during the ionization process.

The CEI technique gives access to several experimental observables, such as the kinetic energy released (KER) in the fragmentation and the relative angles of emission of the fragments, which provide unique information on the 3D molecular structure. It is worth noting that the direct measurement of the complete structure of a carbon monoxide dimer using the CEI technique would require the Coulomb explosion of the two constitutive CO molecules and the coincident detection of the four emitted atomic ions. Although, the cross sections for creation of 4-fold ionized dimers leading to such 4-body fragmentation channels are very small in the present collision system. Moreover, the data analysis procedure is complex because of the strong ambiguity for pairing two sets of carbon and oxygen atomic ions coming from two identical molecules. Thus, only the 2-body and 3-body dissociation channels will be presented in the forthcoming discussion. As described later, the 2-body channels (where only intermolecular fragmentation occurs) give a direct measurement of the intermolecular distance while the molecular orientation can be deduced from the 3-body channels. In the following we will refer to the molecular orientation using a single angle θ because the fragmentation in three bodies does not enable measuring both θ 1 and θ 2 and neither the torsional angle Φ (Fig. 1, lower panel). Note that the θ angle is defined with respect to the van der Waals bond and the oxygen atom. As a result, the O-bonded structure corresponds to θ = 45 • while the C-bonded structure is associated to θ = 135 • .

In the present study, multiply charged dimers are produced following multiple electron capture by slow highly charged ions on neutral dimer targets. The velocity of the Ar 9+ projectiles is about 8 Å/fs (0.37 atomic units) and the geometrical size of the dimer target is of the order of 4.5 Å. For such projectile ions, electron capture is by far the dominant process and occurs in the femtosecond timescale. The subsequent concerted fragmentation of the ionized target also occurs on the femtosecond timescale and ensures the validity of the axial recoil approximation on which relies our Coulomb explosion approach. We focus here on collisions where at least one electron is captured on each molecular site of the CO dimer thus leading to the cleavage of the intermolecular van der Waals bond. Covalent bond breakup also occurs when two or more electrons are captured on one of the CO molecules. The relevant fragmentation channels can be classified into 2-body or 3-body channels depending on either bound or dissociative states of the multi-ionized CO molecule are formed during the collision process. The following 2-body channels have been experimentally identified and are referred to as (m,n) where m and n are the charge states of the emitted CO molecular ions:

(1,1) : (CO) 2+ 2

→ CO + + CO + (2,1) : (CO) 3+ 2 → CO 2+ + CO +
The simultaneous break-up of both the van der Waals bond and of one covalent bond leads to the following 3body channels:

(1,[1,1]) : (CO) 3+ 2 → CO + + C + + O + (1,[2,1]) : (CO) 4+ 2 → CO + + C 2+ + O + (1,[1,2]) : (CO) 4+ 2 → CO + + C + + O 2+ (1,[2,2]) : (CO) 5+ 2 → CO + + C 2+ + O 2+
These channels are referred to as (m,[p,q]) where m=1 is the charge state of the non-dissociating CO + cation and where p and q are respectively the charge states of the carbon and oxygen atomic ions. The simultaneous measurement of several 2-body and 3-body fragmentation channels allows an unambiguous determination of the R e and θ coordinates of the CO dimer from the present data set.

II. EXPERIMENT

The experiment uses the COLd Target Recoil Ion Momentum Spectroscopy technique to perform the coincident detection of the positively charged fragments emitted from collisions of CO dimers with Ar 9+ projectile ions. Both target and projectile beams cross at right angle at the center of the spectrometer. A homogeneous perpendicular electric field of about 40 V/cm is oriented along the vertical X axis and allows the collection of the cations resulting from fragmentation on a micro-channel plate detector coupled to delay line anodes. The 4π collection efficiency is restricted by the 80 mm diameter of the recoil ion detector located 100 mm from the collision point resulting in a limit of 16 eV (resp. 32 eV) for the detection of singly (resp. doubly) charged ions. Before reaching the detector the ions are further postaccelerated to about 3 keV per charge to ensure an optimal detection efficiency of the MCP. Finally, the three dimensional coordinates of the momentum vector of each fragment are computed from their time-of-flight and position on the detector. The highly charged Ar 9+ projectile ions are extracted from an electron cyclotron resonant ion source at a kinetic energy of 135 keV on the ARIBE/GANIL beam line. The ion beam then passes through a 600 µm collimation aperture before colliding with the gas jet. At the exit of the spectrometer, the final charge state of the projectile ions is analyzed thanks to a parallel plates deflector before reaching a dedicated position sensitive detector. Argon projectiles with charge state ranging from 5+ to 8+ give the start for the time-offlight measurement while the intense primary Ar 9+ beam is deflected out of the detector. No selection is imposed on the final charge state of the projectile ion because it shows no influence on the fragmentation dynamics of the ionized target.

The neutral CO dimer targets are produced from a supersonic gas jet with a stagnation pressure of 15 bars, expending through a 30 µm nozzle. A skimmer of 0.5 mm diameter is located 6 mm away from the nozzle and a collimator of 0.5 mm diameter is placed before the collision chamber. The resulting diameter of the gas jet at the collision point is about 1.5 mm. The translational and rotational temperatures of the gas jet are estimated to be below 10 K. However, the vibrational degrees of freedom are only partially cooled down in the supersonic expansion and the vibrational temperature of the jet is expected to be close to the temperature of the nozzle which is here 300 K [15,16]. The proportion of CO dimers within the gas jet is estimated to be about 1 % of the dominant monomer targets and the yield of larger cluster (CO) n≥3 targets is at least one order of magnitude lower than the dimers. As a consequence, an accurate data filtering method is mandatory to extract the relatively small portion of events originating from the relevant dimer break-ups. For this purpose, the coincident measurement of all emitted cationic fragments and the restriction imposed by the momentum conservation law enable to efficiently select the desired fragmentation channels.

In the framework of the Coulomb imaging technique, the intermolecular bond length R e is derived from the KER measurement of the 2-body dissociation channels. Therefore, special attention has been devoted to the absolute momentum calibration of the spectrometer. This calibration requires both time-of-flight and position calibrations along the two transverse directions Y and Z of the spectrometer. The time-of-flight (TOF) calibration is performed by using the centroids of several well identified peaks in the TOF spectrum resulting from the detection of different atomic or molecular ions and allows a precise determination of the spectrometer characteristics (flight length and electric field strength). Additionally, the measured TOF are further corrected to first order from the short flight time of the recoil ions in the post acceleration region. The position calibration along the Y and Z directions of the delay line anodes was adjusted to get the optimal KER resolution, which is only achieved when the three components of each fragment momentum are properly reconstructed. We used the dominant CO 2+ → C + + O + fragmentation of the monomer dication and the reference KER spectrum obtained with a FIG. 2. KER spectra of the monomer dication dissociation CO 2+ → C + + O + . Upper panel: KER spectrum from the present measurement. Lower panel: KER spectrum from electron impact ionization measurement [17] and identification of the corresponding electronic states of the CO 2+ dication [17]. The relative deviation from the reference data is below 1 % over the 7 to 10 eV energy range. Doppler free instrument [17] for tuning the absolute energy calibration of our spectrometer (Fig. 2). Even if our energy resolution does not allow to separate the vibrational levels, the KER peak associated to the 3 Σ + state (KER=7.833 eV [17]) can be used to test the absolute calibration of our apparatus as it contains mainly one vibrational level. Our experimental energy resolution has also been estimated from the width of this KER peak and is found to be ∆E=0.28 eV (FWHM) for a KER of 7.833 eV. Moreover, the contributions of two vibrational levels associated to the 2 1 Σ + state (Fig. 2) have also been deconvoluted and fitted from the high energy peak around 9.5 eV. The results show that our KER calibration is consistent with the reference data from [17] with a precision better than 1 % over the 7 eV to 10 eV range. The resulting uncertainty on the intermolecular distance R e is thus way below 10 % and would allow separating the two predicted C-bonded and O-bonded structures of the CO dimer.

III. EXPERIMENTAL RESULTS

A. 2-body dissociation channels

We first consider the Coulomb explosion of the dimer resulting from a single electron capture on each molecule (CO) 2+ 2 → CO + + CO + . Fig. 3.a.) shows the associated KER spectrum with a single peak centered at KER (1,1) = 3.21 ± 0.04 eV. This value agrees very well with the former experimental measurement by Ding et al. [18] in which the similar dissociation channel of the CO dimers was triggered by multiple ionization with ultrashort intense laser pulses. The presence of a single peak in the KER spectrum indicates no sizable contribution of indirect relaxation processes such as ICD (Intermolecular Coulomb Decay) or RCT (Radiative Charge Transfer). The Coulomb explosion approximation can thus be used to infer the intermolecular distance from the KER spectrum. In this approximation, the equilibrium distance between the centers of mass of the two CO molecules is found to be R

(1,1) e = 4.49 Å. However the CO + ion has a quite large electrostatic dipole moment (µ ≈ 2.75D) and a significant part of the initial potential energy can thus be released as rotational energy. This rotational energy is not accounted for in the KER spectrum deduced from the CO + fragment momenta and leads to an overestimation of the R e value deduced from the pure Coulomb model. The correction of the KER measurement due to the undetected rotational energy will be detailed in section V.B and leads the actual intermolecular distance to be R A small contribution of the (CO) 3+ 2 → CO 2+ + CO + dissociation channel has also been identified (Fig. 3.b). Its low relative intensity results mainly from the low proportion of metastable states (with a lifetime longer than 1µs) of the CO 2+ dication in comparison with the dissociative states leading to the competing 3-body channel (CO) 3+ 2 → CO + + C + + O + . The KER peak is centered at KER (2,1) = 6.37 ± 0.07eV which is two times the KER measured in the CO + + CO + channel as expected within the approximation of the Coulomb explosion. A correction from the rotational energy (see section V.B) leads to an intermolecular distance of R e ∼ = 4.3 Å for the CO 2+ + CO + channel.

The two intermolecular distances inferred from different dissociation channels (R e ∼ = 4.2 Å) seems more likely to correspond to the C-bonded structure predicted by theoretical calculations (see table I). As already discussed in a previous section, this geometry has been calculated to be the global minimum of the PES of the neutral CO dimer and is associated to θ a = θ b = 135 • . The measured KER distributions of Fig. 3 do not show any visible peak corresponding to the intermolecular distance of 3.6 Å expected for the isomeric O-bonded structure. Contrary to previous IR measurements where several subbands were interpreted to arise from two different dimer bond lengths (4.0 and 4.4 Å), we observe here only one intermolecular distance. Our value is close to the calculated distance of the true ground state of the dimer with a C-bonded orientation. Further interpretation of this bond length measurement will be given in the next section especially in regards of the supersonic jet temperature.

B. 3-body dissociation channels

In order to look for a more direct evidence for such a geometrical structure, the 3-body channels will now be investigated. We focus here on several particular 3-body dissociation channels (m=1,[p,q]) of the dimer with the aim to determine the θ coordinate, i.e. the relative orientation of the CO molecules in the dimer. These channels result from a single electron capture on one molecule and a double, triple or quadruple electron capture on the other partner of the dimer. The use of the CEI technique requires that the 3-body break-up occurs simultaneously or at least on a smaller time scale than the relative motion of each fragment ion. Such a specific case will be later referred to as concerted fragmentation. Conversely, the existence of dissociative states of the multi-ionized CO molecule with lifetimes much longer than 10 ps will lead to a so-called sequential fragmentation. Such long-lived states do not exist for multiply charged (CO) n≥3+ cations [19] and sequential fragmentation is thus expected to be observed only in the (1, [1,1]) channel.

1.

The (1, [1,1]) channel: concerted and sequential fragmentation

Since its first observation within a mass spectrometer in 1932 [20], the lifetime of metastable states of CO 2+ has been widely studied both experimentally and theoretically [21][22]. When such metastable states are populated, the late fragmentation of the covalently bond molecule washes most of the information on the initial geometry. In order to identify these events and exclude them from the analysis, we will first study the Dalitz representation of the 3-body fragmentation.

Fig. 4 shows the Dalitz diagram of the 3-body dissociation of the dimer. This two-dimensional graph represents the reduced squared momenta of each fragment ε i = where i is the number of the fragment sorted by ascending order of mass-to-charge ratio. Two main contributions are clearly visible and are assigned respectively to a concerted or a sequential dissociation depending on the lifetime of the transient CO 2+ dication [18,23]. The quasi-horizontal line in the upper part of the graph is associated to the sequential fragmentation. In such a twostep process, the CO 2+ dication breaks up only after the two recoiling molecular ions have reached the asymptotic limit for which their relative Coulomb repulsion vanishes.

As shown in the next section (Fig. 6.b), such a limit is reached after about 10 ps and only metastable states of CO 2+ with lifetimes longer than 10 ps contribute to the sequential dissociation process. The recoiling CO 2+ ion has thus enough time to rotate before fragmenting and completely loses the information on its initial orientation within the dimer. Contrarily, in a concerted fragmentation the CO 2+ dication dissociates within a timescale shorter than 10 fs (see Fig. 6.b). The corresponding events are located close to the bottom of the vertical axis.

Here, the CO + acquires a smaller kinetic energy because of the fast repulsion of the two C + and O + atomic ions following the dissociation of the CO 2+ dication. The exact distribution of these events is directly correlated to the initial structure of the dimer but also to the CO 2+ fragmentation dynamics. Only these concerted fragmentation events contain the desired information about the initial three-dimensional structure of the dimer. This lower part of the Dalitz distribution will later be compared to classical mechanics calculations with the aim to infer the relative molecular orientation in the dimer.

Note that a "V shape" contribution is also visible on the Dalitz plot but with a very small intensity (expending from y D ∼ = -0.2 up to the upper part of the plot). These events are also assigned to a concerted fragmentation and will be discussed in the last section.

The multiply charged (1,[p,q]) channels

If one electron is captured on the first molecule and more than two electrons are captured from the other molecule, the dimer undergoes a concerted 3-body explosion as there are no long-lived states of the CO n+ (n ≥ 3) cations. In the present experiment, events corresponding to n=3 and n=4 have been recorded. The cross sections for multiple ionization decrease with the number of captured electrons and result in low statistics for these highly charged channels. For n=3, two distinct channels compete depending on the charge sharing between the emitted carbon and oxygen ions: (1, [1,2]) and (1, [2,1]). We found that the relative yield of the (1, [1,2]) is approximately five times lower than the dominant (1, [2,1]) channel. This ratio is compatible with previous measurements for monomer targets and results from the lower ionization potentials of the carbon ions [24]. For n=4, the symmetric (1, [2,2]) channel is found to be by far the dominant channel and no significant contribution of the asymmetric channels (1, [1,3]) and (1, [3,1]) has been observed. The preference for symmetric charge sharing in the fragmentation of the CO 4+ ion has also been reported for monomers targets [24] and agrees with our measurement.

The Dalitz plots for the (1, [1,2]), (1,[2,1]) and (1, [2,2]) channels are shown in Fig. 5. For these channels, no se-quential fragmentation is observed but a weak "V shape" contribution can be identified in the (1,[2,1]) channel. The main density distribution stands on the right or left side of the graph depending on the mass-over-charge ratio of each ion. A more detailed description of these graphs will be given when comparing to the simulated data in the next section. 3 where εi(i = 1, 2, 3) are the reduced squared momenta of the fragments, sorted by ascending order of massto-charge ratio. Using this convention, the momentum of the CO + ion is plotted along the vertical axis but the carbon and oxygen ions are plotted the right or left diagonals depending on their respective charge states.

IV. COMPUTATIONAL METHOD

The momentum sharing in the 3-body break-up of the dimer has been investigated using Monte Carlo simulations based on a classical model where each charged fragment is considered as a point-like particle located at its center-of-mass. The complex fragmentation dynamics of the molecular CO n+ (n ≥ 2) has been simply accounted for using the experimental KER spectra. For each simulated event, the classical trajectories of the three emitted ions are then numerically integrated to access the final momentum of each fragment. Despite its simplicity, the resulting model gives a good quantitative description of the dependence of the experimental observables on the geometrical structure of the dimer.

Description of the model

The model assumes a concerted 3-body fragmentation in which both the van der Waals and the covalent bonds break at the same time. The three ions are considered as three point-like particles with zero initial velocity and are initially positioned accounting for the intermolecular distance R e = 4.5 Å, the molecular bond length R CO = 1.13 Å of the neutral CO and the orientation θ of the dissociating CO n+ with respect to the dimer axis (see Fig. 1). Note that the qualitative description of the simulated Dalitz graphs is only weakly sensitive to the bond length and we used here the value R e = 4.5 Å deduced from the pure 2-body Coulomb model (see section III.A). Then, the equations of motion are integrated using a standard Runge-Kutta method and assuming a pure Coulomb repulsion between both atomic C p+ and O q+ ions and the non-dissociating CO + molecular ion. Contrarily, the mutual interaction between the C p+ and O q+ ions is known to be not purely Coulombic as it can be obviously demonstrated by the large KER distributions for these channels. Accounting for the KER distributions by the mean of ab-initio calculations of the potential energy curves would be a tedious task. We thus use a scaled Coulomb energy curve directly inferred from the experimental data. For each simulated event, the total KER is first sampled randomly according to the experimental KER distribution. The 2-body KER for C p+ and O q+ ions is then obtained by subtracting the potential energy due to Coulomb interaction with the remaining CO + molecular ion at a distance R e . The 2body repulsive potential between the C p+ and O q+ ions is then approximated by : V CO (r) = p×q 4π 0

ACO r

where r is the distance between the C p+ and the O q+ ions and where the unit-less parameter A CO is set to obtain the desired KER value. Such a potential curve gives rise to a 2-body KER of p×q 4π 0 ACO RCO = p × q × A CO × 12.74eV for the monomer cation as the bond length is fixed at R CO = 1.13 Å. The parameter is chosen either A CO > 1 or A CO < 1 in order to reproduce the experimental KER spectrum and typical values range from A CO = 0.4 to A CO = 4. The 3-dimensional coupled differential equations of motion are then deduced from the forces acting between each partner:

-→ F 12 = p 4π 0 1 r12 -→ F 13 = q 4π 0 1 r13 -→ F 23 = p•q 4π 0 ACO r23
where the subscript 1, 2 and 3 stand for the CO + , C p+ and O q+ ions respectively.

Finally, the numerical integration interval is set to 1 ns which is a sufficiently long time to ensure that all ions have reached their asymptotic momentum vector limit. Fig. 6 shows three examples of such a 3-body dissociation for the (1,[1,1]) channel, with A CO = 1 but using different initial geometries θ = 90 • , 45 • , 135 • . For all cases, all ions reach their kinetic energy limit after few picoseconds (see Fig. For a qualitative understanding of the 3-body fragmentation, Fig. 7 shows the Dalitz plots associated to the three intended geometrical configurations θ = 90 • , 45 • , 135 • and for KER ranging from 10 to 50 eV (0.39 ≤ A CO ≤ 3.9). First, the figure indicates that, independently of the initial geometry, large kinetic energy released in the dissociation of the molecular CO 2+ dication results in smaller energy of the CO + ions. Indeed, the CO + spends a shorter time in the field of the emitted C + and O + atomic ions and thus acquires less

Convolution with the experimental conditions

To perform a more realistic comparison, the simulation algorithm has been extended in order to include most of the ingredients of the experimental conditions that may impact the data. The main causes responsible for the larger experimental distributions are the vibrational and rotational motion inside the dimer target and, to a smaller extent, the instrumental resolution of the spectrometer. To reproduce as closely as possible the experimental conditions, the 3-body dissociation is now simulated in the electric field of the recoil ion spectrometer and the ions are collected up to the detector. Events are generated using a Monte-Carlo approach. For each dissociation channel, the number of events is taken close to the statistics measured in the experimental data: respectively 2000, 1700, 500 and 500 events for the (1,[1,1]), (1,[1,2]), (1,[2,1]) and (1,[2,2]) channels. For each event, the initial conditions are set randomly and the ions are then propagated up to the detector. In such a way, the simulated data are created as a set of coincident detection of the time-of-flight, Y and Z positions on the detector and are further treated using the same analyzing procedure as for the experimental data, ensuring a fair comparison between experiment and calculation.

Here is a detailed description of the parameters used in the model. First, the initial positions of the dimer were generated using random three dimensional Gaussian distributions with widths (FWHM) ∆x = ∆y = 0.6mm corresponding to the diameter of the ion beam collimator and ∆z = 1.5mm according to the size of the gas jet. The initial velocities of the center of mass of the dimer is dominated by the momentum transferred to the target by the highly charged projectile ion during the collision [25]. These velocity distributions have been inferred from the experimental data and are found to be ∆v x = ∆v y = 450m/s (FWHM) in the transverse plane and about ∆v z = 45m/s (FWHM) in the longitudinal direction. Then, the initial geometry of the dimer is fixed and the initial position of the point-like C + , O + and CO + ions are defined as described in the previous section. The orientation of the dimer with respect to the laboratory frame is then chosen according to an isotropic distribution. With this last assumption, we neglect any angular dependence of the electron capture process leading to these fragmentation channels. Such an angular dependence will however have no influence on the present determination of the geometrical structure of the dimer. To crudely account for the vibration of the molecules inside the dimer, a jitter of ∆R e = 0.9 Å (FWHM) on the intermolecular distance is used as deduced from the width of the KER spectra of the 2-body (CO) 2+ 2 → CO + + CO + channel. Additionally, an arbitrary jitter of ∆θ = 20 • (FWHM) is used around the mean angles θ = 90 • , 45 • , 135 • , 0 • , 180 • . Finally, to reproduce faithfully the experimental KER distribution, we randomly sample the experimental KER spectra associated to each channel and scale the effective repulsion coefficient A CO between the two C p+ and O q+ accordingly.

The coupled motion of each ion is then integrated up to the detector assuming a pure concerted fragmentation and accounting for the homogeneous extraction field E=40 V/m of the spectrometer. Finally, the detector response function has also been included using Gaussian distributions of ∆TOF = 1 ns for the timing resolution and of ∆Y = ∆Z = 0.3 mm for the position resolution.

V. COMPARISON TO EXPERIMENT A. 3-body channels

Simulations have been performed for the four fragmentation channels (1, [1,1]), (1,[2,1]), (1,[1,2]), (1,[2,2]). For each channel, simulations were performed for five initial angular structures of the dimer θ = 90 • , 45 • , 135 • , 0 • , 180 • corresponding to different conformations predicted by theoretical calculations. The resulting Dalitz plots are summarized on Fig. 8 where they are compared to the experimental data. For several channels and initial geometries of the simulation, few events appear as a diffuse background superimposed to the main contribution. These events are due to an incorrect reconstruction during the data analysis procedure and originates from the overlap of the time-of-flight distributions of the C + and O + or the C + and O 2+ partners due to their close massover-charge ratios. A small portion of these events are not efficiently filtered out by the momentum conservation law restriction imposed in the data analysis. They will be ignored in the forthcoming discussion.

The bottom graphs on Fig. 8 represent the projection of the horizontal Dalitz coordinate x D = ε1-ε2 √ 3 and gives the momentum repartition between the two atomic ions C p+ and O q+ . Such one dimensional histograms are more appropriate to perform a quantitative comparison between experiment and calculations, especially when the statistic is low. Fig. 8 will now be used to deduce the three-dimensional structure of the dimer through a detailed comparison to simulated events. Even if the sensitivity to the initial angle θ is limited, the combined information extracted from the four different fragmentation channels enables identifying the initial geometrical structure of the dimer.

-the simulated Dalitz spectra for θ = 90 -the simulated O-bonded structure (θ = 45 • ) does not reproduce the experimental data. It confirms our previous conclusion obtained from the 2-body dissociation (1,1) and (1,2) channels for which the measured dimer bond length does not match with theoretical predictions of the O-bonded structure.

-simulated data corresponding to θ = 0 • and 180 • show very specific Dalitz distributions for all fragmentation channels. The CO + ion acquires a large amount of kinetic energy (events are located in the upper part of the Dalitz diagram) because of the Coulomb repulsion between the CO + ion and the C p+ (θ = 180 • ) or O q+ (θ = 0 • ) ion emitted along the dimer axis. Such events have a very weak contribution in the experimental data, thus excluding a linear geometry of the dimer.

-T-shaped geometries are associated to two distinct θ values (θ ≈ 0/180 • and θ ≈ 90 • ) [4]. They would result in two distinct contributions in the Dalitz plot depending on either the molecule perpendicular or aligned with the dimer axis breaks. The relative amplitude of these two contributions is expected to be about 50 % because the electron capture probability on each molecular site of the dimer does not depend much on its relative orientation with respect to the dimer axis and that no rapid charge transfer between the two molecules is expected before fragmentation. As already discussed, very few events corresponding to θ = 0 or 180 • are observed in the experiment thus excluding a large contribution of T-shaped geometries.

The above detailed analysis of the 3-body fragmentation channels seems to reveal that the CO dimer structure consists mainly of two molecules oriented perpendicularly to the dimer axis (θ = 90 • ). The weak "V shape" observed in the (1, [1,1]) and (1,[2,1]) channels seems to arise from a small contribution of geometries in which the fragmenting molecule is aligned with the dimer axis. However, for θ = 90 • , the simulation does not reproduce the experimental widths of the Dalitz distributions which may indicate that the molecular orientation in the dimer θ has a wider distribution than accounted for in the simulation.

Surprisingly our data show no evidence for the Cbonded or O-bonded structures previously predicted by the calculations and associated to the two isomers observed in infrared spectroscopy measurements [1,2,3]. As discussed in [1], these two isomers are not fully distinct but rather overlapping and are associated to a wide angular extent. Transitions between these two conformations have also been observed in infrared spectroscopy presuming for a wide range of angular orientation ranging from C-bonded to O-bonded structure. Similar behaviors could also be deduced from the calculated PES of the dimer due to the low barrier between the two main minima. Vibrational excitation allows a dis-rotatory motion from the global C-bonded minimum towards the local O-bonded minimum while passing through parallel conformations. However according to the PES calculation, the rotatory motion is supposed to be accompanied by a stretching of the dimer because each angular orientation is associated to a different intermolecular distance.

As already mentioned, the actual vibrational temperature of the gas jet could be as large as 300 K (about 200 cm -1 ≈ 25 meV). The amplitude of the corresponding motion of each CO molecule in the dimer may be very wide [1] and obviously much larger than the arbitrary jitter of ∆θ = 20 • used in the previous simulations. A precise description of this motion and of the corresponding dimer conformations is by far out of the scope of our classical model. For the sake of simplicity, we ran a new set of simulations in the most extreme approximation of a fully random initial orientation of the two molecules with respect to the dimer axis. This approximation includes inherently a mixture of any of the previous initial angles (θ = 90 • , 45 • , 135 • , 0 • , 180 • ) but with a relative weight given by the sine dependance of the solid angle i.e. with a largely favored contribution of angles close to 90 • . The resulting Dalitz plots are shown on Fig. 9 for the four fragmentation channels.

The experimental spectra are very well reproduced within such an approximation of a fully random initial orientation of the two CO molecules. The centroids of the Dalitz projections are similar to the ones obtained previously for θ = 90 • but the widths are significantly increased due to the contribution of other angles. It also confirms that the "V shape" is associated to initial angles close to θ ∼ = 0 or 180 • . The very nice quantitative agreement of the simulated events with our data indicates that, rather than a well defined orientation, the molecules have a quasi-random orientation in the dimer. This observation is partly consistent with the previous results of infrared spectroscopy where the two isomers were also assumed to have a wide angular distribution. Moreover, our supersonic expansion method for the production of CO dimers is similar to the one used in [1] and similar jet temperatures are thus expected. Even if the translational and rotational degrees of freedom are efficiently cooled down in the expansion, the vibrational temperature is large enough to allow highly excited vibrational states and results in a very large angular distribution. Within our experimental resolution, the vibrational motion is so large that it is compatible with a fully random orientation of the molecules inside the dimer.

B. 2-body channels

As shown in section III.A, the dimer intermolecular distance can be deduced from KER spectra of the 2-body (1,1) and (2,1) channels. The pure Coulomb model gives about R e = 4.5 Å for the two channels but the value has to be corrected from the rotational energies of the two emitted molecular ions. During the dimer dissociation, a rotational motion of both ions is induced by the repulsion between the two dipoles of the CO + and/or CO 2+ ions. After dissociation, the resulting final rotational frequency depends on the initial conformation of the dimer (bond length and angular orientation). According to the previous analysis of the 3-body channels, the initial molecular orientation in the dimer is almost random and we used a Monte Carlo approach to average over all geometrical structures.

We thus adapted the previous model and extend it to account for the molecular rotational motion in the 2-body dissociation. The four atomic centers of the dimer are assumed to be point like particles. The partial charge on the carbon and oxygen atoms inside the CO + ions are deduced from the electrostatic dipole moment of the CO + ion. We used a repartition of respectively 90% and 10% on the carbon and oxygen atoms inside the CO + molecular ion. For the (2,1) channel, we assumed the charge to be equally shared between the two atoms as we found no data on the electrostatic dipole moment of the metastable CO 2+ dication in the literature. A pure Coulomb repulsion is used between each atomic ions from two different molecular sites (no intramolecular repulsion) and the binding intramolecular force is described by a restoring force proportional to the displacement along the molecular axis. The spring constant is chosen to reproduce vibrational wave numbers of about 2200 cm -1 . This constant has very little influence on the results as only a negligible part of the initial energy is converted into intramolecular vibrational energy. Finally, the initial structure of the dimer is sampled using a Monte Carlo method by using a dimer bond length of 4.5 Å with a jitter of 0.9 Å (FWHM) and assuming a pure random orientation of the two molecules. This numerical integration of the ion motions is much more time consuming than in our previous model of the 3body dissociation because of the fast vibrational motion of the two molecular ions. As a consequence, the ions are not propagated up to the detector but the calculation is stopped after 50ps where we consider that the Coulomb repulsion between the two molecular ions vanishes. After dissociation, the resulting rotational frequency ranges from 0 to about 2.5 ps -1 depending mainly on the initial orientation of the molecules with respect to the dimer axis. For each event, the total rotational energy of the two ions is calculated as the difference between the initial electrostatic potential energy of the system and the sum of the kinetic energy of the center-of-mass of the two molecular ions. For the (1,1) channel, we found that in average about 0.25 eV of the initial energy is converted into rotational energy. The experimental KER spectra is thus underestimated by 0.25 eV as our KER measurement includes only the energy of the center-of-mass of each CO + molecular ion. Adding this rotational energy to the experimental KER leads to a total energy of 3.46 eV which corresponds to an equilibrium intermolecular distance of the CO dimer of R (1,1) e = 4.17 Å. The un-certainty on the average rotational energy (0.25 eV) is hard to quantify but we estimate that its contribution to the error bars on R e is about 0.2 Å (FWHM). A similar method has been applied to the (2,1) channel but assuming an equal charge sharing between the two atoms in the CO 2+ dication. We found that the rotational motion of the two molecular ions carries an energy of about 0.36 eV for this channel. The corrected total energy is then 6.73 eV and the corresponding bond length is about R

(2,1) e = 4.3 Å but with a large uncertainty due to the lack of information on the CO 2+ dipole moment. As already discussed, the bond length derived from the two channels fits very well with the predicted true ground state of the CO dimer.

According to theoretical calculations, the minimum energy of each angular conformation is reached for different dimer bond length ranging from 3.6 (O-bonded) to 4.3 Å (C-bonded). We thus could expect very large KER distributions in the 2-body channels while the widths of our KER peaks (Fig. 3) do not show any significant contribution of the shortest bond length. Our interpretation of the 2-body and 3-body channels seems then not fully consistent with the shape of the PES of the CO dimer. This apparent contradiction may arise from the production of high excited vibrational states in the dimer gas jet. The high vibration excitation may partially destroy the correlation between angular orientation and intermolecular separation predicted in the calculations as we observe mainly the longest bond length corresponding to the ground state C-bonded conformation.

VI. CONCLUSION

We measured the 2-body and 3-body dissociation of CO dimers following multiple electron capture by slow highly charged Ar 9+ ions. We employed a Coulomb explosion imaging approach based on the axial-recoil approximation to retrieve the initial three dimensional structure of the dimer from the coincident measurement of the momentum vector of each emitted ions. Molecular orientation inside the dimer has been probed using the concerted fragmentation events of four different 3body channels (1,[p,q]) and compared to 3-body simulated events assuming different initial geometries. This experimental method combined with a dedicated classical trajectory model enabled to get a direct and unambiguous determination of the dimer geometry. The structure of the CO dimer is found to be very diffuse and the measured Dalitz plots are found to be compatible with a purely random orientation of each CO molecule in the dimer. This result is somewhat consistent with infrared spectroscopy measurements as it confirms the very wide angular extent of the dimer conformation that may be attributed to a high vibrational temperature of the gas jet. According to the calculated PES, a dis-rotatory path covering both C-bonded and O-bonded geometries as well as intermediate parallel conformations could be responsible for such a large angular distribution at this temperature. The mean intermolecular distance has also been deduced from the KER distribution of the 2-body channels. After correction from the rotational energy acquired during dissociation, we found a bond length of about R e = 4.2 Å. This distance corresponds to the predicted true ground state of the CO dimer with a planar C-bonded geometry. Unexpectedly we found no evidence of the second isomer associated to a much shorter bond length and previously observed in infrared spectroscopy measurements. The relatively high temperature of the gas jet may be responsible for the presence of highly excited vibrational states for which the wave packet is centered around the global minimum of the PES (about 4.2 Å) but with a quasi random orientation of the two CO molecules.

FIG. 3 .

 3 FIG. 3. KER spectra of the 2-body a. (1,1) and b. (2,1) dissociation channels. For both spectra, the background contribution is due to random coincidences and is subtracted using a linear function and the red line is a Gaussian fit of the experimental data. The expected position of the KER peaks corresponding to theoretical predictions Re = 4.3 Å (C-bonded) and Re = 3.6 Å (O-bonded) are represented by dashed black lines.

  subscript i stands respectively for the C + , O + and CO + recoiling ions) along the three axis of the diagram. The corresponding Cartesian coordinates are calculated as : x D = ε1-ε2 √ 3 and y D = ε 3 -1 3

FIG. 4 .

 4 FIG. 4. Dalitz plot of the (1,[1,1]) dissociation channel and identification of the concerted and sequential fragmentation processes.

FIG. 5 . 2 √ 3

 523 FIG. 5. Dalitz diagrams for the (1,[2,1]), (1,[1,2]) and (1,[2,2]) channels. The Cartesian coordinates are: xD = ε 1 -ε 2 √ 3 and yD = ε3 -13 where εi(i = 1, 2, 3) are the reduced squared momenta of the fragments, sorted by ascending order of massto-charge ratio. Using this convention, the momentum of the CO + ion is plotted along the vertical axis but the carbon and oxygen ions are plotted the right or left diagonals depending on their respective charge states.

  6.b).

FIG. 6

 6 FIG. 6. a), c) and d) : calculated ion trajectories of the 3-body dissociation of the (1,[1,1]) channel for θ = 90 • , 45 • , 135 • . b) : Evolution of the kinetic energy and total KER as a function of time for θ = 90 • . The asymptotic limit is reached after about 10 ps.

  kinetic energy. Secondly, for a given KER of the CO 2+ dication, the CO + acquires a significantly larger kinetic energy in the case of the C-bonded or O-bonded (θ = 45 • or 135 • ) geometries because the CO + ion is pushed away by the C + or O + ion emitted towards him. Finally, the dots associated to the θ = 45 • initial orientation are located on the right side of the points for θ = 90 • because the O + atomic ion experienced a smaller Coulomb repulsion due to its central position within the fragmenting dimer. A similar argumentation justifies that the dots for θ = 135 • stand on the left side of the Dalitz plot. Comparing with the experimental data, where a large majority of events are located on the left part of the Dalitz plot (Fig. 4), the calculated 45 • (O-bonded) structure seems to be excluded since it does not match the experimental momentum repartition. On the other hand, both the 135 • (C-bonded) and 90 • structures may reproduce the experiment.

FIG. 7 .

 7 FIG. 7. Dalitz representation of the reduced squared momenta in the concerted 3-body dissociation of the (1,[1,1]) channel for θ = 90 • , 45 • , 135 • and for kinetic energy release ranging from 5 to 50 eV (0.39 ≤ ACO ≤ 3.9).

  • and θ = 135 • are quite similar except for the(1,[2,1]) channel where significantly different distributions are observed. A close comparison of the x D distributions clearly shows a better agreement of the experimental data with the θ = 90 • structure. For all fragmentation channels, the centroid of the simulated θ = 90 • data matches almost perfectly with the experiment while the θ = 135 • curves are significantly shifted to the right or left side of the measured value.

FIG. 8 . 2 √ 3 .

 823 FIG. 8. Experimental and simulated Dalitz plots for the (1,[1,1]), (1,[2,1]), (1,[1,2]) and (1,[2,2]) channels. For the experimental (1,[1,1]) channel, only the concerted events are shown. For each channel, simulations have been run with initial orientations of θ = 90 • , 45 • , 135 • , 0 • , 180 • assuming a concerted fragmentation. Bottom : projection of the horizontal Dalitz coordinate xD = ε 1 -ε 2 √ 3 . The vertical dashed line indicates the centroid of the experimental distribution for each fragmentation channel.

FIG. 9 . 2 √ 3 .

 923 FIG. 9. Experimental and simulated Dalitz plots for the (1,[1,1]), (1,[2,1]), (1,[1,2]) and (1,[2,2]) channels. For the experimental (1,[1,1]) channel, only the concerted events are shown. For each channel, simulations have been run with a random initial orientation and assuming a concerted fragmentation. Bottom : projection of the horizontal Dalitz coordinate xD = ε 1 -ε 2 √ 3 . The vertical dashed line indicates the centroid of the experimental distribution for each fragmentation channel.

TABLE I .

 I Structural parameters for the planar C-bonded and O-bonded geometries of the neutral CO dimer. The relative orientation angle θ with respect to the dimer axis and the intermolecular distance are defined on Fig.1.

	Structure		C-bonded	O-bonded
		expt.	calc.	expt.	calc.
		[1]	[5,6]	[1]	[5,6]
	θ( • )	-	134.2/134.79	-	61/65.18
	Re( Å)	4.4	4.33/4.331	4.0	3.67/3.620

VII. ACKNOWLEDGMENTS

Experiment was performed at Grand Accélérateur National d'Ions Lourds (GANIL) by means of CIRIL Interdisciplinary Platform, part of CIMAP laboratory, Caen, France. The authors want to thank the CIMAP and GANIL staff for their technical support.


Pedro Ramaciotti Morales

Jean-Philippe Cointet
email: jeanphilippe.cointet@sciencespo.fr

Gabriel Muñoz Zolotoochin
email: gabriel@mat.uc.cl

Unfolding the dimensionality structure of social networks in ideological embeddings

Keywords: Network scaling, graph embedding, ideology, attitudinal data, party systems Anti-redistribution (salience) Favors religious principles Anti-islam rhetoric (salience) * Opposes economic interventionism Opposes redistribution Anti-multiculturalism (salience) Opposes multiculturalism Econ. ideology party blurriness EU integration (salience)

Traditionally, public opinion on different issues of public debate has been studied through polls and surveys. Recent advancements in network ideological scaling methods, however, have shown that digital behavioral traces in social media platforms can be used to mine opinions at a massive scale. This has yet to be shown to work beyond one-dimensional opinion scales, which are best suited for two-party systems and binary social divides such as those observed in the US. In this article, we use multidimensional ideological scaling for coupled with referential attitudinal data for some nodes. We show that opinions can be mined in a multitude of issues: from social networks, embedding them in ideological spaces where dimensions stand for indicators of positive and negative opinions, towards issues of public debate. This method does not require text analysis and is thus language independent. We illustrate this approach on the Twitter follower network of French users leveraging political survey data.

I. INTRODUCTION

Opinions may have different forms and functions, which makes them difficult to conceptualize and operationalize. An important type of opinion is that of evaluative opinions: that is, to be for or against something [START_REF] Bem | Beliefs, attitudes, and human affairs[END_REF]. Evaluative opinions are often operationalized as attitudes: an individual disposition towards an attitudinal object (e.g., person, institution, issue, event, bill, policy position). Attitudes can also be held towards complementary attitudinal objects. A classical example are attitudes towards liberal and conservative values: positive attitudes towards one set of values imply negative attitudes towards the other, so that individuals can be placed on attitudinal scales ranging from the most liberal to the most conservative positions. Attitudes are, contingently on other factors, important determinants of behavior [START_REF] Ajzen | Attitude structure and behavior[END_REF], [START_REF] Ajzen | The influence of attitudes on behavior[END_REF]. The traditional method for estimating people's opinions is the administration of surveys or polls (e.g., the ANES poll in the US, or the Eurobarometer in Europe). Besides polling, different behavioral traces can be observed to infer the attitudinal positions of individuals. A classical example is found in This work has been funded by the French National Agency for Research under grant ANR-19-CE38-0006 "Geometry of Public Issues" (GOPI) and CIVICA's "European Polarisation Observatory" (EPO). International Conference on Advances in Social Networks Analysis and Mining (ASONAM '21), November 8-11, 2021, Virtual Event, Netherlands 978-1-4503-9128-3/21/11/$15.00 © 2021 the work of Poole and Rosenthal [START_REF] Poole | A spatial model for legislative roll call analysis[END_REF], in which they estimate attitudes of parliamentarians in the US towards liberal and conservative values using Bayesian inference on roll call data. Recently Barberà [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF] used similar methods to infer the liberal or conservative attitudes of millions of US Twitter users with statistical inference on observed friendship -who follows whom-networks. This method postulates a model for the formation of a friendship network based on homophily (people with similar attitudes follow each other online), and that users' opinions are accounted for with a single latent ideological variable. The method then uses an observed network structure to perform an inference to estimate the ideological parameter. Importantly, these methods do not rely on textual data, making them language-independent. Empirical validation using external data (such as self-declared political affiliation) has shown that the recovered attitudinal variable in the US coincides with attitudes towards liberal and conservative values.

These methods for the inference of attitudes using social networks, however, have been much less successful in European and other settings [START_REF] Barberá | Understanding the political representativeness of twitter users[END_REF]. This is due to the multi-issue, multi-party socio-political underlying systems that structure public debate [START_REF] Hix | Dimensions of politics in the european parliament[END_REF], and ultimately social networks such as Twitter. In this article we use similar network scaling methods on large social network topological structures, and political science expert survey data with attitudinal positions of very few referential social network nodes. We show that, using both sources of data, it is possible to extract attitudinal positions for several issues for large portions of social networks. The result is an ideological embedding procedure for social networks in which dimensions stand as indicators for attitudes towards different issues, such as taxation, immigration, left-right cleavages, and trade protectionism, to name a few. The resulting spaces are ideological in the sense that the position of a user along each dimension provides information about the attitude of that user towards a set of combined issues. We illustrate this procedure with French Twitter data, and we propose benchmarks to test the validity of the approach. We also discuss its potential uses in several fields, such as: the study of populism, polarization, and the effects of Recommender Systems.

II. RELATED WORK

A diversity of methods for ideology estimation exist [START_REF] Imai | Fast estimation of ideal points with massive data[END_REF]. Many methods rely on textual analysis [START_REF] Gentzkow | Text as data[END_REF], [START_REF] Groseclose | A measure of media bias[END_REF], and are thus language-and context-dependent (e.g., people express themselves differently on different platforms, and a model capturing ideology from parliamentary debates [START_REF] Rheault | Word embeddings for the analysis of ideological placement in parliamentary corpora[END_REF] will necessarily differ from one predicting political slant in tweets [START_REF] Stefanov | Predicting the topical stance and political leaning of media using tweets[END_REF]). Other methods rely only on relational traces, producing ideological scaling (spatializations) from topological relations: e.g., who is friends with whom, or votes for or clicks on what. The NOMINATE method by Poole and Rosenthal [START_REF] Poole | A spatial model for legislative roll call analysis[END_REF] is a classical example. Bond and Messing were among the first to apply ideological scaling methods to large social network graph data [START_REF] Bond | Quantifying social media's political space: Estimating ideology from publicly revealed preferences on facebook[END_REF]. The work presented in this article is related to a similar network scaling method, proposed by Barberà [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF] for one-dimensional scaling. This scaling computes a single latent ideological parameter φ i for every user i, following an homophily probabilistic law adjusted for activity and popularity:

P (A ij = 1|α i , β j , γ, φ i , φ j ) = logit -1 α i +β j -γ|φ i -φ j | 2 ,
where A ij = 1 when user i follows user j, α i and β j are the "activity" (tendency to follow others) and "popularity" (tendency to be followed) of users i and j, φ i and φ j are their ideological parameters, and γ is a normalization constant. When computed in the bipartite network of the US parliamentarians and their followers on Twitter, using Markov Chain Monte Carlo methods, the ideological parameters φ i were shown to act as indicators of attitudes towards liberals and conservatives. The inference of ideological parameters has been shown [START_REF] Lowe | Understanding wordscores[END_REF], [START_REF] Carroll | An equivalence relation between correspondence analysis and classical metric multidimensional scaling for the recovery of euclidean distances[END_REF] to be replaceable by Correspondence Analysis (CA) [START_REF] Greenacre | Correspondence analysis in practice[END_REF] on Twitter data [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF]. CA also allows for the inference of multi-dimensional ideological parameters, embedding users in ideological spaces where positions are informative of their attitudes [START_REF] D'esposito | On the use of multiple correspondence analysis to visually explore affiliation networks[END_REF]. Recent works [START_REF] Morales | Your most telling friends: Propagating latent ideological features on twitter using neighborhood coherence[END_REF], [START_REF] Cointet | What colours are the yellow vests? an ideological scaling of facebook groups[END_REF] have suggested that these multi-dimensional parameters might be related to attitudes towards several issues of public debate, beyond the classical one-dimensional attitude scale from leftright or liberal-conservatives cleavages. In this article, we further explore the extraction of multi-dimensional attitudinal indicators suggested in [START_REF] Morales | Your most telling friends: Propagating latent ideological features on twitter using neighborhood coherence[END_REF]. In contrast with attitudinal embeddings relying on text analysis or news media citations [START_REF] Baumann | Modeling echo chambers and polarization dynamics in social networks[END_REF], ideological embeddings relaying only social networks structures have the potential to be language-and contextindependent, thus allowing, for example, comparisons between countries.

III. SOCIAL NETWORK DATA

To illustrate the extraction of attitudinal indicators using external referential attitudinal data we use French Twitter networks 1 . We consider the set of the 831 (out of 925) French Members of Parliament (MPs) present on Twitter who are 1 Data declared the 19 March 2020 at the registry of data processing at the Fondation Nationale de Sciences Politiques (Sciences Po) in accordance with General Data Protection Regulation 2016/679 (GDPR) and Twitter policy. affiliated to 10 parties and their followers2 . This collection was conducted in May 2019, and resulted in 4.487.430 unique followers. In this bipartite network, some MPs are followed by a handful of users (min. degree = 38) while others are followed by most users that follow any MP (max. degree = 2.241.986). On the other hand, many users in this network (2.279.199) follow only one MP, with the most diverse user following 757 MPs (see Fig. 1). To filter out inactive or bot accounts, and accounts without enough ideological referential connections, we follow the criteria proposed in [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF] and consider only followers that follow at least 3 MPs, and that have at least 25 followers. We also removed followers with a repeated set of followed MPs to obtain 368.831 followers to ensure the full rank of the adjacency matrix.

IV. HOMOPHILY EMBEDDING

We represent this bipartite subgraph as an adjacency matrix A ∈ {0, 1} |368.831|×|831| . Next, we produce a reduceddimensionality representation of 831 MPs and the 368.831 followers using the coordinates of accounts in the Principal Components (PCs) of the CA of the adjacency matrix (see Fig. 2). Fig. 3 shows the inertia of the PCs of CA, and the density of the followers in the space spanned by the first 2 PCs of this reduced-dimensionality space. When most of the inertia is concentrated in the first dimensions or components, a spatial representation in those dimensions is a suitable representation of the topological data in the sense that a random graph computed using such spatialization and the chosen probabilistic model would be similar to the original network [START_REF] Roberts | Correspondence analysis of two-mode network data[END_REF]. As seen in Fig. 3, only a few dimensions hold relatively more importance in explaining the topological network data.

V. EXTERNAL REFERENTIAL ATTITUDINAL DATA

To inspect the relation between PCs and attitudes towards different issues of the public debate that might be influencing the network structure, we need to identify reference points in the Twitter network (and thus also in the latent embedding space). We then inspect the attitudinal positions of these references as given by external referential attitudinal data (e.g., surveys and polls). To do this, we compute the position of French political parties in this space as the centroids of the positions of the Twitter accounts of parliamentarians of these parties (see Fig. 4). We compare the position of parties according to our CA dimensions with the attitudinal positioning of parties on scales for different issues as provided by the 2019 Chapel Hill Expert Data (CHES) [START_REF] Bakker | 2019 chapel hill expert survey[END_REF]. This attitudinal data source is compiled using the responses of a survey administered to 421 political scientists, in which they place European political parties on scales from 0 to 10 for 51 different issues. The positioning in our ideological spaces is related, to different degrees, to the positioning by some of the attitudinal scales available in the CHES data. We measure the relation between positioning along the dimensions of the ideological space and the attitudinal positioning on issues given by CHES by computing Pearson correlations. Fig. 4 illustrates two example CHES attitudinal dimensions that are highly correlated with the positioning of parties in dimensions PC1 and PC2. In these examples, positions along PC1 are -negatively-correlated with attitudes towards trade protectionism: the higher a party is on the PC1 axis, the more opposed it is towards trade protectionism. Similarly, positions along PC2 are -positively-correlated with attitudes towards left-or right-leaning "ideological stances" (as referred to in the CHES survey): the higher a party is on the PC2 axis, the more favorable it is towards right-wing ideological stances. This is an example of the aforementioned complementary attitudinal objects, as parties with lower positions on the PC2 axis are opposed to right-leaning ideological stances, and thus favorable to left-leaning stances.

C i W r L K l l z k N 7 A y U k K 1 a V H x B B z 1 E Y E g R g C O E J O z D g a C n D R s W Y u K 6 G B O X E P J 0 n O M e B d K m l M U p w y F 2 S N 8 B 7 d o Z G 9 J e e Q q t Z n S K T 2 9 C S h O 7 p I k o L y G s T j N 1 P N X O i v 3 N e 6 w 9 1 d 1 G 9 H c z r 4 B Y i U t i / 9 J N M v + r U 7 V I 9 H G k a / C o p l g z q j q W u a S 6 K + r m 5 p e q J D n E x C n c o 3 h C m G n l p M + m 1 g h d u + q t o + N v O l O x a s + y 3 B T v 6 p Y 0 Y P v n O K d B o 1 K 2 D 8 q V s / 1 S d S s b d R 7 b 2 M E e z f M Q V Z y g h j p 5 X + E R T 3 g 2 T o 1 r 4 9 a 4 + 0 w 1 c p l m E 9 + W 8 f A B 5 O W R i Q = = < / l a t e x i t > • • •

VI. IDEOLOGICAL SPACE

To link the dimensions of our space with issues of public debate present in the CHES data, we compare the positions of political parties according to our PCs, with their position in the attitudinal CHES dimensions (scoring parties in a 0 to 10 scale, ranging from very opposed to very favorable to each issue). To compare a PC with a CHES dimension, we compute the Pearson correlation of party positions. We compare each one of the first 15 PCs (as suggested by Fig. 3) with all available CHES dimensions. Only the first 3 PCs show correlations with CHES dimensions, and these correlations show that each PC is associated (up to a correlation significance of α = 0.05, marked in dark blue in Fig. 5) with a subset of CHES dimensions.

PC1 is positively correlated with positive attitudes towards the EU, suggesting that the higher the value of the PC1 position of a user, the more positive their views are towards the EU. PC1 is also associated with opposition to redistribution, interventionism, and market regulation. It is also associated with high levels of left-right economic policy dissent (inside parties), and with relevance granted to Russian interference in politics. Negative PC1 positions are correlated with positive views on economic protectionism, and a high level of importance is placed on people-elites cleavages and redistribution. These issues are related to attitudes towards globalization in European settings, and we call it the "Local-Global" (LG) axis. PC2 is associated with positions on issues widely attributable to left-right cleavages in France, including: left-right economic and ideological positions, rural-urban cleavages, religious principles, rights of minorities, authoritarianism, and the balance between crime-fighting and civil liberties among others. We call this second dimension the "Left-Right" (LR) axis. Attitudes towards issues related to internationalization (e.g., trade protectionism, EU integration) have long been recognized as growing in importance in structuring individual preferences (e.g., in voting behavior [START_REF] Grossman | Economic internationalization and the decline of the left-right dimension[END_REF]) when compared with the importance of attitudes towards issues distinguished in left-right cleavages (e.g., privatization, taxation, wealthfare spending). The third PC to exhibit correlations with CHES dimensions is PC3, and is associated with views on the relative importance between leaders and members within parties, with economic policy blurriness, but mostly with importance granted to European integration, multiculturalism, and immigration restrictions. We call this third dimension the "Immigration and multiculturalism" (IM) axis. This third axis acts as an indicator of the importance granted to these issues (salience): lower positions in PC3 relate increased importance.

Our axes span an ideological space in the sense that they position users according to attitudes towards a set of correlated issues. While the notion of ideology refers to "fuzzy" [START_REF] Van Dijk | Ideology: A multidisciplinary approach[END_REF] set different concepts (including normative ones relating to power, e.g., the choice of rulers [START_REF] Lane | Political ideology: Why the american common man believes what he does[END_REF] or the justification of power [START_REF] Mcclosky | Conservatism and personality[END_REF]), we chose to use this term in its descriptive dimension:

as an "organization of opinions, attitudes and values" [START_REF] Adorno | The authoritarian personality[END_REF], a "structure of attitudes" [START_REF] Campbell | The american voter[END_REF], or as a description of high attitude consistency [START_REF] Converse | The nature of belief systems in mass publics[END_REF]. When a single variable is informative of a set of attitudes towards some issues because they display high spatial correlation, we call it "ideological dimensions". These associations were computed and justified, however, using the positions of parliamentarians, and we now seek to validate the definition of our LG, LR, and IM axes using the Twitter bios of their 368.831 followers, in which users briefly describe themselves. Twitter bios are short texts in which users describe themselves and are shown on the profile page and profile preview of each user. For each one of the three axes we select two topics, and for each topic we define a minimalist dictionary to classify the bios of followers. Our goal is to show that, even when applying a minimal text analysis, our three ideological axes distribute users (and not only parliamentarians) according to the meaning proposed for these dimensions. For the LG axis we classify users as mentioning the "EU" (if they include the strings "eu" or "europe") and as mentioning the cleavage between "People and Elites" (if they include the strings "peuple", meaning people in French, "élite", or "politicien"). For the LR axis we classify users as mentioning the "Left-wing" (if they include the string "gauche", meaning left in French) and as mentioning the "Right-wing" (if they include the string "droite", meaning right in French). For the IM axis we classify users as mentioning the "Islam" (if they include the strings "musulman", meaning muslim in French, or "islam") and as mentioning the "Multiculturalism" (if they include the string "multiculturalisme"). In the context of the French public debate, questions of multiculturalism and immigration policy rapidly revolve around a debate on Islam. In the strings that define a topic, we included all possible variants and misspellings. For the topics "EU", "Left", and "Right" we excluded bios with negative sentiment using a minimalistic approach by tagging the presence of the strings "anti", "contre", and "déteste". For the other topics we did not filter by sentiments, as we are trying to detect the importance or salience of the topic along the dimensions (which does not involve positive or negative opinion). Fig. 6 shows the proportion of users that use these topics in their bios and the Clopper-Pearson confidence interval of this proportion at a α=0.05 CI. As many studies suggest, left-right cleavages have a structuring role in behavior (e.g., voting behavior [START_REF] Aldrich | The use of the left-right scale in individual's voting decisions[END_REF]) and in particular on French Twitter [START_REF] Briatte | Recovering the french party space from twitter data[END_REF] (LR axis). In our dataset, however, the most structuring ideological dimension (LG) relates to globalization and European integration; it is the first PC that is related with these issues. In the rest of this article we will focus on the first two dimensions: PC1 and PC2.

VII. PARTY SYMPATHIZERS

A second way of looking at the position of the followers is by examining the relation between those that declare sympathies towards a party and their position in space with respect to those of the party itself. While it is difficult to account for partisanship, we propose a simple approach based on non- negative mentions of parties in the Twitter bio or profile description. To do so, we again propose again a minimalistic approach to classifying users and sympathizers of a given party, based on a few strings for each one of the ten political parties in our dataset:

• EELV (Europe Écologie -Les Verts): "eelv", "les verts";

• LFI (La France Insoumise): "insoumis", "lfi";

• LREM (La République en Marche): "lrem", "en marche"; • MoDem (Mouvement Démocrate): "modem", "mouvement démocrate"; • LC (Les Centristes): "centristes";

• PCF (Parti Communiste Franc ¸ais): "pcf", "communiste";

• PRG (Parti Radical de Gauche): "radical de gauche", "parti radical", "prg"; • PS (Parti Socialiste): "ps", "socialiste";

• RN (Rassemblement National): "rn", "rassemblement national".

As before, we also consider singular and plural, masculine and feminine declensions when possible, as well as upperand lower-case versions of our keywords. We also consider hashtag for keywords made of a single word (e.g., "pcf" and "#pcf"). Finally, as in the previous section, we filter out possible negative references using the same set of minimal identifiers of negative sentiment. Fig. 7 shows the position of party sympathizers in comparison with that of the ensemble of followers and the position of the political parties computed as the mean position of their MPs.

VIII. DISCUSSION AND CONCLUSIONS

In this article we have shown that network topological data (to the exclusion of textual data) and external attitudinal data can be used to embed social networks in ideological spaces where dimensions stand for indicators of attitudes towards sets of issues of public debate. We have illustrated this procedure using the French Twitter network, using the accounts of parliamentarians, and the CHES data on attitudinal positions for European political parties. This allowed us to embed the Twitter network in an ideological space spanned by three dimensions, positioning users according to their views on globalization and European integration, left-right positions, and views regarding immigration and multiculturalism.

Ideological spaces have many potential applications. They can be used to study ideological trajectories in time [START_REF] Cointet | What colours are the yellow vests? an ideological scaling of facebook groups[END_REF], and to study issues related to the meaning of ideological axes, such as the relative importance of left-right divides and polarization regarding globalization [START_REF] Grossman | Economic internationalization and the decline of the left-right dimension[END_REF]. They can also be used to study the dimensional structure (number and relative importance of dimensions) in different digital arenas [START_REF] Benoit | The dimensionality of political space: Epistemological and methodological considerations[END_REF], and party systems online [START_REF] Bakker | Complexity in the european party space: Exploring dimensionality with experts[END_REF]. A third line of applications involves the study of the effect of Recommender Systems in phenomena such as polarization, now conceptualized in geometrical spaces [START_REF] Morales | Auditing the effect of social network recommendations on polarization in geometrical ideological spaces[END_REF], beyond traditional connectionist approaches. These applications must consider the representativity of the users of a given digital social network in the population [START_REF] Barberá | Understanding the political representativeness of twitter users[END_REF]. The study of ideological configurations and dynamics of Twitter in itself presents interesting possibilities due to its importance in setting the political and media agenda.

FollowersFig. 1 .

 1 Fig. 1. Cumulative Distribution Function (CDF) of the degree of two types of nodes of the bipartite network of French MPs (in red) and their followers (in blue).

  -T w it te r' s 2 nd P ri n c ip a l C o m p o n e n t PC3 -Twitter's 3rd Principal Component Twitter account P C 1 -T w it te r' s 1 st P ri n c ip a l C o m p o n e n t PC3 < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 t U X 3 R 4 1 3 Z u I V 4 w 8 C H 4 P 8 6 J M G S Y = " > A A A C y X i c j V H L S s N A F D 2 N r 1 p f 9 b F z E y y C q 5 I U U Z c F N 4 K b C v Y B b Z F k O q 2 x e Z m Z i L W 4 8 g f c 6 o + J f 6 B / 4 Z 0 x B b W I T k h y 5 t x 7 z s y 9 1 4 1 9 T 0 j L e s 0 Z M 7 N z 8 w v 5 x c L S 8 s r q W n F 9 o y G i N G G 8 z i I / S l q u I 7 j v h b w u P e n z V p x w J 3 B 9 3 n S H x y r e v O G J 8 K L w X I 5 i 3 g 2 c Q e j 1 P e Z I o h o d 1 o u k u

Fig. 2 .Fig. 3 .

 23 Fig. 2. Illustration of the homophily embedding through which the bipartite network of the French MPs on Twitter and their followers are represented in a multidimensional geometrical space. The bipartite network is represented as an adjacency matrix (A), to which Correspondence Analysis is applied to produce lower-dimensional spatial representations (B).

Fig. 4 .

 4 Fig. 4. Positioning of parties in the reduced-dimensionality space spanned by PC1 and PC2 of the CA of the bipartite Twitter network, computed as means of the positions of parliamentarians (top), and examples of two external party attitudinal positions (CHES) that are correlated with first two PC.

Fig. 5 .

 5 Fig. 5. Pearson correlations of the positioning of French parties on Twitter and according to CHES data for the first three Principal Components, and for all CHES dimensions, each associated with an issue of public debate. * marks 0.05 confidence and ** marks 0.01 confidence levels.

Fig. 6 .

 6 Fig.6. Proportion of French Twitter users referring to different topics on their bios according to their positions along the three axes of ideological space.

Fig. 7 .

 7 Fig.7. Position of the users identified as sympathizers of each one of the ten parties in our dataset (identified with blue crosses +), and the position of the parties computed as the mean position of its MPs (shown in yellow dots •).

Obtained from http://www2.assemblee-nationale.fr/deputes/liste/ reseaux-sociaux for deputies, and http://www.senat.fr/espace presse/ actualites/201402/les senateurs sur twitter.html for senators.


Henri

B Borderie

S Piantelli

E Bonnet

R Bougault

A Chbihi

J E Ducret

J D Frankland

E Galichet

D Gruyer

M Henri

M La Commara

N Le Neindre

I Lombardo

O Lopez

L Manduci

M Pârlog

R Roy

G Verde

M Vigilante

Negative heat capacity for hot nuclei using formulation from the microcanonical ensemble INDRA Collaboration

Keywords: XX, XX, XX No code given

Negative heat capacity for hot nuclei using formulation from the microcanonical ensemble INDRA Collaboration

An important challenge of heavy-ion collisions at intermediate energies is the identification and characterization of the nuclear liquid-gas phase transition in hot nuclei, which has been theoretically predicted for nuclear matter [1,2,3,4,5]. At present one can say that huge progress has been made even if some points can be deeper investigated [6,7,8,9,10]. Statistical mechanics for finite systems appeared as a key issue to progress, revealing specific first-order phase transition signatures related to the consequences of the local convexity of the entropy [11,12,13,14,15,16,17]. By considering the microcanonical ensemble with energy as extensive variable, the convex intruder implies a backbending in the temperature (first derivative of entropy) at constant pressure [18] and correlatively a negative branch for the heat capacity (second derivative of entropy). Experimentally, these two converging signatures have been observed in hot nuclei from different anal-Correspondence to: borderie@ipno.in2p3.fr yses [19,20,[START_REF] Neindre | Proceedings of the XXXVIII International Winter Meeting on Nuclear Physics[END_REF][START_REF] Borderie | [END_REF]23] of homogeneous event samples. It is important to recall here that signals of phase transition in hot nuclei are only meaningful at the level of statistical ensembles constructed from the outcome of carefully selected collisions [8]. Another consequence of the entropy curvature anomaly manifests itself when systems are treated in the canonical ensemble. In this case a direct phase transition signature is the presence of a bimodal distribution of an order parameter [24] such as the charge (size) of the largest fragment (Z max ) of multifragmentation partitions [25]. As far as microcanonical heat capacity calculation is concerned, a question still subsists. It concerns one hypothesis made in the method used up to now: the same temperature was associated with both internal excitation and thermal motion of emitted fragments, which is not physically obvious if one remembers that the level density is expected to vanish at high excitation energies [26,27,28,29]. Moreover it was shown in simulations compared to data that widths of the fragment velocity spectra cannot be reproduced with the hypothesis of a common temperature for intrinsic and kinetic degrees of freedom [30].

In the present letter we will give heat capacity information for experimental data from microcanonical formulae and by ignoring the hypothesis of a single temperature, which requires the introduction of a limiting temperature for fragments.

We first briefly recall the method proposed for measuring heat capacity using partial energy fluctuations with a microcanonical sampling [31,32,18]. The prescription is based on the fact that, for a given total energy, the average partial energy stored in a part of the system is a good microcanonical thermometer, while the associated fluctuations can be used to construct the heat capacity. From experiments the most simple decomposition of the thermal excitation energy is in a kinetic part, E k , and a potential part, E pot (Coulomb energy + total mass excess). These quantities have to be determined at freeze-out and consequently it is necessary to trace back this configuration on an event by event basis. The true configuration needs the knowledge of the freeze-out volume and of all the particles evaporated from primary hot fragments including the (undetected) neutrons. Consequently some working hypotheses are used, constrained by specific experimental results (see for example [33]). Then, the experimental correlation between the kinetic energy per nucleon E k /A and the thermal excitation energy per nucleon E * /A of the considered system can be obtained event by event as well as the variance of the kinetic energy σ 2 k . Note that E k is calculated by subtracting the potential part E pot from the thermal excitation energy E * and consequently kinetic energy fluctuations at freeze-out reflect the configurational energy fluctuations. In the microcanonical ensemble with total energy E * the total degeneracy factor is simply given by the folding product of the individual degeneracy factors W k = exp(S k (E k )) and W pot = exp(S pot (E pot )) where S k (S pot ) is the entropy of the kinetic (potential) part. One can then define for the total system as well as for the two subsystems the microcanonical temperatures and the associated heat capacities C k and C pot . If we consider now the kinetic energy distribution when the total energy is E * we get

P E * (E k ) = exp(S k (E k ) + S pot (E * -E k )). ( 1
)
Then the most probable kinetic energy E k is defined by the equality of the partial microcanonical temperatures

T k (E k ) = T pot (E * -E k
) and E k can be used as the microcanonical thermometer. An estimator of the microcanonical temperature of the system can be obtained by inverting the kinetic equation of state:

< E k >= M i=1 a i T 2 + 3 2 (M -1) T (2)
The brackets indicate the average on events with the same E * , a i is the level density parameter and M the multiplicity at freeze-out. In this expression the same temperature is associated with both internal excitation and thermal motion of fragments. An estimate of the total microcanonical heat capacity is extracted using three equations.

C k = δ < E k /A > δT , (3)
is obtained by taking the derivative of < E k /A > with respect to T . Using a Gaussian approximation for P E * (E k ) the kinetic energy variance can be calculated as

Aσ 2 k ≃ T 2 C k C pot C k + C pot ; (4)
Eq. ( 4) can be inverted to extract, from the observed fluctuations, an estimate of the microcanonical heat capacity:

( C A ) micro ≃ C k + C pot ≃ C 2 k C k - Aσ 2 k T 2 . ( 5
)
From Eq. ( 5) we see that the specific microcanonical heat capacity (C/A) micro becomes negative if the normalized kinetic energy fluctuations Aσ 2 k /T 2 overcome C k . It is interesting to note that the constraint of energy conservation leads in the phase transition region to larger fluctuations than in the canonical case where the total energy is free to fluctuate. This is because the kinetic energy part is forced to share the total available energy with the potential part: when the potential part presents a negative heat capacity the jump from "liquid" to "gas" induces strong fluctuations in the energy partitioning.

Direct formulae have been proposed in Ref. [34] to calculate heat capacity but never used to extract information from data. They are derived within the microcanonical ensemble by considering fragments interacting only by Coulomb and excluded volume, which corresponds to the freeze-out configuration. Within this ensemble, the statistical weight of a configuration c, defined by the mass, charge and internal excitation energy of each of the constituting M c fragments, can be written [34,19] as

W c (A, Z, E * , V ) = 1 M c ! χV Mc Mc n=1 ρ n (ǫ n ) h 3 (mA n ) 3/2 × 2π Γ (3/2(M c -d)) 1 (detI) (2πK) (3/2)(Mc-d)-1 (mA) 3/2 , (6)
where A, Z, E * and V are respectively the mass number, the atomic number, the thermal excitation energy and the freeze-out volume of the system. E * is used up in fragment formation, fragment internal excitation, fragmentfragment Coulomb interaction and thermal kinetic energy K. I is the inertial tensor of the system whereas χV Mc stands for the free volume or, equivalently, accounts for inter-fragment interaction in the hard-core idealization. ρ n (ǫ n ) represents the fragment level density (see [35] for the complete expression). d takes respectively values 0, 1 and 2 for energy, energy plus linear momentum and energy plus linear and angular momenta conservations.

In the following d will be fixed to 1 to ensure coherence with simulations which are used. Taking into account that S = ln Z = ln c W c , the microcanonical temperature is deduced from its statistical definition [34]:

T = ∂S ∂E * -1 = 1 c W c c W c (3/2M c -5/2)/K -1 (7) = (3/2M c -5/2)/K -1 . (8)
The notation refers to the average over the ensemble states. The heat capacity of the system, C, is related to the second derivative of the entropy by the equation ∂ 2 S/∂E * 2 = -1/CT 2 . Thus, one can evaluate the second derivative of the system entropy versus E (Eq. ( 9)) or alternatively the heat capacity C (Eq. ( 10))

∂ 2 S ∂E * 2 = (3/2M c -5/2)(3/2M c -7/2) K 2 - (3/2M c -5/2) K 2 (9) C = 1 -T 2 (3/2M c -5/2)(3/2M c -7/2) K 2 -1 (10)
These two quantities only depend on two parameters M c and K which must be estimated at freeze-out.

In Refs. [36,30] we presented simulations which correctly reproduce most experimental observables and estimate freeze-out properties in a fully consistent way. They concern hot nuclei which undergo multifragmentation formed in central collisions and selected by event shape sorting (quasifused systems, QF, from 129 Xe+ nat Sn, 32-50 AMeV) [37]. We will use the event by event properties at freeze-out which were inferred to calculate the required quantities M c and K.

Experimental data were collected with the 4π multidetector INDRA described in detail in Ref. [38,39]. Accurate particle and fragment identifications were achieved and the energy of the detected products was measured with an accuracy of 4%. Further details can be found in Ref. [40,41,42].

The method for reconstructing freeze-out properties from simulations [36,30] requires data with a very high degree of completeness, (measured fraction of the available charge ≥93% of the total charge of the system), crucial for a good estimate of Coulomb energy. QF sources are reconstructed, event by event, from all the fragments and twice the charged particles emitted in the range 60 -120 • in the reaction centre of mass, in order to exclude the major part of pre-equilibrium emission [43,37]; with such a prescription only particles with isotropic angular distributions and constant average kinetic energies are considered. In simulations, dressed excited fragments and particles at freeze-out are described by spheres at normal density. Then the excited fragments subsequently deexcite while flying apart. All the available asymptotic experimental information (charged particle spectra, average and standard deviation of fragment velocity spectra and calorimetry) is used to constrain the four free parameters of simulations to recover the data at each incident energy: the percentage of measured particles which were evaporated from primary fragments, the collective radial energy, a minimum distance between the surfaces of products at freeze-out and a limiting temperature for excited fragments. All the details of simulations can be found in Ref. [36,30]. The limiting temperature, related to the vanishing of level density for fragments [28], was mandatory to reproduce the observed widths of fragment velocity spectra. With a single temperature (internal and kinetic temperatures equal) the sum of Coulomb repulsion, collective energy, thermal kinetic energy directed at random and spreading due to fragment decays accounts only for about 60-70% of those widths. By introducing a limiting temperature, which corresponds to intrinsic temperatures for fragments in the range 4-7 MeV (see figure 1 of [19]), the thermal kinetic energy increases, due to energy conservation, thus producing the missing percentage for the widths of final velocity distributions. The agreement between experimental and simulated velocity spectra for fragments, for the different beam energies, is quite remarkable (see figure 3 of [30]). Relative velocities between fragment pairs were also compared through reduced relative velocity correlation functions [44,45] (see figure 4 of [30]). Again a good agreement is obtained between experimental data and simulations, which indicates that the retained method (freeze-out topology built up at random) and the deduced parameters are sufficiently relevant to correctly describe the freeze-out configurations, including volumes.

For hot nuclei produced in central collisions the total excitation energy, E tot , differs from the thermal one due to the presence of a radial collective expansion energy, E R , and E tot = E * + E R . To derive event by event at freezeout E tot , E * , the multiplicity, M c (M f o ) and the total kinetic energy, K, of the multifragmenting hot nuclei, the following equations are used.

The limiting temperature for the fragments, T lim , is introduced according to the formalism of [28], which corresponds to the following definition for the intrinsic temperature of the fragments, T f rag :

1 T f rag = 3 2 < K f o > + 1 T lim ( 11
)
where < K f o > is the average kinetic energy of fragments and particles at freeze-out. The equation used for calorimetry and to derive the sharing between internal excitation energy and kinetic energy on the event by event basis is the following:

Mcp k=1 K k cp + ∆B cp + M f o n < K f o > +M evap n θ f rag + ∆B n = (M f o -1) < K f o > + M f k=1 a k θ 2 f rag + ∆B f o + V f o Coul + E R = E tot + ∆B hn , (12)
where θ f rag is equivalent to the temperature T f rag in an ensemble average. ∆B, hn, K, M , cp, n, and f o stand respectively for mass excess, hot nucleus, kinetic energy, multiplicities, charged products, neutrons and freeze-out.

E R = M f o k=1 ( r k R0 ) 2
A k E 0 is the radial collective expansion energy (R 0 is the rms of fragment and particle distances to centre at the freeze-out volume, E 0 the radial expansion energy at R 0 and r is the distance of the considered particle/fragment of mass A from the centre of the fragmented source). V f o Coul is the Coulomb energy of the configuration for freeze-out previously determined. The chosen level density parameter a k = A k 10 M eV -1 , where A k is the mass of the k th fragment, well corresponds to that expected for a typical primary fragment [46]. By introducing Eq. ( 11) in Eq. ( 12) we obtain a third degree equation in < K f o >, from which we can deduce the energy sharing between the internal excitation energy of the fragments and the total thermal kinetic energy at freezeout K = (M f o -1) < K f o >. K is shared at random between all the particles and fragments at freeze-out under constraints of conservation laws. Information on initial angular momenta of multifragmenting QF hot nuclei are unknown, which explains why d=1 is used for Eqs. ( 9) and (10). Systematic error on E * was estimated around 1 AMeV.

Values of heat capacity and second derivative of the entropy versus thermal excitation energy E * have been calculated respectively from Eqs. ( 10) and ( 9) for QF hot nuclei with Z restricted to the range 80-100 to suppress tails of the distributions at different beam energies. The average over the ensemble states have been assimilated to an average over "event ensembles" sorted into E * bins. A binning of 0.5 AMeV was chosen to have a sufficient number of events in each bin in order to reduce statistical errors. Figure 1 shows the results. Error bars correspond to systematic plus statistical errors; systematic errors were evaluated by varying the free parameters of simulations within their limits defined by a χ 2 procedure [30]. The left part of the figure shows the results for the direct calculation of C/A. Negative heat capacity is observed on a rather large thermal excitation energy range and the second diverging region is more visible than the first one. As the second derivative of the entropy is a very small quantity (from ∼ 2.10 -4 to ∼ 3.10 -6 ), we have kept the presentation made in [34] i.e. A 2 ∂ 2 S/∂E * 2 for fig. 1 -right part; A is replaced by the average mass, < A >, on the considered E * bin. This quantity, which has small error bars in regions with values close to zero, better defines the E * domain of negative heat capacity. Positive values are measured in the range 6.0 -10.0 AMeV. The related microcanonical temperatures calculated with Eq. ( 8) are displayed in fig. 2. Considering the error bars, they are rather constant around 17 -18 MeV in the E * range where negative heat capacity is observed. With the large multiplicities observed in the present study, microcanonical temperatures are close to classical kinetic temperatures (see figure 3 of [19]). Last point, the statistical nature of the carefully selected event samples, i.e. the independence of the entrance channel, was verified and fig. 3 shows the main contributions of the different bombarding energies to the signal for each E * bin.

Negative heat capacity is thus confirmed for systems with A around 200 without any constraint, which demonstrates once again the robustness of this signal and supports the predictions made in [18] i.e. observation at constant pressure or at constant average volume. It is important to stress here that negative heat capacity cannot be observed at constant volume. For the liquid-gas transition in hot nuclei the volume is not fixed but multiplicity and partition-dependent, which means that on the theoretical side one is forced to consider a statistical ensemble for which the volume can fluctuate from event to event around an average value [8].

As compared to previous estimates of heat capacity [START_REF] Neindre | Proceedings of the XXXVIII International Winter Meeting on Nuclear Physics[END_REF][START_REF] Borderie | [END_REF], there is a significant difference in the E * range for negative values: 6.0±1.0 -10.0±1.0 AMeV in the present study and <4.0±1.0 -6.0±1.0 AMeV in the previous one. For quasifusion hot nuclei selection the same shape event sorting was used. The degree of completeness was different (93% here to be compared to 80% before) but it does not affect significantly the thermal excitation energy per nucleon. The method to reconstruct freeze-out properties was also different. But the main difference seems to be related to the average freeze-out volume. In previous estimates the average freeze-out volume was kept constant at 3 times the volume at normal density (3 V 0 ) over the whole thermal excitation energy range whereas in the present study it varies from 3.9 to 5.9 V 0 (see fig. 4). A direct consequence in the approximate method is an increase of Coulomb energy and consequently a decrease and a distortion with excitation energy of < E k > obtained by subtracting the potential part (Coulomb energy + total mass excess) to the thermal energy. To verify this reasoning the approximate method has been used. From our simulation, for each E * bin, we have calculated < E k > (see Eq. ( 2)) and derived an apparent single temperature, T S , needed to build the normalized kinetic energy fluctuations, Aσ 2 k /T 2 S , to be compared to C k (see Eqs. (3) and ( 5)). Figure 5 shows that heat capacity becomes negative in the E * range 5.5±1.0 -9.0±1.0 AMeV, i.e. when Aσ 2 k /T 2 S overcomes C k . This clearly confirms that the main difference, as compared to previous estimates, comes from different average freeze-out volumes. We also note a small decrease of the E * domain as compared to the present study, which possibly comes from the previous method. Last information, the apparent single temperature, T S , exhibits a monotonic behaviour, varying from 5.2 to 9.7 MeV over the whole E * range and values increasing from 6.5 to 8 MeV on the domain of negative heat capacity.

In conclusion, heat capacity measurements have been revisited without approximation and by ignoring the hypothesis of a single temperature associated with both internal excitation and thermal motion. For those measurements microcanonical formulae and data reconstructed at freeze-out with the help of a simulation have been used. Negative heat capacity has been confirmed for hot nuclei with A around 200 in the coexistence region of the phase transition.

E* (A MeV)

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .

 12345 Fig. 1. Heat capacity (left) and second derivative of the entropy (right) versus thermal excitation energy E * . Error bars correspond to systematic plus statistical errors (see text).