N M Souza-Neto

A Y Ramos

H C N Tolentino

Y Joly

Depth dependent local structures in CoPt thin films

X-ray absorption spectroscopy (XAS) has been used to clarify the thicknessdependent magnetic properties in nanometric CoPt films. We get benefit from the variation of the sampling depth with the grazing angle to investigate the variations of the local order within the film. In order to properly resconstruct the 3D information the experiments were performed either in the in-plane as in the out-of-plane geometries and supported by ab initio calculations. A depth dependence in the chemical order is revealed and the magnetic behavior is interpreted within this framework.

Introduction

Improvement of low-cost high-storage density magnetic media is an emerging technological challenge [1]. A clear understanding of the macroscopic magnetic properties requires a knowledge of its dependence with layer thicknesses and the complex microstructural effects frequently localized at the interface with the substrate or the surface of the films. Such effects can only be studied using experimental techniques able to peer selectively along the sample's depth. Analysis techniques based on X-ray diffraction (XRD) and giving the long range order as a function of the probed thickness have been successfully developped in the last past decades [2,3]. The versatility in the geometries and detection schemes enables the association of X-ray Absorption Spectroscopy (XAS) to advanced experimental setups, providing renewed original insights to the material science. Grazing incidence (GI) XAS is already used worldwide [4] and its sensitivity in probing in-depth by tuning the grazing incidence is becoming an explored issue [5,6].

Under specific growth conditions, the equiatomic CoPt nanometric films develop a chemical ordering along the c axis (L1 0 tetragonal phase) in a direction perpendicular to the film surface plane, giving rise to perpendicular magnetic anisotropy (PMA). Many structural parameters are determinant for the films quality. Fine characterization of the stoichiometry and the structural order is mandatory, as well as a proper determination of the surface states. Moreover, anisotropic chemical order at short range scale is sufficient to promote PMA [7,8]. A crucial issue is then to fully recover the 3D-information, either at the short range as at the long range scale.

Here we focus on CoPt films grown by magnetron sputtering following an experimental procedure promoting the growth of the L1 0 chemically ordered phase normally to the film surface [9]. The magnetic properties of the CoPt studied films exhibit singular thickness dependence. The overall perpendicular magnetic anisotropy (PMA) is very large in thin samples (<10nm), partially lost in samples with intermediate thickness and almost recovered in the larges samples (>200nm). The scope of the study was to elucidate the origin of the drop in the PMA, and specially whether this drop was due to inhomogeneous disorder or to in-depth structural heterogeneity. As the sampling depth is strongly dependent on the incident angle, near the critical angle for total reflection [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF], we used GIXAS with controlled incidence to get information about the local order depth profiles in samples with different thicknesses. XANES and XMCD measurements revealed a inhomogeneous disorder with the modification of the chemical order in the film plane [5]. This showed the capability of the method, that has been recently completed by the development of an approach based on the recursive Parrat method to quantitatively deconvolute the thickness dependencies of the XANES information [6]. Owing to the support of ab initio calculations, we will show, however, that in plane measurements are insufficient to provide an univocal reconstruction of the 3D chemical local order in CoPt films. The complete depth profile is attained by combining in plane and out of plane resolved GIXAS measurements.

Experimental and procedures

The studied 50nm thick CoPt film was grown by magnetron sputtering following an experimental procedure promoting the growth of the L1 0 chemically ordered phase normally to the film surface. Details on sample preparation and characterization of the films can be found elsewhere [9].

Grazing incidence x-ray absorption measurements were carried out using the linear polarization produced by the undulator beamline 4-ID-D of the Advanced Photon Source. The sample was placed in the center of rotation of a 8-circle diffractometer to facilitate measurements of XANES spectra as a function of the grazing angle in the geometries with the electric field aligned parallel (in-plane IP) or perpendicular (out-of-plane OP) to the surface of the film. In the series of measurements for both geometries the XANES data were collected with grazing incidence angles from 0.2 to 1 degrees with steps of 0.05 degrees. The accuracy of the grazing incidence alignment is better than 0.02 degrees, with angular motor resolution of 0.0001 degrees. The fluorescence photons from Co K edge were collected with a photodiode and double checked with a Si-drift one element SSD detector.

Ab initio simulations of XANES spectra are calculated using the code FDMNES [START_REF] Joly | [END_REF] in the multiple scattering theory within the muffin tin approximation. CoPt clusters with 6 Åradius have been considered. XANES spectra have been performed on the CoPt alloy in the chemically ordered L1 0 in two polarizations: with the electric field parallel to the L1 0 [001] direction (c-XANES ) and with this field in the (100) plane (a-XANES). The XANES in the fcc structure has been calculated using the atomic positions obtained by Monte Carlo simulations [12] on a 2nm-large cluster (400 atoms). The calculated spectra have been further convoluted with a Lorentzian shape function to account for the core-hole and final state lifetime, and the experimental resolution.

Results and discussion

The ab initio calculated c-XANES and a-XANES correspond respectively to OP and IP measurements in the ideal case of the film perfectly oriented with L1 0 chemical order normal to the film surface. The two spectra (the two upper curves, in figure 1) show significant differences. In the ordered structure L1 0 structure, Pt and Co planes alternate along the [001] axis. Pt neighbours to the central Co atom are only found in the [001] direction. Only c-XANES contain their contribution, while only Co neighbours contribute to the a-XANES signal. The difference between the two spectra are similar to those identified in a CoPt alloys with increasing Pt content [13,[START_REF] Hlil | th International Conference on X-Ray Absorption Fine Structure (XAFS14[END_REF] and can be interpreted in the same way. In the c-XANES, the increase of the white line B accounts for the drop in occupied states associated to a small charge transfer from Co 3d to Pt 5f orbitals. The presence of a splitted structure (B -C) just above the edge is a known experimental signature of fcc structure and arises from band structure effect. The splitting is found in ab initio calculations for cluster size including the fifth neighbouring shell of the fcc structure, ie , in the case of CoPt, for cluster size above 6.1A. The C structure intensity is largely independent on the short range chemical order. The A shoulder is essentially due to Co 4p band. The isotropic spectra is the weighted average of c-XANES and a-XANES contributions and corresponds to polycrystalline samples with L1 0 structure but no preferential orientation. The XANES spectra has also been calculated on the isotropic CoPt fcc phase. We note that the isotropic XANES in the ordered and disordered phase are very similar (figure 1, the two lower curves), and have the same dominant features as in a-XANES. Consequently a perfectly ordered phase L1 0 structure will only be differenciated by XAS from the fcc structure if the direction of the electric field is near the 001 direction. For polycrystals this differenciation is possible only if a dominant number of crystals has its [001] axis close to the direction of the electric field. It is then clear that XAS measurements performed along a single polarization are insufficient to give a unambigous information about the chemical order and our previous study [5] should be completed.

Figure 2a shows XANES spectra collected in the in-plane geometry. The critical angle for total reflection in the sample is about 0.51 deg at 7750 eV. At this energy the penetration depth -taken as the depth where the incident beam intensity is attenuated by a factor 1/e, -is 2 to 5nm for grazing angle below 0.5 degree and higher than 20nm for angles above 0.6 degree. The XANES spectra collected at the lowest angle are only related to the upper layer of the film, while essentially the deeper layers are probed for the largest anglesFor the lowest values of the grazing angles the relative intensity of the first main resonance(B) is higher than for the second one (C). This intensity drops for an incident angle θ comprised between 0.5 and 0.55 degrees, i.e. close to θ C . The two intensities become equivalent. This angular evolution is similar to that previously obtained. Equivalent intensities of the B and C structures are a known experimental signature of the fcc structure. In the fcc structure the B and C peaks are of almost equal intensities. Ab initio calculations show that similar relative intensities should be expected in L10 with electric field oriented along the normal to the (100) plane ( figure 1a, a-XANES and fcc). Higher B resonance is only observed for electric field oriented parallel to the L1 0 [001] axis (c-XANES). The higher B resonance at low angle should be associated to the a presence of a L1 0 phase with the [001] axis in the film plane, in the upper layers of the sample. This feature drops at higher angles, indicating that for the buried atoms -i.e. close to the substrate-this order is not kept.

At this stage, various interpretations of the results are still possible. The layer close to the substrate may be mainly chemically disordered (fcc). However it can also have the chemically ordered L1 0 structure with the [001] axis normal to the surface plane. In order to check this later possibility, a series of measurement has been performed in the OP geometry (figure 2b). It shows the same angular evolution, and a drop at the same angular value as for the IP geometry. The XANES spectra obtained for angles above 0.55 degree are identical in the two orientations. There is no anisotropy in the XANES spectra. We shown that a perfect L1 0 structure -or well textured one -would yield to anisotropy in the XANES spectra, the lack of anisotropy in the XANES spectra allows to withdraw such structural anisotropy in the buried layer. The structure of this layer is then either chemically disordered ( fcc) or formed by L1 0 grains with essentially random orientation, the technique being not able to discriminate among these two cases. Moreover we observe that the changes in the A and B features are more marked for OP measurements (figure 3). It appears that the XANES taken in the OP geometry show a larger c-XANES character: higher B structure and lower A shoulder. So we can conclude that the upper layers of the films contain ordered grains, but more grains are oriented with the c axis normal to the film plane.

The presence of a layer without structural anisotropy close to the substrate explains the lowering the PMA performance in the 50nm thick sample. The growth conditions have been optimized to promote the growth of the L10 phase oriented with c perpendicular to the film plane,-and it is indeed for thin samples (below 10nm). The growth, on the substrate of a layer without structural anisotropy, acting as a "dead layer" for the PMA, is then surprising and must probably be related to dynamic processes in the growth by magneton sputtering. However the presence of the upper L10 layer partially texturized with c perpendicular to the film plane, explains the remaining PMA in the sample. It also explains the recovering of the PMA for larger samples, when the active upper layer becomes large when compared to the "dead" buried layer.

14thFigure 1 .Figure 2 .

 12 Figure 1. Calculated XANES spectra for CoPt chemically ordered and disordered structures.

14thFigure 3 .

 3 Figure 3. XANES above and below the critical angle θ c 0.50 deg for measurements IP (a) and OP(b) .

Acknowledgements

Work at Argonne is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC-02-06CH11357.


José Penuelas

Pascal Andreazza
email: pascal.andreazza@univ-orleans.fr

Caroline Andreazza-Vignolle

Christine Mottet

Maurizio De Santis

Hélio C N Tolentino

Real-time icosahedral to fcc structure transition during CoPt nanoparticle formation

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

Introduction

Supported transition-metal nanoparticles (NPs) present growing interest in many areas of material science because of their catalytic [1], optical and magnetic properties [2]. In the nanometer scale, the particle size plays a significant role in their structures and as a consequence in their chemical and physical properties. The size-dependent evolutions of the structure and structural transitions of nanoparticles have been the subject of fundamental and applied research interests [3]. Thus, the equilibrium structure of a nanometer-sized particle may differ from the bulk. In the case of most transition metals, the atomic structure is based on the spheres packing in a densest arrangement, i.e. a perfect tetrahedral packing (face-centered cubic or hexagonal close-packed). However, ultrasmall particles with icosahedral or decahedral structures were commonly observed since the pioneer works of Ino [4] on gold. These non-crystalline structures are composed of distorted tetrahedral units where adjacent tetrahedral faces are connected by a twinning plane. The resulting morphologies favor the low energy (111)-like close-packed facets and induce large internal strains. These non-crystalline morphologies exhibit fivefold symmetry incompatible with the translational symmetry of crystalline lattices. Their stability depends on a critical balance between surface and volume contributions to the total energy and consequently decreases as the size increases. The icosahedral morphology should be the most favored at small size, while fcc morphologies like truncated octahedron are expected for larger sizes; decahedra could be favored in the intermediate range [3]. In the case of bimetallic nanoparticles, not only the size but also the composition can influence their structures [5][6][7]. The classical sequence of stability could be modified by segregation and finite size effects. In addition, kinetic processes involved during their elaboration can induce metastable structures such as large icosahedra or decahedra [8,9] as well as metastable chemical arrangements such as three-shell onionlike bimetallic NPs [10]. The existence of such particles suggests that the conditions under which their growth occurred have led to non-equilibrium structures. In addition to kinetic effects, surface contamination and contact effect with external support or matrix have been identified as factors inducing a change of the particle structure and morphology [11,12]. As a consequence, the knowledge of the NPs formation mechanisms is of prime importance since their properties are directly connected to their structure.

In this article, we focus on morphological and structural investigations of supported CoPt NPs in the ultrasmall size range 1 < diameter < 4nm. The case of CoPt system is particularly interesting due to the high structural sensitivity of the magnetic properties of CoPt equi-atomic composition nanoparticles [13,14]. We used surface sensitive x-ray scattering methods as Grazing Incidence Small-Angle X-ray Scattering (GISAXS) and Grazing Incidence X-ray Diffraction (GIXD) to follow in situ and in real-time the nucleation and growth processes. As compared to Transmission Electron Microscopy (TEM) [15] which is a technique restricted to the particle observation in a limited area, GIXD and GISAXS techniques probe complementarily the order and morphology at the atomic scale of nano-objects on a statistical average over the whole sample. In addition, due to the high surface to volume ratio and the high reactivity of Co, in situ analysis under ultrahigh vaccum (UHV) is absolutely required to prevent an influence of the environment on the morphology or structure of nanoparticles , even in the case of clusters protected by a coating or embedded in a matrix. The real-time aspect of the experiment is of particular interest to study dynamical processes during each stage of the growth (nucleation, growth and coalescence). Many studies by small-and/or wide-angle scattering have been done for monometallic clusters as Au [16][17][18] or Ag [19]. To our knowledge few papers present bimetallic NPs x-ray scattering investigations in real-time and in situ conditions. We report in this paper the temperature effect during the growth, in term of atoms and particles mobility and in term of structural evolution. The choice of a high growth temperature (500˚C) was motivated by the interest to enhance atoms mobility and consequently the structural rearrangement. A structural analysis method based on the Debye equation is presented coupled with cluster model calculations. These Monte Carlo simulations of relaxed CoPt cluster structures (disordered truncated octahedral, decahedral and icosahedral) were performed using a semi-empirical tight-binding potential to interpret diffraction spectra and structural transition. Through these investigations, we identify specific morphologies and structures and follow the size-dependent and temperature-dependent evolutions of the chemically disordered CoPt NPs.

Experimental

The experiments were performed at BM32 beamline located at the European Radiation Synchrotron Facility (ESRF) using the UHV setup allowing molecular beam epitaxy growth combined to in situ and in real time GISAXS and GIXD measurements [20,21]. Figure 1(a) shows a schematic view of the experiment geometry. The two-dimensional GISAXS pattern was recorded using a CCD camera mounted at a sample-detector distance of 1m. Due to the much higher scattering intensity at q = 0 because of the specularly reflected beam and the diffuse reflectivity in comparison to the GISAXS signal at finite values for q , both tungsten beam-stop and aluminum absorber foils were used in order to optimally match the measured intensity with the given dynamic range of the CCD detector. A standard scintillator detector, which could be positioned in the scattering direction (fixed exit angle α f = α i and scan angle δ), was used for the GIXD intensities measurements. This GIXD geometry ensures that the length of the beam path within the sample, which is relevant for absorption, is constant during the measurements. In both cases, due to the low size range of particles and low quantities of deposited metal, the x-ray grazing incidence angle were selected in order to reduce the scattering background from amorphous substrate.

Co and Pt atoms were co-deposited by condensation in the UHV chamber using electron beam evaporation on prepared substrate. The base pressure was 1 × 10 -10 mbar, this value increased to 3 × 10 -10 mbar during the deposition. The substrates used for cluster condensation were amorphous SiO 2 coated with amorphous carbon (a-C) layer of 10nm. These substrates were chosen to limit the cluster-substrate interactions and to allow comparisons with our previous TEM results obtained on a-C deposited nanoparticles [15]. Substrates were degassed at 650˚C with a ramp rate of 10˚C/min to vaporize water and some organic species. Co and Pt deposition rates were calibrated to obtain a global composition close to Co 50 Pt 50 with constant flux during all growth stages. These rates (0.26 × 10 15 atoms/cm 2 /hour) were selected as weak to limit the kinetic effects on the bimetallic NPs during the growth. The temperature fixed to 500˚C was calibrated by a thermocouple and a pyrometer during the deposition.

Results and discussion

Modeling and analysis procedure

The morphological parameters of the nanoparticles as diameter D, height H and correlation distance between particles Λ were extracted from GISAXS experimental data by comparison with simulations using IsGISAXS software [22] in the Distorded Wave Born Approximation. The GISAXS intensity of an assembly of islands is proportional to the form factor of the nanoparticles and the interference function, which is the Fourier transform of the NP position autocorrelation function. For a size distribution system as observed in TEM [Fig. 1(b)], the local monodisperse approximation could be applied in the simulation, then the total scattered intensity was calculated as the incoherent summation of the intensities of domains with monodisperse size. For the form factor, the best results were obtained with a spherical shape that can be truncated by taking an aspect ratio H/D < 1. A narrow Gaussian size distribution with constant relative geometrical standard deviation σ(D)/D = 0.3 according to our previous TEM results [15] was used for each simulation. In order to analyze 2D patterns [see Fig. 2(a)], two cuts of the intensity in the q and q ⊥ directions (determined by δ and α f , respectively) are simultaneously fitted to determine the evolution of the dimensional parameters [19]. Preliminary to the GISAXS analysis, we performed ex-situ TEM observations on a sample prepared in the same UHV set-up and with same deposition parameters (Fig. 1(b)). The substrate was a self-supported amorphous carbon membrane prepared in the same conditions than the top layer of the silica substrate. Statistical measurements gave an average diameter of 2.5nm with a Gaussian size distribution in accordance with selected parameters given above.

The diversity of particle structures obtained in nanometer scale as described in the Introduction don't allow a simple identification from diffraction data as in a bulk material. One alternative is to calculate the wide angle scattering pattern of predicted finite structures of nanoparticles in a size range deduced from GISAXS data analysis. The scattering intensity for a model cluster is calculated quantitatively on the basis of the Debye equation [START_REF] Warren | X-Ray Diffraction[END_REF][START_REF] Reinhard | [END_REF]26] from a list of the coordinates of all atoms (Co and Pt types) contained in each kind of structure. Polarization evolution effects, preliminary calculated, were neglected due to their weak evolution with q. We have used three types of structures in our cluster models: truncated octahedral (TOh) based on face-centered-cubic structure, decahedral (Dh) and icosahedral (Ih) according respectively with Wulff [27], Marks [28] and Ino [4] constructions. To determine the atomic structure of our samples, we used either single size and single cluster type to fit the diffraction patterns, or a weighted sum of the intensities from several sizes or structures [START_REF] Reinhard | [END_REF]. Models composed of clusters with complete shells were found to be sufficient for data analysis in a size range between 50 and 2000 atoms according to GISAXS results. Monte Carlo (MC) simulations of equiconcentration CoPt cluster structures (TOh, Dh and Ih illustrated in the insets of Fig. 4) were performed using a semi-empirical tight-binding potential [29]. The models are relaxed at a temperature (between 410 and 550C) close to the experimental value (500˚C). Earlier studies [30] have shown that the use of pure geometrical structures induces significant changes in the scattering pattern as important as particle size decreases. Figure 3(a) illustrates this tendency in the case of a fcc TOh model of geometric and MC relaxed nanoparticles of 405 atoms (around 2nm in size). The variation of atom positions in the simulated structure with respect to geometric model is due to structural and thermal relaxations that must be taken into account in the case of NPs. In addition, Figure 3(b) shows the evolution of broad diffraction peaks located at Bragg angles of TOh fcc crystalline models, which become sharper with increasing cluster size (in the 1.5-3nm size range). Finally, the wide-angle scattering profiles of the three different structures (TOh, Dh and Ih) for a similar size (2.5nm) are clearly distinguishable (see Figure 3(c)) and thus, allow the discrimination between the different models [START_REF] Reinhard | [END_REF]. The diffraction technique is rather sensible to the symmetry than to the size or to the local atomic distortions.

GISAXS and GIXD analysis

Real-time GISAXS measurements were performed during the deposition on amorphous carbon surface heated at 500˚C with a frame interval of 30 minutes. Figure 2 shows the representative evolution of 2D patterns which contain informations on the size, the shape and the correlation distance of the nanoparticles. The 2D GISAXS patterns present two lobes of scattered intensity which are already observed from one hour of deposition. However, a quantitative analysis was possible only from 2h of deposition time. The progressive shift of the maximal intensity position to the low q values and the lobe extension (Fig. 2(a-c)) indicate an evolution of the form factor and of the interference function, simultaneously. From this evolution with time, we deduced the evolution of D and H shown in Figure 2(d). These parameters increase with the deposition time according to theoretical predictions in the case of Volmer Weber growth. The NPs diameter and height (D,H) of truncated sphere shape is found to increase from (2nm,1.4nm) at 2h deposition time to (3.2nm,3.1nm) at the end of the deposition experiment (Fig. 2(d)). The faster increasing of height compared to the diameter is classically observed for high coverage rate on the substrate surface, when a direct capture of atoms on the NPs top is favored [31]. In our case, the coverage rate remains low (15-20%). Thus, this shape evolution from a flat to isotropic shape is certainly due to the high substrate temperature. This temperature favor the atom mobility in the clusters which leads to an equilibrium morphology as the experiment duration increase. Concerning the mean inter-particle distance Λ, the value remains quasi constant from 4h deposition time until the end of the experiment. At earlier stages of the growth, the results suggest that after a nucleation stage corresponding to a Λ decreasing, a dynamic coalescence occurs corresponding to the increasing of the mean inter-particle distance (between 2 and 4h).

Simultaneously with GISAXS investigations, diffraction patterns were obtained during the CoPt deposition (Fig. 4(a)) showing the scattering contribution of the nanoparticles. The "background" pattern displays mainly the scattering contribution of the amorphous substrate. To obtain the scattered signal due to the deposited metal, the data were normalized (see examples in Fig. 4(b)) taking into account the absorption effect in the substrate and the background subtracted [32]. Thus, the resulting metal contribution (see fig. 4(c-f))) can be compared with simulated GIXD spectra to provide a fit of the CoPt experimental patterns. For the highest deposition times (Fig. 4(a)), the diffraction patterns features are in agreement with the different peaks of the fcc structure with an interatomic distance of 2.65 Å, close to the A1 CoPt bulk value, proving the alloying of Co and Pt. This experimental value is consistent with the MC simulated model (2.68 Å). Indeed, this latest value correspond also to the interatomic distance obtained by Monte Carlo simulation in bulk phase with the same interatomic potential [29]. At the early stages of the deposition, the results are not in agreement with a fcc crystalline structure but rather with a non-crystalline atomic arrangement. For the medium time deposition (5h), Fig. 4(c),(d) and (e) show the superposition between calculated and experimental According to semi-empirical calculations [3], on pure Pt and Co metal clusters, as sizes increase the icosahedral range is dropped in favor of decahedral structures, and approaching to the bulk limit the fcc structure (TOh) becomes the most favorable. In the case of pure Pt clusters, for example, stable decahedra cover a size range for which lower limit is placed at N ∼ 75. In the case of pure Co, the decahedral range starts at N ∼ 400. In the CoPt disordered equi-atomic case [29], the calculation results indicate that the Dh range begins at N ∼ 100. As regards to the transition from the Dh to fcc motif, calculations suggest that decahedra and fcc models are competing already at N ∼ 600, so the transition could take place in the 600 < N < 1000 size range. Our results showing a structural evolution with the NPs size, are consistent with these tight binding simulations but without Dh structure formation in a intermediate size range, as previously observed in pure cluters [25]. However, the icosahedral structure observed at the early stage of NPs formation shows a Ih stability at larger size N ∼ 400 than predicted by equilibrium calculations. Our experiments were performed with a low deposition rate, but nevertheless during the deposition under non equilibrium conditions. In addition, the GISAXS results concerning the evolution of the shape and interparticle distance suggest a coalescence process during the early stage of the growth. Such events can affect the structure by making highly defective clusters, with morphological disruption and large number of stacking faults [33] to allow the NPs transformation to the minimum energy morphology, i.e. the TOh shapes at larger sizes [34]. After the coalescence stage, the fcc resulting morphology was locked in and the size increase is dominated by atom by atom growth mechanism.

Conclusion

To conclude, we have shown that combined GISAXS and GIXD techniques can allow a fine morphological and structural analysis of CoPt nanoparticles in the 1-4nm size range. However, these experiments must be supported by wide angle scattering calculations using cluster models. Thus, Monte Carlo simulations of relaxed CoPt cluster structures (disordered truncated octahedral, decahedral and icosahedral) were performed using a semi-empirical tight-binding potential, to interpret diffraction pattern. Real time scattering experiments were performed during the formation of supported equi-atomic composition CoPt nanoparticles obtained by UHV deposition. Our results show that in the nanoparticles assembly, icosahedral structure dominates at early stage (small particle sizes) of the growth process and fcc structure appears at larger sizes. The structural change occurs because of competition between surface and volume energy components as predicted by MC simulations. However, this structural transition was favored by the coalescence between small icosahedral nanoparticles.

Fig. 1 .

 1 Fig. 1. (a) Schematic drawing of the scattering measurements geometry. Monochromatic x-ray beam of 18 keV were irradiated on the sample at a grazing angle αi of 0.07˚. The scattered intensity is recorded as a function of the in-plane angle α f with respect to the surface plane and of the in-plane angle δ. These angles allow defining the reciprocal space coordinates q ⊥ and q , respectively perpendicular and parallel to the sample surface. (b) TEM micrograph of NPs CoPt sample annealed at 500˚C. The inset displays the size distribution histogram.

Fig. 2 .

 2 Fig. 2. Measured GISAXS patterns of sample grown at 500˚C at different growth times: (a) 3, (b) 6 and (c) 9 hours. The white dotted line is a visual guide line to show the shift of the interference function maximum. (d) graph shows the evolution of diameter D, height H and correlation distance between particles Λ, extracted from GISAXS experimental data (q ⊥ and q cross sections).

Fig. 3 .

 3 Fig. 3. Calculated GIXD spectra of (a) geometric and Monte Carlo relaxed (TOh) truncated octahedron CoPt clusters of 405 atoms (2nm),(b) a series of CoPt TOh clusters of increasing size, 201, 405, 807 and 1289 atoms corresponding to 1.5, 2, 2.5 and 3nm in diameter, (c) TOh, decahedral (Dh) and icosahedral (Ih) models of clusters around 2.5nm in diameter. The insets shows the snapshots of different MC simulated clusters (Co and Pt atoms are represented as light and dark spheres respectively).

Fig. 4 .

 4 Fig. 4. (a) Evolution of the GIXD patterns as a function of time (t in hours) from 1 to 10h. (b) Three examples of normalized patterns compared with background obtained at 500˚C before the deposition. (c), (d), (e) comparison between NPs contribution experimental pattern at 5h deposition time and calculated patterns of relaxed Ih, Dh and TOh models, respectively. (f) best fit obtained with a mix of Ih and TOh models


Franc ¸ois Emilie Franceschini
email: franceschini@lma.cnrs-mrs.fr

T H Yu

Guy Cloutier

Ultrasound characterization of aggregated red blood cells: towards in vivo application

Ultrasonic backscattered signals from blood contain frequency-dependent information that can be used to obtain quantitative parameters reflecting the aggregation state of red blood cells (RBCs). The difficulty to use this frequency-dependent information in vivo is due to the attenuation caused by intervening tissue layers that distorts the spectral content of backscattering properties from blood microstructures. We propose an optimization method to simultaneously estimate tissue attenuation and blood structure factor. In an in vitro experiment, we obtained satisfactory estimates with relative errors below 25% for attenuations between 0.115 and 0.411 dB/cm/MHz and D <10 (D the aggregate diameter expressed in number of RBCs).

Introduction

For the detection and characterization of tissues, quantitative ultrasound techniques using the radio frequency (rf) backscattered signals have received broad interest for the past 30 years. One approach is to use the magnitude and the frequency dependence of the rf backscatter spectrum in order to quantify the tissue structures such as the size, acoustic impedance, and concentration of the scatterers. Many in vitro and in vivo experiments have been performed to demonstrate the utility of this approach for characterizing of eye, liver, kidney, prostate and breast. Recently, the frequency dependence of the ultrasound backscattering coefficient (BSC) was studied to assess the level of red blood cell (RBC) aggregation. Two parameters describing RBC aggregation, the packing factor and mean aggregate diameter, were estimated from the Structure Factor Size Estimator (SFSE). [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] The difficulty to use the SFSE in vivo is that the spectral content of backscattered signals is also affected by attenuation caused by intervening tissue layers between the probe and the blood flow. More generally, ultrasound scatterer size estimation techniques for tissue characterization (such as liver, kidney, prostate or breast) has had limited success in clinical practice because of tissue attenuation effects. 2,3 Some groups [START_REF] Roberjot | Absolute backscatter coefficient over a wide range of frequencies in a tissue-mimicking phantom containing two populations of scatterers[END_REF][START_REF] He | Application of stochastic analysis to ultrasonic echoes -Estimation of attenuation and tissue heterogeneity from peaks of echo envelope[END_REF][START_REF] Oosterveld | Ultrasound attenuation and texture analysis of diffuse liver disease: methods and preliminary results[END_REF] developed measurement methods to evaluate the frequencydependent attenuation in order to compensate a posteriori the backscattered power spectrum. Our goal is to explore an other strategy for in vivo measures of RBC scatterer sizes. Recently, we propose an optimization method providind an estimate of total attenuation and blood structural parameters simultaneously, termed the Structure Factor Size and Attenuation Estimator (SFSAE). [START_REF] Franceschini | Simultaneous estimation of attenuation and structure parameters of aggregated red blood cells from backscatter measurements[END_REF] This method consists in fitting the spectrum of the backscattered radio-frequency (rf) echoes from blood to an estimated spectrum by a modified SFSE model. Herein, the SFSAE is improved and in vitro experimental evaluation of the SFSAE is performed.

Background: Structure Factor Size and Attenuation Estimator

The technique allowing to estimate simultaneously blood structural parameters and total attenuation has been described in detail in Ref. 6 and is summarized here.

Blood can be considered as a very dense suspension of particles (i.e. red cells) having strong interactions (collision, attraction, deformation, flow dependent motions). We develop a theoretical model of blood ultrasound backscattering based on the particle approach. [START_REF] Mo | Theoretical models of ultrasonic scattering in blood[END_REF][START_REF] Twersky | Low-frequency scattering by correlated distributions of randomly oriented particles[END_REF] Thsi approch consists of summing contributions from individual RBCs and models the RBC interaction by a particle paircorrelation function. The model proposed in Ref. 1 has been modified to predict the theoretical backscatter coefficient from blood:

6 BSC theor (k) = mσ b (k)S(k)A(k) ( 1
)
where k is the wave vector, m is the number density of RBCs in blood estimated by measuring the hematocrit H by microcentrifugation (m = H/V s , where V s is the volume of a RBC), σ b is the backscattering cross section of a single RBC, S is the structure factor describing the spatial organization of RBCs, and A is the frequency-dependent attenuation function. The backscattering cross-section σ b of a weak scattering particle small compared to the wavelength (Rayleigh scatterer) can be determined analytically as follows:

σ b (k) = 1/(4π 2 )k 4 V 2 s γ 2 z
, where γ z is the variation of impedance between the RBC and its suspending medium (i.e. the plasma). The structure factor S is the Fourier transform of the pair-correlation function 8 g and is approximated by its second-order Taylor expansion 1 in k as

S(k) = 1 + m (g(r) -1) e -2jkr dr ≈ W - 12 5 (kR) 2 . (2)
In this last equation, W is the low-frequency limit of the structure factor (S(k)| k→0 ) called the packing factor. 8,9 R is the radius of 3D RBC aggregates assumed to be isotropic. We introduce D = R/a as the isotropic diameter of an aggregate (expressed in number of RBCs) with a the radius of one RBC sphere-shaped model of volume V s . The attenuation function A is given by: A(k) = e -4α 0 f , where f is the frequency and α 0 is the attenuation coefficient (in dB/MHz) defined by: α 0 = i α i e i , where α i and e i are respectively the intervening tissue layer attenuations (in dB/cm/MHz) and thicknesses. We thus assume that the attenuation increases linearly with the frequency: α(f ) = α 0 f . The measured backscatter coefficient was computed as

BSC meas (k) = BSC ref (k) P meas (k) P ref (k) . (3)
In Eq. (3), the mean backscattered power spectrum P meas was obtained by averaging the power spectra of 20 backscattered echoes from blood. The mean power spectrum P ref was obtained from a reference sample of non-aggregated RBCs at a low hematocrit of 6% (i.e. Rayleigh scatterers). 1 0 In this case, 20 echoes were also averaged. This reference sample was used to compensate the backscattered power spectrum P meas for the electromechanical system response, and the depthdependent diffraction and focusing effects caused by the US beam.

The packing factor W , aggregate diameter D and total attenuation along the propagation path α 0 were determined by matching the measured BSC meas given by Eq. ( 3) with the theoretical BSC theor given by Eq. ( 1). For this purpose, we searched values of W , D and α 0 minimizing the cost function F (W, D, α 0 )=||BSC meas -BSC theor || 2 . In all studied cases, the cost function seemed to have a unique global minimum, as was observed by plotting the cost function surface F (W, D) with varying values of α 0 (see Fig. 1 in Ref. 6). For the optimization problem, we defined a set of lower and upper bounds on the variables (W , D, α 0 ) so that the solution is searched in the range: 0≤ W ≤100, 0≤ D ≤50 and 0≤ α 0 ≤1 dB/MHz. We chose to reject the solution (W , D, α 0 ) having an estimated diameter D very small compared to 0.1.

In vitro experiment in a Couette flow device

Blood prepation

Fresh porcine whole blood was obtained from a local slaughter house, centrifuged and the plasma and buffy coat were removed. Two blood samples were then prepared: (i) a H6 reference sample, which was a 6% hematocrit non-aggregating RBCs resuspended in saline solution; and (ii) a 40% hematocrit T40 test sample, which consisted of RBCs resuspended in plasma to promote aggregation.

Experimental set up

US measurements were performed in a Couette flow system to produce a linear blood velocity gradient at a given shear rate. The schematic configuration of the experience is shown in figure 1.

The system consisted of a rotating inner cylinder with a diameter of 160 mm surrounded by a fixed concentric cylinder of diameter 164 mm. A 60 mL blood sample was sheared in the 2 mm annular space between the two coaxial cylinders. The US scanner (Vevo 770, Visualsonics, Toronto, Canada) equipped with the RMV 710 probe was used in B-mode. The single-element focused circular transducer had a center frequency of 25 MHz, a diameter of 7.1 mm and a focal depth of 15 mm. We operated at a sampling frequency of 250 MHz with 8 bits resolution (Gagescope, model 8500CS, Montreal, Canada). The probe was mounted in the side wall of the fixed outer cylinder and was positioned to have its focal zone at the center of both cylinders. To ensure ultrasonic coupling, the hole within the outer stationary cylinder (containing the probe) was filled with a liquid agar gel based mixture. When solidified, this gel was cut to match the curvature of the cylinder to avoid any flow disturbance. The gel was a mixture of distilled water, 3% (w/w) agar powder (A9799, Sigma Chemical, Saint-Louis, MO), 8% (w/w) glycerol and a specific concentration of 50 µm cellulose scattering particles (S5504 Sigmacell, Sigma Chemical, Saint-Louis, MO) that determined the attenuation coefficient. Five experiments were performed with five mixtures having Sigmacell (SC) concentrations varying from 0% to 1% (w/w). Since skin is one of the most attenuating tissue layers during in vivo scanning, phantoms were prepared in order to have attenuations closed from skin attenuation. The 0% concentration constituted the non-attenuating gel and the four other mixtures mimicked skin attenuations.

Attenuation measurements

The attenuation coefficients of the reference (0% SC) and of the others skin-mimicking phantoms α sc were determined by using a standard substitution method. A transducer with center frequency of 25 MHz (Vevo 770, Visualsonics, Toronto, Canada) was used in transmission/reception with a reflector on the opposite side of the phantom for reflection measurements. Reflected signals were recorded both with and without the agar gel sample in the acoustic path. The attenuation coefficient was then estimated using a log spectral difference technique. For each concentration of SC, six regions were scanned for averaging purpose. Thicknesses of skin-mimicking phantoms e sc were fixed to 10 mm. As shown in the table 1 summarizing results, attenuation coefficients of skin-mimicking phantoms were in the same range as the human dermis (0.21 dB/MHz at 14 -50 MHz considering a 1 mm dermis thickness [START_REF] Raju | High-frequency ultrasonic attenuation and backscatter coefficients of in vivo normal human dermis and subcutaneous fat[END_REF] ).

Attenuations of the blood α blood sheared at different shear rates were also measured in the reflection mode using the same experimental configuration shown in figure 1. The gel had a 0% SC concentration and the rotating inner cylinder was used as the reflector. Table 2

Measurement protocol

Prior to each measurement, the T40 blood was sheared at 200 s -1 during 30 s to disrupt RBC aggregates. The shear rate was then reduced to residual values of 5, 10, 20, 30 and 50 s -1 during 90 s to reach an equilibrium in the state of aggregation in the sheared blood sample. After that, for each shear rate, 20 B-mode images were constructed for 80 s. For each line of the B-mode images, echoes were selected with a rectangular window of length 0.4 mm at twenty depths every 0.04 mm (i.e. with 90% overlap between windows). For each depth, the power spectra of the backscattered echoes were averaged over 20 acquisitions (corresponding to the 20 acquired images) to provide P meas . This protocol was repeated five times with the five agar-based phantoms. Then, the T40 blood was removed and the H6 sample was introduced in the Couette device. The H6 sample was sheared at 50 -1 and coupled with the 0% SC concentration agar gel. Echoes were windowed as for the H40 sample at the same depths and their power spectra were averaged over 20 acquisitions to obtain P ref .

Reference measurements with the 0% SC concentration phantom

The experiment with the 0% SC phantom was realized in order to have reference results on packing factors W ref and aggregate diameters D ref obtained from the classical SFSE. [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] These parameters were assumed to be true values of packing factors and aggregate diameters for all shear rates, and will be compared in the next section with packing factors and diameters estimated by the SFSAE and by the SFSE when skin-mimicking phantoms are used.

It is important to emphasize the fact that the H6 reference sample was also measured with the 0% SC phantom. The phantom attenuation, although small with no SC, therefore affected equivalently both spectra P meas and P ref in Eq. ( 3). The resulting measured backscatter coefficient BSC ref was thus not biased by attenuation. .sr -1 ) 10

Results

Reference parameters with the SFSE

Parameters evaluated with the SFSAE

Typical results of the SFSAE minimization procedure for the different agar phantoms at a shear rate of 10 s -1 and 50 s -1 are given in Fig. 3. For both shear rates, it can be observed that more the total attenuation increases, more the backscattering coefficient amplitude decreases at all frequencies and more the frequency dependence of the backscattering coefficient changes. One can also notice that the parameters W and D from the SFSAE are very similar to the reference parameters W ref and D ref , as well as the total attenuation α 0 from the SFSAE similar to the reference total attenuation. The reference total attenuation α ref corresponds to α sc e sc + α blood e blood where α sc and α blood are the skin-mimicking phantom attenuation and the blood attenuation estimated in the reflection mode as shown in section 3.3.

For each residual shear rate, parameters W , D and α 0 were estimated by the SFSAE. Figure 4 summarizes these results. In this figure, the relative errors for each parameter correspond to:

(W -W ref )/W ref , (D -D ref )/D ref and (α 0 -alpha ref )/α ref .
Except for the shear rate 5 s -1 with the skin-mimicking phantom having the smallest attenuation (0.25% SC), the SFSAE gave quantitatively satisfactory estimates of W and D with relative errors below 25%.

Parameters evaluated with the SFSE with compensation for blood attenuation

The packing factor W comp and the diameter of the aggregates D comp were also evaluated by compensating the backscatter coefficients in the SFSE with the value measured in reflection. Results are presented in Fig. 5. The relative errors are below 25% for all shear rates and all skinmimicking phantoms.

Conclusions

The accuracy of the estimates obtained with the SFSAE was as satisfactory as those obtained with the SFSE with attenuation-compensation (i.e when a priori are known about the attenuation). For both methods, relative errors for W and D were below 25%, except for one value corresponding at the shear rate 5 s -1 with the skin-mimicking phantom having the smallest attenuation (0.25% SC). In this last case, the SFSAE gave less accurate estimates (relative errors around 50% for W and α 0 ). The SFSAE seems to reach its limit of applicability for large aggregate sizes: typically D ref = 10.11 (i.e. kR=2.8)

Nevertheless, the SFSAE has the major advantage to be easily applicable in vivo because of the simultaneous estimation of the blood structural properties and total attenuation (contrary to the SFSE attenuation-compensation method, needing the attenuation and thickness of the tissue intervening layers to be known).

Figure 1 :

 1 Figure 1: The Couette flow system

Figure 2

 2 Figure 2 reports results on W ref and D ref from the SFSE with compensation for blood attenuation in the case of no gel attenuation. It can be observed that the amplitude of the backscattering coefficient as well as the estimation of the parameters on W ref and D ref decrease when the shear rate increases (i.e. the level of aggregation becomes smaller).

Figure 2 :

 2 Figure 2: Backscatter coefficients for blood sheared at different residual shear rates and measured with the 0% SC concentration phantom, and corresponding fitting with the classical SFSE with no compensation for attenuation.

Figure 3 :

 3 Figure 3: Backscatter coefficients for blood sheared at 10 s -1 and 50 s -1 , and measured with each of the five phantoms. The corresponding fitted models are the SFSE for the 0% SC phantom, and the SFSAE for the four other skin-mimicking phantoms (0.25, 0.5, 0.75 and 1% SC).

Figure 4 :Figure 5 :

 45 Figure 4: (a) Values of W , D and α 0 (in dB/MHz) for different residual shear rates estimated by the classical SFSE for the 0% SC concentration and by the SFSAE for the four skin-mimicking phantoms. (b) Corresponding relative errors.

Table 1 :

 1 summarizes results. Estimated values of the attenuation coefficients of the reference (0% SC) and of the others skin-mimicking phantoms (using a log spectral difference technique)

	SC Sigmacell attenuation α sc
	(%)	(dB/MHz)
	0.25	0.115
	0.5	0.219
	0.75	0.320
	1	0.412

Table 2 :

 2 Estimated values of the blood attenuation sheared at different shear rates in the Couette

	flow device (using a log spectral difference technique)
	Shear rates Blood attenuation α blood
	(s -1 )	(dB/MHz)
	5	0.053
	10	0.036
	20	0.024
	30	0.016
	50	0.013

Acknowledgments

This work was supported by the Canadian Institutes of Health Research (grants ♯MOP-84358 and CMI-72323), by the Heart and Stroke Foundation of Canada (grant ♯PG-05-0313), and by the National Institutes of Health of USA (grant ♯RO1HL078655). Dr Cloutier is recipient of a National Scientist award of the Fonds de la Recherche en Santé du Québec. We are also thankful to Dr F. Destrempes for his helpful discussion on the optimization tool.


Mostapha Diss
email: mostapha.diss@unicaen.fr

Vincent Merlin
email: vincent.merlin@unicaen.fr

ON THE STABILITY OF A TRIPLET OF SCORING RULES *

Keywords: Self-selectivity, Stability, Instability, Scoring Rules, Impartial Culture Assumption

When choosing a voting rule in order to make subsequent decisions, the members of a committee may wish this rule to be self-selected when it is the object of a choice among a menu of different possible voting rules. Such concepts have recently been explored in Social Choice theory, and a menu of voting rule is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this paper the menu constituted by the three well-known scoring rules {Borda, Plurality, Antiplurality}. Under the Impartial Cultures Assumption (IC), which proposes an a priori model to estimate the likelihood of the profiles, we will derive a probability for the stability of this triplet of voting rules.

Introduction

Since long, Social Choice theory has tried to study and model the construction of collective choices upon individual choices. The applications of this theory are widely spread since the actual options may be social states, political candidates, investments, etc. But the choice always concerns alternatives (political candidates for example) and, until recently, little attention has been paid to the choice of the voting rule itself. [START_REF] Koray | Self-selective social choice functions verify arrow and gibbardsatterthwaite theorems[END_REF], [START_REF] Koray | Charactrization of self-selective social choice functions on the tops-only domain[END_REF], [START_REF] Barberà | Choosing how to choose: Self-stable majority rules and constitutions[END_REF], [START_REF] Barberà | Self-selection consistent functions[END_REF] and [START_REF] Houy | De la stabilité des procédures et constitutions en théorie du choix social[END_REF] have been the first to tackle that question and enrich, thereby, the traditional field of the Social Choice Theory. The individuals are thus able to choose how to choose and compare the voting rules on the basis of 1 their properties. These authors show how that enrichment allows to introduce new positive axioms and new possibility and impossibility results.

One of the new principle introduced by this literature is the notion of selfselectivity. Consider a set of possible voting rules. A voting rule is self-selective (or self stable) if there is no alternative rule that beat the given voting rule if the given voting rule is used to choose between the rules. Let us try to illustrate this concept through some examples. Imagine that the votes are made according to the majority rule: when there is a majority favorable to the majority voting rule, the majority decision is self-selective. On the other hand, if the dictator is favorable to the majority voting rule, dictatorship is not self-selective. This exactly what happened in Spain: the first decision imposed by the King Juan Carlos was to get rid of his dictatorship in order to establish a democracy. The issue is then to know whether self-selective voting rules exist.

Though Barberà and Jackson have applied these ideas to the set of quota rules, until now, no paper has tried to apply this concept to a set of famous voting rules, the scoring voting rules. The objective of this paper is to consider the triplet that forms the three most well known scoring rules: the Borda method (B), the Plurality rule (P) and the Antiplurality rule (A). In this paper, we will also depart from the consequentialism hypothesis which have been assumed by most of the authors, Nicolas Houy being sole exception. Thus, individual will judge and evaluate the scoring rules on the basis of their intrinsic properties, without taking into account the fact that the choice of a voting rule may be instrumental for a voter, in order to increase the chances of the alternatives he prefers.

Though the previous examples have illustrated the notion of self-selectivity for a given voting rule, the concept becomes more subtle when it comes to judge the stability of a menu of possible decision rules. We will consider the definition proposed by [START_REF] Houy | De la stabilité des procédures et constitutions en théorie du choix social[END_REF]: A set of voting rules is (weakly) stable if there always exists at least one self-selective rule at any profile. Notice however that stronger definitions can be adopted, taking into account the fact that the dynamic of designations should always drive the results to a self-selective voting rules (for more details on the different concepts of stability, see [START_REF] Houy | De la stabilité des procédures et constitutions en théorie du choix social[END_REF]).

Even with this definition of stability, it is not sure whether any set of rules is stable: In fact, Nicolas Houy proposed necessary and sufficient conditions for the stability of a set of neutral 1 rules. However, the conditions he proposed are very general, and he did not seek to apply his concepts to a set of "classical" rules, like the scoring rules, which are not only neutral, but also anonymous 2 We will first show that the triplet {Borda, P lurality, Antiplurality} may not be stable at some preference profile. Next, we will evaluate the probability that the triplet {Borda, P lurality, Antiplurlity} is stable in a very weak sense: at each profile, there should at least one self-selective rule.

The likelihood of stability of this voting triplet necessarily depends upon the hypothesis that are set upon the behavior of the voters. We will assume that each individual have the same probability to pick any of the possible preference types. This hypothesis is known as the Impartial Culture model (IC) in Social Choice literature. We will outline a more concise definition of that hypothesis in section 2.

Regarding the technical aspects, we will use a method based upon the Schläfli's formula in order to derive the desired probabilities. This technique has been widely used in various articles these last years. See [START_REF] Saari | The likelihood of dubious election outcomes[END_REF], [START_REF] Tataru | On the relationship of the condorcet winner and positional voting rules[END_REF] and [START_REF] Merlin | L'agrégation des préférences individuelles: les règles positionelles itératives et la méthode de Copeland[END_REF]al. (2000, 2002).

The paper will be organized as follows: After reminding some basic definitions dealing with preferences, scoring rules, and probability models in Section 2, we will go through the different aspects of the stability of the {Borda, Plurality, Antiplurality} triplet and the different cases in Section 3. Section 4 is devoted to the computations. Finally section 5 presents our conclusions.

Notations and Definitions

The Preferences

Throughout all the paper, we will suppose that every voter can rank the set of candidates (here, voting rules) without ties. Preferences will then be linear orders. We will denote by abc the fact that candidate a is preferred to candidate b who is himself preferred to candidate c. In addition to that we will suppose that the voters are honest, i.e. they reveal their true preferences.

We will assume that the set X = {Borda, Plurality, Antiplurality} is the menu proposed to the n individuals. The 3! = 6 types (with strict and transitive preferences) of preferences are displayed in Table 1 and enumerated from 1 to 6, n i being the number of agents having the preference type number i:

B (n 1 ) ABP (n 2 ) BAP (n 3 ) BP A (n 4 ) P BA (n 5 ) P AB (n 6 )
The existence or not, of a self-selective rule will thus be determined by the study of the situation represented by the vector ñ = (n 1 , ..., n 6 ) which describes the number of electors according to their preferences, with 6 i=1 n i = n. The set of possible profiles for a size n population is then:

D n = {ñ = (n 1 , ..., n 6 ), 6 i=1 n i = n}
As we may consider variable electorates, we will define the voting rules on:

D = ∪ ∞ n=1 D n

Scoring Rules

A voting rule will is thus a mapping from D into 2 X \ ∅. At this stage, we assume that a voting rule is a social choice correspondence, that is, it associate to each profile at least one (winning) alternative. A scoring voting rule is a social choice correspondence which depends upon the positions the alternatives occupy in the preference orderings of the individuals taking part in the collective decision. In other terms, each individual gives P 1 points to the candidate, whom he considers first, P 2 to the next, until P m points for the last candidate, providing that P 1 ≥ P 2 ≥ ... ≥ P m and P 1 > P m . The candidate who has received the greatest amount of points over the whole population is elected. Our work will only be dealing with the case where three options are considered, i.e. m = 3. Hence, for any -1 ≤ s ≤ 1, let us consider the following family of point vectors W s = (2, 1 + s, 0); This family describes all the possible scoring rules for the case of 3 options. In particular, W s is the Plurality rule for s = -1, the Borda rule for s = 0 and the Antiplurality rule if s = 1. In fact, We will study the stability of the triplet {B, W -s , W s }, that will allow us to get easily results for the triplet {Borda, P lurality, Antiplurality} for the case s = 1.

With the preferences described in Table 1 and a vector ñ = (n 1 , n 2 , n 3 , n 4 , n 5 , n 6 ), the application of the Borda rule, defined by the vector W s = (2, 1, 0), on the triplet {Borda, P lurality, Antiplurality} gives us the following scores:

   S(W 0 , ñ, A) = 2(n 1 + n 2 ) + (n 3 + n 6 ) S(W 0 , ñ, B) = 2(n 3 + n 4 ) + (n 2 + n 5 ) S(W 0 , ñ, P ) = 2(n 5 + n 6 ) + (n 1 + n 4 ) (1)
with S(W 0 , ñ, A) being the score of A obtained from the application of the rule s = 0 to the vector ñ = (n 1 , n 2 , n 3 , n 4 , n 5 , n 6 ). More generally, the application of the rule defined by W s = (2, 1+s, 0) to the same triplet allows us to find the following scores:

   S(W s , ñ, A) = 2(n 1 + n 2 ) + (1 + s)(n 3 + n 6 ) S(W s , ñ, B) = 2(n 3 + n 4 ) + (1 + s)(n 2 + n 5 ) S(W s , ñ, P ) = 2(n 5 + n 6 ) + (1 + s)(n 1 + n 4 ) (2)
We thus find the antiplurality case for s = 1. The application of the rule given by W -s = (2, 1 -s, 0) provides the following expressions for the scores:

   S(W -s , ñ, A) = 2(n 1 + n 2 ) + (1 -s)(n 3 + n 6 ) S(W -s , ñ, B) = 2(n 3 + n 4 ) + (1 -s)(n 2 + n 5 ) S(W -s , ñ, P ) = 2(n 5 + n 6 ) + (1 -s)(n 1 + n 4 ) (3)
We thus find, the plurality case when s = 1.

The Impartial Culture Assumption

The Impartial Culture hypothesis (IC) is one of the most used hypothesis in the Social Choice literature in order to describe a priori the likelihood of each particular profile. This hypothesis has been introduced for the first time in literature for the social choice by [START_REF] Guilbauld | Les théories de l'intérêt général et le problème logique de l'agrégation[END_REF] in order to study the Condorcet paradox. The basic idea of the Impartial Culture IC model is that each voter selects randomly and independently his preferences from a uniform distribution on the set of possible types. Hence, all the profiles have the same apparition probability and the occurrence of a voting situation ñ = (n 1 , ..., n m! ) is given by:

P r(ñ) = n! n 1 !n 2 !...n m! ! ( 1 m! ) n (4)
In the case of 3 alternatives, each voter picks any of the preference types displayed on Table 1 with probability 1 6 ; we obtain P r(ñ) = n! n 1 !n 2 !...n 6 ! (6) -n . A reader who wants to improve his knowledge about the use of probability assumptions in voting theory may consult [START_REF] Berg | On probability models in voting theory[END_REF] or [START_REF] Gehrlein | Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences[END_REF][START_REF] Gehrlein | Condorcet's Paradox, Theory and Decision Library C[END_REF].

3 The Stability of {Borda, Plurality, Antiplurality}

A Definition for Stability

We will now present the definition of stability which we will then apply to the triplet {Borda, Plurality, Antiplurality}. Under the hypothesis of a universal domain, if one proposes to a society a set of neutral and different voting rules, could we be sure that the preferences will lead to a self-selective rule chosen by the individuals? In a global aspect, the notion of stability concerns the choice of one rule among many of them, independent of initial preferences. Moreover, on applying this notion, one rule may be considered as preferred to another if it is more stable within a given initial menu of voting rules.

Definition 1 Consider a set of voting rules E, and a profile of linear preferences on these voting rules. A voting rule is self-selective at some profile if the choice of the voting rule is this given rule for this profile. The set E is weakly stable if, at any profile, there exists at least one self-selective voting rule.

Let us consider two examples to highlight these ideas. We will first consider a society of 6 individuals. We propose to that society the menu of rules E = {Borda, Plurality, Antiplurality}. The individual preferences are displayed in Table 2. For this example, only the Borda rule is self-selective because the application of Borda gives out Borda as a result (9 points for B, 5 for P and 4 for A). On the other hand, the application of the Plurality and Antiplurality gives out Borda as a result.

However, Table 3 shows that the menu {B, P, A} may not be stable. With the preferences displayed in Table 3, we observe that the Antiplurality rule selects the Plurality rule (14 points for P, 8 for A and 6 for B), the Borda count picks out the Plurality rule (9 points for P, 8 for A and 4 for B) which itself leads to the choice of the Antiplurality rule (8 points for A, 4 for P and 2 for B). Thus, our objective is to try to calculate the stability probability of this set of 3 voting rules under the Impartial Culture Hypothesis. To fulfil this programme, we must first enumerate all the possible relationships among the three scoring rules.

The Different Cases

In this study, we will first assume that the size of the population is large, which will make the probabilities of tied outcomes extremely unlikely3 . Thus, the application of each rule to the triplet {Borda, Plurality, Antiplurality} only gives the three possible singletons {Borda}, {P lurality} or {Antiplurality} as a result. In fact, taking aside the unlikely cases where the results is a tie, we will obtain (3) 3 = 27 possible cases. We can partition the 27 cases into 6 categories, represented in the Table 4.

Case 1 in details

Let us consider, for example, the inequalities that characterize the first case presented on Table 5. Since B gives A, A gives P, and P gives B, we will study the system (5), and let s go to 1.

               I 1 : S(W 0 , ñ, A) > S(W 0 , ñ, P ) I 2 : S(W 0 , ñ, A) > S(W O ñ, B) I 3 : S(W -s , ñ, B) > S(W -s , ñ, P ) I 4 : S(W -s , ñ, B) > S(W -s , ñ, A) I 5 : S(W s , ñ, P ) > S(W s , ñ, B) I 6 : S(W s , ñ, P ) > S(W s , ñ, A) (5)
This set of inequalities is equivalent to:

               I 1 : n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 I 2 : 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 I 3 : (-1 + s)n 1 + (1 -s)n 2 + 2n 3 + (1 + s)n 4 + (-1 -s)n 5 -2n 6 > 0 I 4 : -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 I 5 : (1 + s)n 1 + (-1 -s)n 2 -2n 3 + (-1 + s)n 4 + (1 -s)n 5 + 2n 6 > 0 I 6 : (-1 + s)n 1 -2n 2 + (-1 -s)n 3 + (1 + s)n 4 + 2n 5 + (1 -s)n 6 > 0 (6)
In the same way, we can easily write down the different inequalities for the other 26 cases. In order to limit the efforts and make our computations easier, we may state several remarks 2 . With the help of elementary mathematical operations, we can easily demonstrate that 6 cases are impossible (5 th , 6 th , 13 th , 18 th , 22 th et 26 th case). The 5 th case is discussed as an example in Appendix I. In order to facilitate the resolution, notice that many cases can be re-written in the form of reduced systems, with 4 inequalities only instead of 6. Let us consider the first case being defined by the system (6). Easy computation demonstrate that I 4 + I 6 + 2I 1 = I 3 and I 4 + I 6 + 2I 2 = I 5 . We can thus reduce our system of inequalities into (7):

       I 1 : n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 I 2 : 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 I 4 : -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 I 6 : (-1 + s)n 1 -2n 2 + (-1 -s)n 3 + (1 + s)n 4 + 2n 5 + (1 -s)n 6 > 0 (7)
We will present in Appendix II the other reduced systems (1 st , 2 nd , 9 th , 10 th , 11 th , 14 th , 17 th , 20 th , and the 27 th case). For the remaining cases, we will use an indirect method. In order to calculate the probability of the 3 rd case, we will use the results obtained for the 1 st and the 18 th cases. Denote by P r(X) the probability of an event X. Thus:

P r(case 1) = P r(A → P, B → A, P → B) (8)
A → P means that the application of the Antiplurality rule to the triplet {B, P, A} gives out the Plurality as a results. We may thus write that: P r(A → P et B → A) = P r(case 1) + P r(case 3) + P r(case 18) (9)

We will see further on that we will be able to calculate P r(A → P et B → A). But we already know that P r(case 18) = 0. We can finally write that: P r(case 3) = P r(A → P et B → A) -P r(case 1) (10)

We will be using the same method in order to simplify the calculus of the probability of each remaining cases (4 th , 7 th , 8 th , 12 th , 15 th , 16 th , 19 th , 21 th , 23 th , 24 th , 25 th ). The details for all the cases are given in Appendix III. In fact, we will need to calculate the likelihood of the cases displayed in Table 6 where, each time, 4 inequalities are sufficient. 4 Results : probabilities computations

Case 1 in details

The method used in order to compute the probability of voting events has been developed by [START_REF] Saari | The likelihood of dubious election outcomes[END_REF], and further developed by Merlin, Tataru andValognes (2000, 2002) and [START_REF] Valognes | The impact of indifferent voters on the likelihood of some voting paradoxes[END_REF]. Under IC assumption, each voter is equally likely to pick independently any of the six preference types. By using Central Limit Theorem which implies, when n tends towards infinity, the following formula in R holds for n i :

P r(a < (n i -n 6 ) σ √ n < b) → 1 √ 2π b a (e -t 2 2 dt) , n → ∞ (11)
The measure of

(n i -n 6 ) σ √ n noted µ( n i -n 6 σ √ n ) verify the following relation in R: µ( n i -n 6 σ √ n ) → 1 √ 2π e -t 2
2 λ, where λ is the Lebesgue measure on the one dimensional hyperplane. In addition, the resolution of our systems requires an application of this theorem in R 5 :

Let m = (m 1 , m 2 , ...m 6 ) T =    n 1 . . . n 6    -    n 6 . . . n 6    σ √ n (12)
The Central Limit Theorem in R 5 implies, when n tends towards infinity, that :

µ(m) → 1 ( √ 2π) 5 e -|t| 2 2 λ ( 13
)
where t = (t 1 , t 2 , ..., t 6 ) ∈ R 6 ; |t| 2 = t 2 1 + ... + t 2 6 and λ is the Lebesgue measure on the five dimensional hyperplane (t 1 + ... + t 6 = 0). As stated above, the fact that the Antiplurality designates Plurality, that Borda designates Antiplurality and that Plurality designates Borda is characterized by the four inequalities of the system (7):

To compute the probability of this voting situation we will use the following reasoning. m verify the inequalities of the system (7) if and only if ñ = (n 1 ...n 6 ) verify them too. Then we write :

P r(m verif ies(7)) → 1 ( √ 2π) 5 C 1 (e -|t| 2 2 )dλ ( 14
)
where

C 1 = {m ∈ R 5 , m verif ies (7), 6 j=1 m j = 0} (15)
The measure μ ≡

1 ( √ 2π) 5 e -|t| 2
2 λ is absolutely continuous and radially symmetric. Hence, computing P (m satisf ies (7)) reduces to finding the measure of the cone C 1 . That will be done using the Schläfli's formula (See [START_REF] Kellerhals | On the volume of hyperbolic polyhedra[END_REF] and [START_REF] Milnor | Hyperbolic Geometry: the first 150 years[END_REF]). This formula gives the differential volume on the set of spherical n simplexes:

dvol n (C 1 ) = 1 (n -1) 1≤j<k≤n vol n-2 (F j ∩ F k )dα jk ; vol 0 = 1 (16)
where α jk is the dihedral angle formed by the facets F j and F k of C 1 , and n indicates the simplex dimension. [START_REF] Saari | The likelihood of dubious election outcomes[END_REF] were the first to present an application of this formula in voting theory in order to compute the probabilities of some voting events. We present in this work the adaptation of this formula for the probability computation of the stability of {Borda, Plurality, Antiplurality} triplet.

Each facet F i of the cone C 1 corresponds to one of the inequality I i in equation ( 7):

       F 1 : n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 F 2 : 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 F 3 : -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 F 4 : (-1 + s)n 1 -2n 2 + (-1 -s)n 3 + (1 + s)n 4 + 2n 5 + (1 -s)n 6 > 0 (17) Let -→ v 1 , - → v 2 , -→ v 3 , - → v 4 the normal vectors to the hyperplanes F 1 , F 2 , F 3 , F 4 , pointing inside the cone C 1 :        -→ v 1 = (1, 2, 1, -1, -2, -1) -→ v 2 = (2, 1, -1, -2, -1, 1) -→ v 3 = (-2, -1 -s, 1 + s, 2, 1 -s, -1 + s) -→ v 4 = (-1 + s, -2, -1 -s, 1 + s, 2, 1 -s) (18)
The relation :

-→ v i . -→ v j = -→ v i . -→ v j . cos(π -α ij ) = --→ v i . -→ v j . cos(α ij )
enables us, in addition, to find the value of the 6 following angles α ij :

α 12 = 2π 3 (19) α 13 = α 24 = arccos ( 3 2 √ 3s 2 + 9 ) (20)
α 14 = α 23 = arccos ( 3 √ 3s 2 + 9 ) (21)
α 34 = arccos ( s 2 -3 2(s 2 + 3) ) (22)
which implies that:

dα 12 = 0 (23) dα 13 = dα 24 = √ 3 s (3 + s 2 ) √ 9 + 4s 2 (24) dα 14 = dα 23 = √ 3 3 + s 2 (25) dα 34 = - 4 √ 3s (3 + s 2 ) √ 9 + 10s 2 + s 4 (26) The vectors -→ v 1 , -→ v 2 , -→ v 3 , -→ v 4 are perpendicular to the vectors -→ v 5 = (1, 1, 1, 1, 1, 1) and - → v 6 = (1, -1, 1, -1, 1, -1).
Then we can calculate the vertex P 123 = F 1 ∩ F 2 ∩ F 3 by solving the system:

P 123 :                n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 = 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 = 0 -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 = 0 (-1 + s)n 1 -2n 2 + (-1 -s)n 3 + (1 + s)n 4 + 2n 5 + (1 -s)n 6 > 0 n 1 + n 2 + n 3 + n 4 + n 5 + n 6 = 0 n 1 -n 2 + n 3 -n 4 + n 5 -n 6 = 0 (27)
We derive the following results:

P 123 :                n 1 = 2 n 2 = -1 n 3 = -1 n 4 = 2 n 5 = -1 n 6 = -1 (28)
By solving similar linear systems, we obtain the other vertices of the cone C 1 to be equal to:

P 124 :                n 1 = 1 n 2 = -2 n 3 = 1 n 4 = 1 n 5 = -2 n 6 = 1 P 134 :                n 1 = 1 + s n 2 = -2 n 3 = 1 -s n 4 = 1 -s n 5 = -2 n 6 = 1 + s P 234 :                n 1 = 2 n 2 = -1 + s n 3 = -1 + s n 4 = 2 n 5 = -1 -s n 6 = -1 -s (29)
Also we compute:

vol(F 1 , F 2 ) = ( P 123 , P 124 ) = 1 3 π (30) vol(F 1 , F 3 ) = vol(F 2 , F 4 ) = ( P 123 , P 134 ) = arccos ( 3 2 √ 9 + 3s 2 ) (31) vol(F 1 , F 4 ) = vol(F 2 , F 3 ) = ( P 124 , P 134 ) = arccos ( 3 √ 9 + 3s 2 ) (32) vol(F 3 , F 4 ) = ( P 234 , P 134 ) = π -arccos ( -3 + s 2 2(3 + s 2 ) ) (33)
We can now easily apply the Schläfli 's formula:

dvol(C) = 2vol(F 1 , F 3 )dα 13 + 2vol(F 1 , F 4 )dα 14 + vol(F 3 , F 4 )dα 34 (34)
We find:

dvol(C) ds = 2 √ 3 s arccos ( 3 2 √ 9+3s 2 ) (3 + s 2 ) √ 9 + 4s 2 + 2 √ 3 arccos ( 3 √ 9+3s 2 ) 3 + s 2 - 4 √ 3 s (π -arccos ( -3+s 2 6+2s 2 )) (3 + s 2 ) √ 9 + 10s 2 + s 4 (35)
By Dividing the formula by (n -1) = 2 and 2π 2 (the volume of the unit sphere in R 4 ) we obtain: The graphical representation (Figure 1) shows the integration for 0 ≤ s ≤ 1. We may find : P r(case 1) = P r(s = 1) = 0.00299346. We will use the same method to find out the other probabilities. We give out all the corresponding probabilities in the Table 7.

P r(case 1, s) = 1 4π 2 s 0 2 √ 3 t arccos( 3 2 √ 9+3t 2 ) (3 + t 2 ) √ 9 + 4t 2 + 2 √ 3 arccos( 3 √ 9+3t 2 ) 3 + t 2 - 4 √ 3 t (π -arccos( -3+t 2 6+2t 2 )) (3 + t 2 ) √ 9 + 10t 2 + t 4 dt

General Results

In fact, when examining carefully the systems, we observe that we only need to obtain three other expressions for the probabilities P r(cas 1, s) to derive all the other probabilities. Then probabilities for each possible cases are displayed in Table 7. Table 8 sums up the results according to the categories.

We have defined a stable set as being a set which always has a self-selective rule. We may thus state the following proposition:

Proposition 1 : Consider a large population, where each voter has a probability of 1/6 to choose one of the six possible linear orders over the set E = {Borda, Plurality, Antiplurality} of voting rules. Then the set E is stable for 84.49% of the profiles. We may also learn a few other facts from the results displayed in Tables 7 and8. In fact, the instability does not come from cases like example displayed in Table 3. A cycle among the three scoring rules is quite unlikely (less than 0.6% ). The problem arises when rule 1 designate rule 2 while rule 2 designate rule 1, the third one leading either to rule 1 and 2.

As a byproduct, we can also compute the probability that all the scoring rules gives the same results. This is the joint probability of cases 14, 17 and 20. We get back a probability of 0.5346, in accordance with the result previously obtained by Fishburn and Gerhlein with different techniques.

It also possible to use differently the results of Table 7, in order to examine whether a rule is more likely to be self-selective.

The figures in = 33.33% While following the same method we find: P r(B is stable alone or with others) = P r(P is stable alone or with others) = 33.33%. Therefore, we can say that no rule is more stable than another.

Conclusion

The objective of this paper was to illustrate the recently growing literature on the self-selectivity of voting rules with a classical example, by considering the three most famous scoring rules. As one may have guessed, the set {Borda, Plurality, Antiplurality} is not stable, the problem being mainly concentrated on cycles between two rules.

Thus, is there any escape from impossibility theorems when one consider the choice of a voting rule ? Several possibilities could be explored:

-First, we may consider different probability assumptions, considering more homogeneous societies. One possibility would be to consider the Impartial Anonymous Culture, which assumes that all the vectors ñ are equally likely.

This assumption is known to slightly diminish the probability of paradoxical events compared to the IC assumption. Another possibility, quite common in Social Choice theory, is to consider restriction on the preferences, like singledpeaked preferences.

-In this paper, we have considered rules which seldom give tied outcomes. However, Nicolas Houy has pointed out that one of the key to the stability is the existence of a rule that selects often many rules. So, why not trying such a rule, like for example the Copeland method, which always selects alternatives in the top cycle of the pairwise majority rule ? Indeed, for three voters, the set E = {Borda, P lurality, Copeland} is stable if we breaks the ties in favor of the statu quo.

-Another possibility would be to consider consequentialism as a way to model the preference of the voters over the voting rules. Consequentialism implies that two voting rules are considered equivalent by the voters if they lead to the selection of the same alternatives. Thus, preferences may now be weak ordering, and it is known the presence of weak orderings in the preference profile tends the reduce the probability of paradoxical outcomes (see [START_REF] Gehrlein | Condorcet efficiency: A preference for indifference[END_REF] and [START_REF] Merlin | On the likelihood of condorcet's profiles[END_REF]).

The careful exploration of these possibilities will tell us whether there is any hope of finding reasonably stable menu of voting rules that one can propose with confidence to the choice of the voters.

• 11 th case :

B → A, P → P et A → B        n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 (-1 + s)n 1 -2n 2 + (-1 -s)n 3 + (1 + s)n 4 + 2n 5 + (1 -s)n 6 > 0 -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 • 14 th case : B → A, P → A et A → A        2n 1 + (1 -s)n 2 + (-1 + s)n 3 -2n 4 + (-1 -s)n 5 + (1 + s)n 6 > 0 (1 -s)n 1 + 2n 2 + (1 + s)n 3 + (-1 -s)n 4 -2n 5 + (-1 + s)n 6 > 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0 (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 • 17 th case : B → B, P → B et A → B        (-1 -s)n 1 + (1 + s)n 2 + 2n 3 + (1 -s)n 4 + (-1 + s)n 5 -2n 6 > 0 -2n 1 + (-1 + s)n 2 + (1 -s)n 3 + 2n 4 + (1 + s)n 5 + (-1 -s)n 6 > 0 -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 (-1 + s)n 1 + (1 -s)n 2 + 2n 3 + (1 + s)n 4 + (-1 -s)n 5 -2n 6 > 0 • 20 th case : B → P, P → P et A → P        (1 + s)n 1 + (-1 -s)n 2 -2n 3 + (-1 + s)n 4 + (1 -s)n 5 + 2n 6 > 0 (-1 + s)n 1 -2n 2 + (-1 -s)n 3 + (1 + s)n 4 + 2n 5 + (1 -s)n 6 > 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 > 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 • 27 th case : B → B, P → P et A → A        -2n 1 -n 2 + n 3 + 2n 4 + n 5 -n 6 > 0 -n 1 + n 2 + 2n 3 + n 4 -n 5 -2n 6 > 0 (1 + s)n 1 + (-1 -s)n 2 -2n 3 + (-1 + s)n 4 + (1 -s)n 5 + 2n 6 > 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0

Appendix (III)

• P r(case 4) = P r(A B) -P r(case 11)

• P r(case 7) = P r(P B) -P r(case 9)

• P r(case 8) = P r(A → P et B → P ) -P r(case 7) -P r(case 20)

• P r(case 12) = P r(P → B et B → A) -P r(case 1)

• P r(case 15) = P r(P → A et B → B) -P r(case 10)

• P r(case 16) = P r(A → P et B → B) -P r(case 10)

• P r(case 19) = P r(A → B et B → P ) -P r(case 2)

• P r(case 21) = P r(A → A et B → B) -P r(case 27)

• P r(case 23) = P r(A → A et B → A) -P r(case 12) -P r(case 14)

• P r(case 24) = P r(A → A et B → P ) -P r(case 9)

• P r(case 25) = P r(A → B et B → B) -P r(case 15) -P r(case 17)

Appendix (IV)

• A → P et B → A        n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 > 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 • A B        n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 (-1 + s)n 1 + (1 -s)n 2 + 2n 3 + (1 + s)n 4 + (-1 -s)n 5 -2n 6 > 0 • P B        -n 1 -2n 2 -n 3 + n 4 + 2n 5 + n 6 > 0 n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 > 0 (-1 -s)n 1 + (1 + s)n 2 + 2n 3 + (1 -s)n 4 + (-1 + s)n 5 -2n 6 > 0 -2n 1 + (-1 + s)n 2 + (1 -s)n 3 + 2n 4 + (1 + s)n 5 + (-1 -s)n 6 > 0 • A → P et B → P        -n 1 -2n 2 -n 3 + n 4 + 2n 5 + n 6 > 0 n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 > 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 > 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 • P → B et B → A        n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 (-1 -s)n 1 + (1 + s)n 2 + 2n 3 + (1 -s)n 4 + (-1 + s)n 5 -2n 6 > 0 -2n 1 + (-1 + s)n 2 + (1 -s)n 3 + 2n 4 + (1 + s)n 5 + (-1 -s)n 6 > 0 • P → A et B → B        -2n 1 -n 2 + n 3 + 2n 4 + n 5 -n 6 > 0 -n 1 + n 2 + 2n 3 + n 4 -n 5 -2n 6 > 0 2n 1 + (1 -s)n 2 + (-1 + s)n 3 -2n 4 + (-1 -s)n 5 + (1 + s)n 6 > 0 (1 -s)n 1 + 2n 2 + (1 + s)n 3 + (-1 -s)n 4 -2n 5 + (-1 + s)n 6 > 0 • A → P et B → B        -2n 1 -n 2 + n 3 + 2n 4 + n 5 -n 6 > 0 -n 1 + n 2 + 2n 3 + n 4 -n 5 -2n 6 > 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 > 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 • A → B et B → P        -n 1 -2n 2 -n 3 + n 4 + 2n 5 + n 6 > 0 n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 > 0 -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 (-1 + s)n 1 + (1 -s)n 2 + 2n 3 + (1 + s)n 4 + (-1 -s)n 5 -2n 6 > 0 • A → A et B → B        -2n 1 -n 2 + n 3 + 2n 4 + n 5 -n 6 > 0 -n 1 + n 2 + 2n 3 + n 4 -n 5 -2n 6 > 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0 (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 • A → A et B → A        n 1 + 2n 2 + n 3 -n 4 -2n 5 -n 6 > 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 > 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0 (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 • A → A et B → P        -n 1 -2n 2 -n 3 + n 4 + 2n 5 + n 6 > 0 n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 > 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0 (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 • A → B et B → B        -2n 1 -n 2 + n 3 + 2n 4 + n 5 -n 6 > 0 -n 1 + n 2 + 2n 3 + n 4 -n 5 -2n 6 > 0 -2n 1 + (-1 -s)n 2 + (1 + s)n 3 + 2n 4 + (1 -s)n 5 + (-1 + s)n 6 > 0 (-1 + s)n 1 + (1 -s)n 2 + 2n 3 + (1 + s)n 4 + (-1 -s)n 5 -2n 6 > 0

Appendix (IV)

Our results depend on four different probabilities. The other calculations give us similar results. The case N • 1 has been presented previously, it's similar to the followings cases: 2, 9, 10, 11 and 27. Here we present the three other probabilities :

• 14 th case : B → A, P → A and A → A This case is defined by the following system:

       2n 1 + (1 -s)n 2 + (-1 + s)n 3 -2n 4 + (-1 -s)n 5 + (1 + s)n 6 > 0 (1 -s)n 1 + 2n 2 + (1 + s)n 3 + (-1 -s)n 4 -2n 5 + (-1 + s)n 6 > 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0 (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 (36) Let        F 1 : 2n 1 + (1 -s)n 2 + (-1 + s)n 3 -2n 4 + (-1 -s)n 5 + (1 + s)n 6 > 0 F 2 : (1 -s)n 1 + 2n 2 + (1 + s)n 3 + (-1 -s)n 4 -2n 5 + (-1 + s)n 6 > 0 F 3 : 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 > 0 F 4 : (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 (37) and -→ v 1 , -→ v 2 , -→ v 3 , - → v 4 the normal vectors to the hyperplanes F 1 , F 2 , F 3 , F 4 :        -→ v 1 = (2, 1 -s, -1 + s, -2, -1 -s, 1 + s) -→ v 2 = (1 -s, 2, 1 + s, -1 -s, -2, -1 + s) -→ v 3 = (2, 1 + s, -1 -s, -2, -1 + s, 1 -s) -→ v 4 = (1 + s, 2, 1 -s, -1 + s, -2, -1 -s) (38)
We find the value of the 6 following angles α ij :

α 12 = α 34 = 2π 3 (39) α 13 = α 24 = arccos ( -3 + s 2 3 + s 2 ) (40) α 14 = α 23 = arccos ( -3 + s 2 2(3 + s 2 ) ) (41) (42)
which implies that:

dα 12 = dα 34 = 0 (43) dα 13 = dα 24 = -2 √ 3 3 + s 2 (44) dα 14 = dα 23 = -4s √ 3 (3 + s 2 ) √ 9 + 10s 2 + s 4 (45) (46)
Then we can calculate the vertex P 123 = F 1 ∩ F 2 ∩ F 3 by solving the system:

P 123 :        2n 1 + (1 -s)n 2 + (-1 + s)n 3 -2n 4 + (-1 -s)n 5 + (1 + s)n 6 = 0 (1 -s)n 1 + 2n 2 + (1 + s)n 3 + (-1 -s)n 4 -2n 5 + (-1 + s)n 6 = 0 2n 1 + (1 + s)n 2 + (-1 -s)n 3 -2n 4 + (-1 + s)n 5 + (1 -s)n 6 = 0 (1 + s)n 1 + 2n 2 + (1 -s)n 3 + (-1 + s)n 4 -2n 5 + (-1 -s)n 6 > 0 (47)
We derive the following results:

P 123 :                n 1 = -1 n 2 = s+1 2 n 3 = s+1 2 n 4 = -1 n 5 = -s+1 2 n 6 = -s+1 2 (48)
By solving similar linear systems, we obtain the other vertices of the cone C 1 to be equal to:

P 124 :                n 1 = s+1 2 n 2 = -1 n 3 = -s+1 2 n 4 = -s+1 2 n 5 = -1 n 6 = s+1 2 P 134 :                n 1 = 1 n 2 = s-1 2 n 3 = s-1 2 n 4 = 1 n 5 = -s-1 2 n 6 = -s-1 2 P 234 :                n 1 = s-1 2 n 2 = 1 n 3 = -s-1 2 n 4 = -s-1 2 n 5 = 1 n 6 = s-1 2 (49) and -→ v 1 , -→ v 2 , -→ v 3 , - → v 4 the normal vectors to the hyperplanes F 1 , F 2 , F 3 , F 4 :        -→ v 1 = (1, 2, 1, -1, -2, -1) -→ v 2 = (2, 1, -1, -2, -1, 1) -→ v 3 = (1 -s, -1 + s, -2, -1 -s, 1 + s, 2) -→ v 4 = (-1 -s, -2, -1 + s, 1 -s, 2, 1 + s) (57)
We find the value of the 6 following angles α ij :

α 12 = α 34 = 2π 3 (58) α 13 = α 24 = arccos ( 3 2 √ 9 + 3s 2 ) ( 59
)
α 14 = arccos ( 3 √ 9 + 3s 2 ) ( 60
)
α 23 = π -arccos ( 3 2 √ 9 + 3s 2 ) (61) (62)
which implies that: We derive the following results:

dα 12 = dα 34 = 0 (
P 123 :                n 1 = 1 n 2 = 1 n 3 = -2 n 4 = 1 n 5 = 1 n 6 = -2 (69)
By solving similar linear systems, we obtain the other vertices of the cone C 1 to be equal to: and -→ v 1 , -→ v 2 , -→ v 3 , -→ v 4 the normal vectors to the hyperplanes F 1 , F 2 , F 3 , F 4 :

P 124 :                n 1 = 1 n 2 = -2 n 3 = 1 n 4 = 1 n 5 = -2 n 6 = 1
       -→ v 1 = (-1, -2, -1, 1, 2, 1) -→ v 2 = (1, -1, -2, -1, 1, 2) -→ v 3 = (1 -s, -1 + s, -2, -1 -s, 1 + s, 2) -→ v 4 = (-1 -s, -2, -1 + s, 1 -s, 2, 1 + s) (78)
We find the value of the 6 following angles α ij :

α

Figure 1 :

 1 Figure 1: The probability of the 1 st case as a function of s

  calculate the vertexP 123 = F 1 ∩ F 2 ∩ F 3 by solving the system: 2n 2 + n 3 -n 4 -2n 5 -n 6 = 0 2n 1 + n 2 -n 3 -2n 4 -n 5 + n 6 = 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 = 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 (68)

F 1 :

 1 1 , F 2 ) = vol(F 3 , F 4 ) = ( P 123 , P 124 ) 1 , F 3 ) = vol(F 2 , F 4 ) = vol(F 2 , F 3 ) = ( P 123 , P 134 ) = arccos( 1 , F 4 ) = ( P 124 , P 134 ) = π -(Subcase 1) = P r(s = 1) = 0.04027691 The case (A → P and B → A) give us the similar probability than the followings cases: and B → P P → B and B → A A → A and B → B P → A and B → B A → A and B → P • Subcase 2: A → P and B → PThis subcase is defined by the following system:-2n 2 -n 3 + n 4 + 2n 5 + n 6 > 0 n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 > 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 > 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 -n 1 -2n 2 -n 3 + n 4 + 2n 5 + n 6 > 0 F 2 : n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 > 0 F 3 : (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 > 0 F 4 : (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 (77)

  calculate the vertex P 123 = F 1 ∩ F 2 ∩ F 3 by solving the system:-n 1 -2n 2 -n 3 + n 4 + 2n 5 + n 6 = 0 n 1 -n 2 -2n 3 -n 4 + n 5 + 2n 6 = 0 (1 -s)n 1 + (-1 + s)n 2 -2n 3 + (-1 -s)n 4 + (1 + s)n 5 + 2n 6 = 0 (-1 -s)n 1 -2n 2 + (-1 + s)n 3 + (1 -s)n 4 + 2n 5 + (1 + s)n 6 > 0 (87)

Table 1 :

 1 The six preference types among {B, P, A} AP

Table 2 :

 2 An example for stability

	BP A (2)	BAP (2)	P BA (1)	AP B (1)
		Table 3: An example of instability
	BP A (1)	P BA (2)	AP B (4)

Table 4 :

 4 The six possible categories All the 27 cases then fall into one of the six categories (see Table5). The example displayed on Table 2 corresponds to case 17, and the cycle of Table 3 corresponds to case 2.

	Categories	Cycles	Number of self-selective rule
	I	Cycle with 3	0
	II	Cycle with 2	0
	III	Cycle with 2	1
	IV	0 cycle	1
	V	0 cycle	2
	VI	0 cycle	3

Table 5 :

 5 All the possible cases

					the voting rule
				Case N 0 B P	A
				3	A A	P
				4	A A	B
		the voting rule	5	P B	B
	Case N 0 B P	A	6	A B	B
	1	A B	P	7	P B	P
	2	P A	B	8	P A	P
	Table 5.a : Category I	Table 5.b : Category II
					the voting rule
				Case N 0 B P	A
				12	A B	A
				13	P A	A
				14	A A	A
				15	B A	B
		the voting rule	16	B B	P
	Case N 0 B P	A	17	B B	B
	9	P B	A	18	A P	P
	10	B A	P	19	P P	B
	11	A P	B	20	P P	P
	Table 5.c : Category III	Table 5.d : Category IV
		the voting rule
	Case N 0 B P	A
	21	B B	A
	22	B A	A
	23	A P	A
	24	P P	A		the voting rule
	25	B P	B	Case N 0 B P	A
	26	B P	P	27	B P	A
	Table 5.e : Category V	Table 5.f : Category VI

Table 6 :

 6 The cases to consider for probability computations

	P r(A → P and B → A)	P r(A → P and B → B)
	P r(A	B)	P r(A → B and B → P )
	P r(P	B)

P r(A → A and B → B) P r(A → P and B → P ) P r(A → A and B → A) P r(P → B and B → A) P r(A → A and B → P ) P r(P → A and B → B) P r(A → B and B → B)

Table 7 :

 7 The probability for each case

		Case N • Probabilities Case N • Probabilities
		1	0.00299346	15	0.03728344
		2	0.00299346	16	0.03728344
		3	0.03728344	17	0.17821261
		4	0.03728344	18	0
		5	0	19	0.03728344
		6	0	20	0.17821261
		7	0.03728344	21	0.03728344
		8	0.03728344	22	0
		9	0.00299346	23	0.03728344
		10	0.00299346	24	0.03728344
		11	0.00299346	25	0.03728344
		12	0.03728344	26	0
		13	0	27	0.00299346
		14	0.17821261
		Table 8: Probability by category
	Categories	Cycles	Number of self-selective rule Probabilities
	I	Cycle with 3	0	0.00598692
	II	Cycle with 2	0	0.14913376
	III	Cycle with 2	1	0.00898038
	IV	0 cycle		1	0.68377159
	V	0 cycle		2	0.14913376
	VI	0 cycle		3	0.00299346

Table 9 :

 9 General table

	T he set	P robabilities
	{A}	21.85 %
	{B}	25.58 %
	{P }	21.85 %
	{AB}	3.73 %
	{AP }	7.46 %
	{BP }	3.73 %
	{ABP }	0.3 %
	Instability	15.51 %
	alone with 21,85 %. {AB} are stable alone with 3,73 %.
	Thus,

Table 9 means, for example, that the rule {A} (or {P }) is stable P r(A is stable alone or with others) = P r({A}) + P r({AB}) +P r({AP }) + P r({ABP })

Following[START_REF] May | A Set of Independent, Necessary and Sufficient Conditions for Simple Majority Decision[END_REF], a voting rule is neutral is any permutation on the names of alternatives also permutes the social result accordingly. It guaranties that no alternative is favored by the voting rule.

Anonymity has also been defined by[START_REF] May | A Set of Independent, Necessary and Sufficient Conditions for Simple Majority Decision[END_REF]. A rule is anonymous if the social result is unaffected by permutations on the names of the voters. This condition ensures that all the voters are treated equally.

[START_REF] Marchant | The probability of ties with scoring methods : Some results[END_REF] studied the probability of apparition of each tie for scoring voting rules (Borda, Plurality...). He points out some analytical and numerical results, proving that this probability tends towards zero when the number of electors is very large.

We must highlight that several facts can be geometrically demonstrated by using some of[START_REF] Saari | Geometry of voting[END_REF][START_REF] Saari | Basic Geometry of voting[END_REF] important results dealing with the geometry of scoring rules. In order to simplify the presentation, these results are not enclosed in the paper.

ACKNOWLEDGEMENTS

We gratefully thank Nicolas Houy for the comments, remarks and suggestions he made on the different preliminary versions of this paper.

NOTES

Appendix (I)

The 5 th case (B → P, P → A, A → B) is describe by the following system of inequalities:

Evaluating -1 2 (I 3 + I 6 ) enables us to find the inequality

. In comparison with I 2 we can say that the 5 th case is impossible to realize. This result also derives from [START_REF] Saari | Geometry of voting[END_REF][START_REF] Saari | Basic Geometry of voting[END_REF], which states that in the three candidate case, all the scoring rules coincide if and only if the Plurality rule and the Antiplurality rule select the same winner. Cases 6, 13,18, 22 and 26 obey to the same logic.

Appendix (II)

• 1 st case :

Also we compute:

We find:

Finally we obtain:

and P r(case 14) = P r(s = 1) = 0.17821261 The case 14 give us the similar probability than the case 17 and the case 20.

• Subcase 1: A → P and B → A

This subcase is defined by the following system:

We derive the following results:

By solving similar linear systems, we obtain the other vertices of the cone C 1 to be equal to:

Also we compute:

We find:

Finally we obtain: