Vita Ilakovac
email: vita.ilakovac-casses@sorbonne-universite.fr

Adrien Girard
email: adrien.girard@sorbonne-universite.fr

Victor Balédent
email: victor.baledent@universite-paris-saclay.fr

Pascale Foury-Leylekian

Bjoern Winkler
email: b.winkler@kristall.uni-frankfurt.de

Ivan Kupčić
email: ivan.kupcic@phy.hr

Helmuth Berger
email: helmuth.berger@epfl.ch

Alexeï Bosak
email: alexsei.bossak@esrf.fr

Jean-Paul Pouget

Order-disorder type of Peierls instability in BaVS 3

Lattice dynamics of low-dimensional BaVS3 is reported across the metal -insulator Peierls transition occurring at TP = 69 K using a combination of the thermal diffuse scattering TDS of x rays, inelastic x-ray scattering (IXS), and density functional theory calculations. The non detection of a Kohn anomaly points to a unique situation of an order-disorder Peierls instability with a quasielastic critical scattering which has been fully characterized. These observations are discussed in the scope of a Peierls instability dominated by strong electron-phonon coupling and/or non-adiabatic effects.

I. INTRODUCTION

The mechanism of the q-momentum anomaly in the frequency dependence of low frequency phonon branches is established in metals since the pioneering calculation of W. Kohn 1 and its first detection in Pb 2 in the 1960's. But it is only with ab-initio calculations of the phonon spectrum including the electron-phonon coupling that such anomalies could be predicted in more complex systems 3 . In the adiabatic approximation the so-called Kohn anomaly in the phonon spectrum is due to an efficient screening of ion-ion interaction by the conduction electrons which is related to the shape of the Fermi surface 4 . More precisely in the weak electron-phonon coupling limit, the Kohn anomaly in the phonon spectrum directly reflects the singular q-dependent part of the electron-hole or Lindhard response of the electron gas. This singularity is enhanced when the spatial dimension of the electron gas is reduced 4 . Thus for a 1D conductor undergoing a Peierls instability one expects, in the adiabatic approximation and in the perturbative treatment of the electron-phonon coupling, the formation of a giant Kohn anomaly at the 2k F wave vector (k F being the Fermi momentum of the 1D electron gas) 5 . Indeed a sharp 2k F Kohn anomaly has been observed in the Krogmann salt K 2 Pt(CN) 4 Br 0.3 .3H 2 O 6 and in the blue bronze K 0.3 MoO 3 7 . In the latter compound it has been consistently shown that experiments follow quite well the expectations of the weak coupling scenario of the Peierls transition which leads to a displacive type of lattice instability 8 . However the formation of a pronounced 2k F Kohn anomaly is not the general rule for 1D conductors (for a recent review see 9 ). For example the organic conductor TTF-TCNQ exhibits only a weak and broad Kohn anomaly, whose frequency does not soften significantly on approaching the Peierls transition 10 . A somewhat similar situation was reported in 2D metals such as the transition metal dichalcogenide 2H NbSe 2 where an earlier neutron scattering investigation revealed the formation of a broad Kohn anomaly accompanied by a zero energy critical scattering which reflects a pronounced order-disorder type of lattice instability 11 . However, a recent high resolution IXS investigation of 2H NbSe 2 shows that the Peierls phonon mode exhibits in fact both a critical frequency softening and damping enhancement on approaching the Peierls transition 12 . This mixed displacive/order-disorder character of the Peierls transition dynamics is beyond the classical weak coupling -adiabatic scenario. Thus new scenarios which have been proposed rely on q-dependent electron-phonon coupling 13 , strong electron-phonon coupling 14,15 , or nonadiabaticity. The first type of scenario achieves, through anharmonic effects induced by the coupling to electrons, a broad Kohn anomaly, while the second type of scenario favors an order-disorder type of critical dynamics. However, these scenarios do not entirely explain the observed Peierls dynamics of the transition metal dichalcogenides.

Here we report the first unambiguous observation of an order-disorder type of dynamics in the Peierls compound BaVS 3 .

The room temperature (RT) structure of this 3d 1 transition metal compound is shown in Fig. 1. It consists of a hexagonal packing of quasi-one-dimensional (1D) chains of face-sharing VS 6 octahedra directed along the c axis, separated by Ba atoms 16 . Its structural anisotropy suggests properties of a linear chain compound, but with a rather low conductivity anisotropy ratio σ c /σ a ≈ 3. 17 A zigzag deformation of the V chains at T S = 250 K reduces the crystal symmetry from hexagonal to orthorhombic 18 , without significant changes in the resistivity and the para-magnetic properties 17 . A semiconducting behavior is observed below the temperature of the resistiv- ity minimum, T m ≈ 130 K, 16,17,19 and an extra feature shows up in the pre-edge of the V L-edge linear dichroism 20 . This suggests additional electronic transformations which are not yet clearly elucidated but which could be associated with a host-guest space-group adjustement of Ba/S and V sublattices 21 . At T P = 69 K the system undergoes a metal-to-insulator (MI) transition, accompanied by a doubling of the unit cell in the c direction 22 . The driving force of the MI transition is the Peierls instability resulting in the formation of charge density waves (CDWs) along the V-S chains 23 . Spin degrees of freedom are affected at T P 17 , but an incommensurate anti-ferromagnetic order establishes only below T N = 30 K [24][25][26][27] .

Local density approximation (LDA) calculations reveal a peculiar interplay between two different types of active electrons at the Fermi level: two narrow E g bands (localized states) and one dispersive (itinerant states), with a mainly A 1g character, extending along the c * direction [28][29][30] . But the LDA A 1g band is found to be almost complete. This is incompatible with the experimental results pointing to an equipartitial A 1g /E g occupation 23 . The discrepancy is corrected when considerable correlation/exchange effects are included through Dynamic Mean Field Theory (DMFT) [31][32][33] .

The Peierls transition achieves also an ordering between the A 1g and E g orbitals 32,34,35 . In the Peierls ground state occurring below T P , A 1g electrons are paired into magnetic singlets while E g electrons keep their spin free. The E g spins order antiferromagnetically (AF) at T N 25,27 . Under pressure T P drastically drops 36,37 , while T N stays constant 38 . At 1.4 GPa T P and T N merge together and the q P -modulation of the Peierls state becomes incommensurate, while at 1.8 GPa the two ground states conjointly vanish [36][37][38] . The drastic drop of T P has been ascribed to a decrease of the lifetime of the A 1g electron-hole pairs due to their hybridization with the non-paired E g orbitals 27 . Furthermore a quantitative analysis of the drop of T P under pressure suggests that the Peierls transition of BaVS 3 could occur in the non-adiabatic (anti-adiabatic) limit 27,37 . Note that in this limit the dynamics of the Peierls chain calculated for free electrons and for a weak electron-phonon coupling is of the order-disorder type 9 .

Recent combined Thermal Diffuse Scattering (TDS) and Inelastic X-ray Scattering (IXS) investigation of the BaVS 3 lattice dynamics across T S shows that the orthorhombic distortion is triggered by the softening of a Γ phonon 39 . In this work, we present an equivalent study of the metal-insulator Peierls transition. We first present Diffuse Scattering (DS) data in the l = 1/2 plane and corresponding model calculation in order to identify the major phonon contribution. Further, IXS results show that the low energy phonons are optical in nature and that they do not soften. The analysis of the q-width of the DS related to the q P wave vector gives the insight in the evolution of the spatial extension of the fluctuating domains while the energy-width of the IXS central peak points to their life-time evolution. All these results point to an original situation where the structural transition at T S is displacive and phonon-driven, while the Peierls transition at T P is of order-disorder type. This unique results is discussed in terms of the models including strong electron-phonon coupling and non-adiabaticity.

II. EXPERIMENT

Single crystals of BaVS 3 , of a needle-like shape along the [001] axis, were grown by the tellurium flux method 40 . They were slightly polished to obtain lateral dimensions of about 100 µm, in order to maximize the IXS signal and then mounted on a glass fiber and fixed with an epoxy glue. X-ray Diffuse Scattering (DS) experiments were conducted first with the diffractometer built as a side station at the ID28 beamline 41 of the European Synchrotron Radiation Facility (ESRF), with a monochromatic beam of 17.8 keV photons. The DS maps were recorded with a Pilatus 3 X 1M detector in the range 30 -300 K. Closedcycle cryocooler with Kapton windows was used for both IXS and DS measurement. In the case of DS data collection, an internal screen was added, which was rotating in opposite direction relatively to the cryostat movement. It allowed us to suppress the background scattering from both entrance and exit windows. The IXS experiments were conducted at the ID28 beamline of the ESRF, with the high-resolution spectrometer operating at an energy of 17.8 keV. With our setup, the energy resolution was about 3 meV. A correction of 10 K (14 K) is applied in IXS (DS) scans in order to match the temperature of the structural transition with the MI transition determined by conductivity measurements, 69 K 17 . Scattering angle restrictions imposed the choice of representative DS reconstruction as (h,k,3.5). However, different mount allowed us to explore essentially equivalent (h,k,4.5) plane by IXS, thus gaining the intensity. CrysAlis software (Rigaku -Oxford Diffraction) was used for the preliminary diffraction data reconstructions. High quality reconstructions were performed with locally developed software.

III. RESULTS

A. Diffuse Scattering

DS maps

The reconstruction of the (h,0,l) plane of BaVS 3 , measured at 114 K is shown Fig. 2a. At l = n+1/2, is shows diffuse streaks extending in a * direction. They correspond to a hexagonal, honeycomb-shaped diffusion, visible already at RT in the reconstruction of the (h,k,3.5) plane (see Fig. 2b). At low temperature, when approaching T P from above, it disappears and is replaced by a diffuse-scattering spot-pattern which condenses in superstructure satellites at T P , see Figs. 2c-e. Note that the intensity range of the DS taken at low temperature is modified relatively to this at 300 K, in order to make the diffusion pattern more visible. The corresponding figures presented in the same intensity range are shown in the Annexe (see Sec.VII). The intensity of the honeycombshaped diffusion, as measured at the A' point (diffuse hexagon corner), underlines all important changes in BaVS 3 , see Fig. 3. When lowering the temperature, it increases at the orthorhombic transition (T S ), strongly decreases at the minimum of resistivity (T m ), re-increases at the Peierls transition (T P ) and progressively vanishes below T P .

The reduced wave vector of the superstructure satellites (see Fig. 2e) appearing at T P is (0.5, 0, 0.5) in hexagonal settings (space group P6 3 /mmc), which will be used throughout this work. This correspond to the point L of the first Brillouin Zone (BZ), as shown by black points in Fig. 2g. Other satellites are equivalent due to either 3fold rotational axes or the mirror planes of the P6 3 /mmc space group. One third of them are located at the diffuse hexagon edge, and are indicated by open circles in Fig. 2g (L' points). As the honeycomb-shaped diffusion exceeds the first BZ, the behavior of the satellites at the L and L' points are compared throughout the presented work. Remark that in orthorhombic settings (space group Cmc2 1 ), and for a single-twin domain, the satellite is at the reduced wave vector (1, 0, 0.5) [21][22][23] .

Hexagon-shaped diffusion is observed in (h, k, 3.5) and (h, k, 4.5) planes. But it is not visible in the (h, k, 2.5) plane (not shown), where only diffuse spots appear at satellites positions above T P . The increase of the hexagonal DS intensity with l shows that the scattering is of displacive nature, with a predominantly c (z) direction of atomic displacement. In order to understand the origin of the diffuse hexagons, we performed lattice dynamics calculation within density functional theory (DFT) implemented in the CASTEP code 42,43 and following the formalism detailed in 44 . TDS-map calculation of the (h, k, 3.5) plane in the Cmc2 1 space group is shown Fig. 2f. The analysis is presented in detail in the Annexe (see Sec.VII). The calculation shows that the TDS is a result of phonons of an energy of about 6-9 meV, due mostly to an out of phase motion in z-direction of the two Ba ions in the unit cell.

Momentum-space temperature evolution

In order to determine the temperature evolution of the spatial extension of the pretransitional fluctuations above T P , we have measured the profile of the diffuse scattering. This was done at two positions where a satellite appears below T P , and for the position of a diffuse hexagon corner. The satellite L (-0.5, 0, -4.5) and the diffuse hexagon corner A' (-1.0, 0, -4.5) were scanned along the c * direction, as shown by light blue lines in Fig. 2a. The second satellite, L' (3, -1.5, 3.5) was scanned in a * +b * direction (perpendicular to c * ), as shown in Fig. 2c.

The data analysis of the profiles was performed using a Lorentzian function deconvoluted by the experimental resolution. Assuming the applicability of the Ornstein-Zernike correlation function, the half width at half maximum (HWHM) obtained in this procedure gives the inverse correlation length (ξ -1 ) of the spatial , where edges of diffuse hexagons appear. Note that the positions of the points Γ, M, K, M' are not exact, but there is an error of 5% of the reduced wave vector in the h,k-directions. Moreover, in the orthorhombic phase, at least 3 twins coexist. A general q point in experimental data becomes a superposition of nonequivalent points of the reciprocal orthorhombic unit cell. For example, the L point in the hexagonal phase becomes a superposition of T and R points in the orthorhombic phase.

correlations [START_REF] Moret | Structural studies of phase transitions in one-dimensional conductors, Chrystal Chemistry and Properties of materials with Quasi-One-Dimentional Structures[END_REF] . Fig. 4a shows the critical behavior of the inverse correlation length measured at two satellite and the diffuse hexagon corner position. For the diffuse spots at two satellite positions, HWHM decreases to zero at P , in accordance with previous results 23 . The HWHM of the diffuse hexagon corner (A' point) is almost temperature independent as it only slightly decreases when the temperature is lowered.

The generalized susceptibility at a satellite position is given by the intensity maximum of the diffuse scattering weighted by the temperature factor k B T [START_REF]The q dependent diffuse scattering I(q) is proportional to the q Fourier component of the displacement-displacement correlation function < |uq| 2 > which is related in the classical limit to the generalized susceptibility χ(q) by the expression < |uq| 2 > = kBT χ(q) 45 . In the Ornstein approximation χ(q) has a lorentzian shape whose HWHM defines ξ -1[END_REF] . Its reciprocal value shows linear, Curie-Weiss, behavior (see Fig. 4b). The generalized susceptibility measured at the A' point is almost independent of temperature. There is just a broad and shallow dip at T p . DFT calculations performed on the hexagonal and orthorhombic structure are presented in Fig. 6 of the ref. 39 . The comparison with the calculations using the hexagonal structure is not useful here. The low frequency phonon spectrum is strongly perturbed in the vicinity of the Γ point because of the instability leading to the hexagonal to orthorhombic transition. The calculations on the orthorhombic structure show that the phonons in the 0-15 meV energy range are essentially dispersionless besides when approaching the Γ point. In order to make the comparison easier, the corresponding points in the orthorhombic Cmc2 1 space group, used in the DFT calculations, are given in parentheses in Fig. 5.

Temperature evolution of the phonon energy

Fig. 6a-c show IXS energy scans at different temperatures on L, L' and A' points, corresponding to the two satellites and the hexagon-diffuse corner (see Fig. 2g or Fig. 5b). It clearly indicates that the phonon energy does not change with temperature. We can note that upon From these data it is clear that there is no Kohn anomaly at the critical wave vector associated with the MI transition. The Peierls transition in BaVS 3 is not phonon driven. Its character is not displacive, but of order-disorder type.

Quasi-elastic scattering

We are finally interested in the dynamics of the fluctuations which can be deduced from the profile of the IXS central peak measured at the position of a satellite. The central peak profile is analyzed by a Lorentzian function and deconvoluted by the experimental resolution of 2.65 meV (FWHM). Its inverse HWHM measures the lifetime of the pretransitional fluctuations, while its integrated intensity divided by the temperature can be related to the generalized susceptibility. Fig. 8a shows that the energy-HWHM of the central peak decreases almost linearly when approaching T P . The lifetime of the pretransitional fluctuations thus diverges at T P . At the same time, the inverse generalized susceptibility vanishes in a Curie-Weiss-like way, as shown in Fig. 8b. Finally, note that the quasi-elastic scattering measured at the A' point does not exhibit a critical behavior at T P .

IV. DISCUSSION

Experimental results obtained by our DS and IXS investigations presented in Sec.III, show that BaVS 3 exhibits inherent high temperature disorder and a general lattice dynamics which are unusual for a Peierls system. Below we shall detail the uniqueness of these results in separated subsections.

A. The honeycomb-shaped DS

The first unexpected observation was an X-ray diffuse scattering pattern in form of hexagons in l=n+0.5c * planes (n being an integer). It is clearly visible in the temperature range from T P to at least 300 K, as shown in Fig. 2b. Another pattern of diffuse spots, precursor to the Peierls transition, is detected above T P (see Fig. 2d). It will be discussed in the next section (Sec.IV B). The high temperature hexagonal DS is thus due to some lattice disorder which is not directly related to the Peierls instability.

The numerical calculation of the honeycomb-shaped DS (see Annexe) and the DFT calculations of the BaVS 3 orthorhombic structure 39 show that it is mainly caused by an out of phase, c-direction, low frequency motion of the two Ba atoms in the unit cell. Such low frequency phonon spectrum is corroborated by the IXS measurements (see Fig. 5), which will be discussed in Sec.IV C. However, the phonons are not the only contribution to the hexagonshaped diffuse scattering. As shown in IXS scans at A' point (diffuse hexagon corner) represented in Fig. 6c, the (quasi)-elastic contribution is more important than the phonon intensity.

Although there is a relatively good agreement between the measured and calculated DS, there is certainly a more profound explanation for the presence of a so well defined hexagonal shaped diffuse scattering pattern at high temperature in BaVS 3 . Several other systems, with an underlying hexagonal symmetry, show such diffuse scattering [START_REF] Welberry | [END_REF] . For example, hexagonal shaped DS due to cluster short range order has been observed in binary metallic systems 48,49 . Another example are substituted perylene radical cation salts, whose local order can be modeled by a 2D hexagonal-like Ising system with frustrated interactions 50 .

What can be the structural origin of the high temperature Ba disorder in BaVS 3 , giving rise to the hexagonshaped DS? Note that the BaVS 3 structure is composed of succession of hexagonal-packing of BaS 3 layers parallel to the (a,b) plane with V-ions intercalated between them, as shown in Fig. 1. Observed Ba-motion in (mainly) c direction corresponds to an out-of-plane shift from the BaS 3 layers. Reasons for such a shift can be the loss of cohesion in the hexagonal packing of these layers along c, some steric forces causing buckling of hexagonal layers, such as the poorly defined position of V atoms in their octahedral cavities, as proposed by recent structural refinements 21 . It is possible that such DS comes from frustrated interactions acting on Ba sites, resulting in local atomic displacements. This could explain the presence of a non dispersive, Einstein-type, lowest energy phonon branches, which will be discussed in Sec.IV C.

It is not the low crystal quality, but the presence of this inherent dynamical disorder that can explain some properties of BaVS 3 , in particular its relatively low conductivity, or "bad metallicity". Moreover, this peculiar DS, pointing to an inherent disorder, agrees with recent structural refinements 21 . They indicate that, contrary to earlier structural studies, BaVS 3 is in fact a 1D host (V-chain) -guest (hcp BaS 3 matrix) composite system which lock-in to a common space group only below T m = 130 K.

B. Anisotropy and critical diffuse scattering at the Peierls transition

The Peierls transition stabilizes a commensurate modulation with the main Fourier component being the critical wave vector corresponding to the L high symmetry point of the hexagonal structure BZ. In the l=n+0.5c * plane, the superstructure reflections are located either inside the above quoted diffuse hexagons (L point) or at their edge (L' point), as shown in Fig. 2e. Above T P , the satellites transform into anisotropic diffuse spots (Fig. 2d) exhibiting a critical behavior when T P is approached, as previously reported 23 . First, their peak intensity corrected for the thermal population factor has a Curie-Weiss behavior, I m /T ∝ (T -T P ) -1 , as T /I m , plotted in Fig. 4b, vanishes linearly when approaching T P . Moreover, the HWHM of the diffuse spots (q-HWHM) in a chosen direction provides a measure of the inverse correlation length (ξ -1 ) of the Peierls pretransitional fluctuations. This quantity has been measured along the c * direction (ξ -1 c ) and a * + b * direction (ξ -1 ab ). As the CDWs form in V-S chains, running in the c-direction (equivalent to the c * -direction), ξ c (ξ ab ) correspond to correlation lengths in the longitudinal (transverse) direction relatively to the CDW. Both ξ -1 c and ξ -1 ab vanish in a critical way at T P as shown in Fig. 4a. The temperature dependence of ξ -1 c , agrees with the one previously reported in ref. 23 . Fig. 4c reports ξ -1 c in the temperature range where it is always larger than the CDW modulation wavelength, (ξ -1 c < 2k F ). Thus the data have been taken exclusively in the temperature range where the CDW fluctuations in the longitudinal direction are thermally coherent. For the ξ -1 ab , at about 5 K above T P it exceeds 1.48 nm -1 , which corresponds to a = 0.675 nm , the interchain V-V distance. Above this temperature CDW belonging to neighboring chains are not correlated, meaning that the system enters into a true 1D CDW fluctuation-regime. Above 80 K, the temperature dependence of ξ -1 ab deviates from the corresponding measurement in the previous work 23 , as the two measurements have not been performed in exactly the same transverse directions.

Contrary to the diffuse scattering at L(L')-points, the q-HWHM and T /I m temperature dependence measured at the A' wave vector (diffuse hexagon corner) does not exhibit any critical behavior at T P (see open triangles in Fig. 4). This confirms that the two types of DS detected in this study are not directly correlated. Finally, note that the diffuse lines detected until 170 K in the first X-ray diffuse scattering Laue-diagrams of BaVS 3 23 could correspond to sections of the reciprocal space in direction close to the diffuse hexagon edges, which are not directly related to the Peierls instability.

C. Einstein-type phonon modes

The low energy phonon dispersion, measured by IXS at 75K (orthorhombic phase few K above T P ), is presented in Fig. 5. Two phonon branches, at 5-8 meV and 10-13 meV, can be clearly distinguished. The analysis presented in the Annex shows that similar modes are found both at the reciprocal space positions of the diffuse hexagons (A' and L' points) and inside them (A, L, and H points). These two phonon branches exhibit a weak dispersion in the reciprocal space. The data are consistent with the calculated phonon spectrum of the orthorhombic phase 39 . The lower phonon branch corresponds to the calculated phonon bunch, at 7-9 meV, which is particularly dispersionless, with the exception of the region close to the Γ point, the region which is difficult to measure because of the presence of a Bragg reflection. The upper branch coincides with the lower part of the second phonon band, of more dispersive phonons in the 10-20 meV energy range. Low frequency phonon modes of BaVS 3 can thus be described by a model of Einstein oscillators or localized atomic vibrations. This is a quite surprising result because the structure of BaVS 3 is derived from a compact packing where low energy modes are expected to be much more dispersive. This implies the existence of weak inter-atomic coupling or of some competing elastic forces exerted by neighboring atoms. Such a feature can be related to the host-guest structure of BaVS 3 above T m = 130 K.

D. Origin of the order-disorder Peierls dynamics

IXS energy scans at L and L' positions of the critical wave vector of the Peierls transition of BaVS 3 are shown in Fig. 7a,b. One recovers the two low frequency phonon bands previously discussed. Our second unexpected observation is that the position of their maxima, at about 6 meV and 13 meV, respectively, does not change with temperature. Fig. 7c summarizes the temperature dependence of the energy of the two phonons and reveals the absence of the Kohn anomaly in the phonon spectrum. There is thus no critical phonon energy softening when the Peierls transition is approached. This finding agrees with the phonon spectrum calculation in the orthorhombic phase 39 which does not reveal any negative phonon frequency.

Instead, there is a critical growth of a zero-energy response at L and L' points, shown by the increase of the intensity of the central peak (CP), when approaching T P , see Fig. 8b. Indeed, its energy-integrated intensity I i diverges at T P in a Curie-Weiss way, as T /I i vanishes linearly at the Peierls transition. This is, consequently, the same dependence as for T /I m , derived from the intensity of the 2k F Peierls pretransitional diffuse scattering at L and L' points (see Fig. 4b). In addition, the zero-energy response width progressively decreases when lowering the temperature. This shows that the nucleation of clusters happens with a mean lifetime τ which increases when approaching T P . This corresponds to a diffusive dynamics with critical slowing down of cluster fluctuations.

The critical dynamics of BaVS 3 is typical of orderdisorder transition such as have been found in H-bond ferroelectrics 51 . In such phase transitions the low temperature structure is achieved by the critical growth of quasi-static clusters, whose size is given by the correlation length ξ, while their lifetime τ is related to the inverse width of the zero-energy response. In this process the phase transition is attained by nucleation of domains of the low temperature structure inside the high temperature matrix. On a local scale, each atom is located in a double-well potential with the two minima indexed by an Ising variable (pseudo-spin). The local dynamics is controlled by a competition of the atomic thermal jump over the potential barrier and the inter-atomic coupling, described by coupled pseudo-spins 52 . This order-disorder transition exhibits a damped oscillator dynamics 51,52 . The finding of an order-disorder dynamics relying on the presence of local atomic displacements which form clusters is consistent with the observation of an Einstein-like phonon spectrum in BaVS 3 .

There are two classes of models which lead to an orderdisorder type of the Peierls transition: the first class assumes a strong electron-phonon coupling (EPC) 15,53 , while the second class assumes that the Peierls transition occurs in the non-adiabatic regime 9 .

Models assuming a strong EPC are very similar. As proposed by Yu and Anderson 53 , the strong EPC can result in a double-well potential via a polaronic capture mechanism between electrons and ions. In this model, applied to A15 superconductors, the local anharmonic potential is obtained via an interaction of electrons with a non-dispersive optical mode (Einstein mode). The choice of an ion between the two minima of the potential at a particular site i is described by an Ising variable σ i . The CDW state is achieved by the ordering of ions in a lattice of double-well sites. In this model there is similarity between the strong EPC limit of the Peierls instability and the Jahn-Teller effect. Similar features are obtained in strongly EPC chain models developed by Aubry and coworkers 14 . In the model of Gor'kov 15 , applied to transition metal dichalcogenides, an arbitrary ionic movement is considered and different contributions to local elastic potentials are explicitly evaluated. Strong EPC results in trapping of the electron-cloud in the vicinity of an ion i and the creation of a double-well potential. It leads to the Peierls energy gain and fixes the order of magnitude of the CDW gap. The inter-site coupling between the σ i 's is described by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction which oscillates with the 2k F momentum between the pseudo-spins.

In both models assuming the strong EPC, a formation of distinct chemical bonds is expected on a local scale, which manifests itself by a sizable change of inter-atomic distances in the Peierls modulated phase. Across the MI transition of BaVS 3 , the amplitude of modulation of V-V bond distance is of the order of 0.1 Å, 54 which is twice larger than the one (0.05 Å) in the case of the CDW of the blue bronze 9 . In the Peierls ground state of BaVS 3 , there is also a strong deformation of the sulfur octahedra surrounding each of the four nonequivalent V sites, resulting in a variation of V-S distances between different V sites which can be as large as 0.2 Å21,54 . In addition, below T P , the coordination of the V changes from 6 (V-S 6 octahedron) to 5 (V-S 5 square base pyramid) 21 . Such a feature could be also understood as the result of a collective Jahn-Teller instability.

The consequence of the strong variation of V-V and V-S distances induced by strong EPC is the modulation of transfer integrals and the modification of the occupancy of A 1g and E g orbital states on V sites 33,54 . It could have a key role in the localization process of these orbitals. Even if the mechanism is different, it plays a role similar to the electron-electron coupling, inducing an inter-orbital A 1g /E g charge transfer 31 . However the strength of these interactions has not been clearly determined. Note that in the case of strong EPC the concepts of Fermi surface and of nesting are no longer relevant. However, the possibility of an inter-band nesting condition within A 1g /E g1 manifold has been deduced from Angle Resolved Photoemission (ARPES) measurements. 55 An order-disorder Peierls transition is also expected in the non-adiabatic limit in the case of extremely rapid phonon fluctuations compared to the lifetime time of the electron hole pairs (τ eh ), i.e. for a critical phonon frequency such that Ω 2k F > 1/τ eh 9 . In that case the fluctuations of long living 2k F electron-hole pairs cannot screen the inter-atomic interactions in order to form a Kohn anomaly. This situation has not yet be considered in the literature. However, the decrease of the Peierls transition of BaVS 3 under pressure has been interpreted as due to a decrease of τ eh due to a strengthening of the hybridization between the CDW-setting A 1g orbitals and the spin-carrying E 1g orbitals. Also the quantitative fit of T P versus pressure leads to 1/τ eh ≈ 4 meV at ambiant pressure 37 . This leads to a central peak HWHM of 2.3 meV (=4/7 τ eh , according to ref. [ 9 ]) which agrees with the high temperature measurements of Fig. 8a.

We have considered two possible mechanisms of Peierls transition which could account for the order-disorder type of dynamics for the Peierls transition of BaVS 3 . Either the Peierls transition occurs in the strong EPC limit or in the non-adiabatic limit. The two mechanisms are not exclusive. It is interesting to remark that in the literature theories involving a strong EPC do not consider the effect of non-adiabaticity. On the other hand, nonadiabatic processes do not necessarily require a strong EPC. Thus at the present stage of our investigation it is difficult to state which mechanism is appropriated for BaVS 3 .

E. Comparison with other Peierls-like systems

Among the Peierls systems the finding of an orderdisorder Peierls dynamics in BaVS 3 is unique. Another candidate is possibly VO 2 , another non-standard Peierls system [START_REF] Pouget | Basic aspects of the metal-insulator transition of the vanadium dioxide, VO2: a critical review[END_REF] . In more standard 1D Peierls systems such as the Krogmann salt, K 2 Pt(CN) 4 Br 0.3 x 3H 2 O, 6 and the blue bronze K 0.3 MoO 3 , 7 a well pronounced 2k F Kohn anomaly is observed above T P . The same is in the 2D CDW metals 2H-NbSe 2 12 and the tellurides DyTe 3 , TbTe 3 56 and ZrTe 3 57 . A confusing situation occurs for 1D transition metal trichalcogenides where the Kohn anomaly was not detected. From this non-detection, it has been argued that the Peierls transition of NbSe 3 occurs in the strong EPC limit 58 . However, recent phonon spectrum calculation predict the existence of a giant Kohn anomaly in TaS 3 with a negative frequency at 2k F momentum 59 .

On the other hand, in many systems where the spin-Peierls instability occurs, the transition takes place in the non-adiabatic limit 9,60,61 . In such systems, an orderdisorder critical spin-Peierls dynamics is predicted [62][63][64] and observed in inorganic CuGeO 3 , 65 and organic (TMTTF) 2 PF 6 , 66 spin-Peierls compounds.

V. CONCLUSION

Our TDS and IXS measurements show that the Peierls transition in BaVS 3 is an order-disorder transition with damped oscillator dynamics. The two possible models explaining this type of Peierls instability include either a strong electron-phonon coupling or a non-adiabaticity, the two mechanisms being non-exclusive. At the present state of our study it is difficult to state which mechanism is appropriated for BaVS 3 .

VI. ACKNOWLEDGMENT

The help of Aleksandra Chumakova during the experiment as well as continuous discussion with Lázsló Forró and Alla Arakcheeva, particularly concerning recent structural results, are gratefully acknowledged.

VII. ANNEXE

The phonon mode analysis is performed in the l=3.5 plane along the white dotted lines in Fig. 9a:

• A' 1 → A' 2 [(5, -2, 3.5) → (5, -3, 3.5)], correspond-
ing to a diffuse hexagon edge.

• A' 1 → A' 3 [(5, -2, 3.5) → (3, -2, 3.5)], crossing a hexagon center.

The first/second scan crosses reciprocal space positions L'/L, respectively, where satellites appear below T P .

The equivalent portion of the reciprocal space in the reconstructed l=3.5 plane is shown in Fig. 9b. Fig. 10a,b shows the dispersion of low energy phonons in the two scanned directions. In the A' 1 → A' 2 direction, the phonons are dispersionless (see Fig. 10a), while along A' 1 → A' 3 , a small energy variation is observed close to the L-point (Fig. 10b).

Table .I shows phonon energy and TDS intensity depicted in selected points of the reciprocal space, along A' 1 -A' 2 and A' 1 -A' 3 directions. Low energy phonons of about 7-9 meV strongly participate to the TDS intensity.

Table .II gives normal coordinates of six lowest phonon modes. Important components are underlined by bold characters. It shows clearly a prominent Ba-motion, the contribution of S and particularly V being at least 5-10 times smaller. Modes 1 and 3 are mostly an out-of-phase movement of the two Ba atoms in the unit cell. The first, which has particularly important contribution to the TDS intensity, concerns mostly a z-axis movement, even if the the y-axis motion is not negligible. Modes 2/4/6 are, respectively, important x-axis-out-of-phase/yaxis-in-phase/z-axis-in-phase motions of Ba-atoms. I. Calculated energy (E in meV) and TDS intensity (I in arb. units) of first ten phonon modes, as classified by their energy. The analysis is performed in four points in the (h, k, 3.5) plane, which are indicated in Fig. 9a. Six of them belong to the lowest-energy phonon band, which contributes strongly to the TDS intensity, as highlighted by bold characters.

Finally, for the sake of comparison with the Fig. 2b-e, the reconstructions of the (h, k, 3.5) plane at 300 K, 114 K, 74 K and 69 K, presented in the same intensity range, are shown Fig. 11. It shows that the honeycombshaped diffusion is hardly visible at low temperature if the intensity is not magnified.

at. atom mode 1 mode 2 mode 3 nb. type x y z x y z x y z 1 S -0.018638 0.045221 0.005759 0.001864 -0.000473 -0.012033 -0.017230 0.071573 0.031629 2 S 0.012955 0.006649 0.010771 0.035247 -0.025505 0.020174 0.034880 -0.099225 0.000762 3 S -0.011107 0.008052 0.010000 0.028942 0.015266 -0.015181 -0.035594 -0.100014 0.000770 4 S 0.019254 0.045498 0.006630 -0.001752 -0.004019 0.016281 0.016890 0.071239 0.031032 5 S 0.000240 0.035703 -0.033482 -0.011860 -0.001400 -0.000408 -0.000199 0.048889 -0.049679 6 S 0.001243 0.007513 0.018790 0.044826 -0.002350 -0.001278 -0.000820 -0.031234 0.011595 7 V -0.000209 0.007619 -0.035802 -0.006179 -0.000126 0.001224 0.000015 0.006375 0.006592 8 V 0.000012 -0.015077 -0.003328 -0.000416 0.000901 0.000083 -0.000004 0.004271 -0.000796 9 Ba -0.006414 0.252127 -0.670397 -0.161270 0.001964 0.032208 -0.000365 0.572453 0.149188 10 Ba 0.006997 -0.131141 0.670494 0.169363 -0.003925 -0.026332 -0.000107 -0.732379 -0.237025 at. atom mode 4 mode 5 mode 6 nb. type x y z x y z x y z 1 S 0.025130 -0.063164 -0.004178 -0.021247 0.015127 0.009098 -0.002786 0.033906 0.000852 2 S 0.019749 -0.060383 0.027532 0.006365 0.000293 0.007071 0.010326 -0.010352 0.007651 3 S -0.019977 -0.059950 0.027062 0.003714 -0.003397 -0.008965 -0.009612 -0.009688 0.007838 4 S -0.025640 -0.063808 -0.003476 -0.016370 -0.010477 -0.004794 0.003838 0.035239 0.001556 5 S -0.000040 -0.014336 -0.011538 -0.021412 0.001843 -0.001384 0.000770 0.010397 0.006744 6 S 0.000576 -0.043381 -0.043496 0.015734 -0.001214 0.000535 0.000649 -0.007179 0.004498 7 V 0.000001 0.003980 -0.008552 -0.000598 0.000416 -0.001099 -0.000016 0.014587 0.000744 8 V -0.000103 0.006367 0.004775 0.002839 -0.000308 -0.000288 0.000080 0.005872 0.041715 9 Ba 0.006294 -0.588395 -0.129169 -0.053814 -0.019800 -0.016180 -0.000322 -0.005712 -0.708018 10 Ba 0.000950 -0.570885 0.059024 -0.044477 -0.005120 0.015988 -0.000659 0.086928 -0.695822

Table II. Six lowest-energy-mode normal coordinates.

Figure 1 .

 1 Figure 1. Room temperature hexagonal structure of BaVS3 , seen along the c axis, presenting a top BaS3 plane, middle V-layer and a bottom BaS3 layer. aH , bH are for the hexagonal unit cell wave vectors. According to recent structural refinements 21 the Ba/S and V subsystems are not adjusted in terms of space group at RT.

Figure 2 .

 2 Figure 2. Reconstruction of the (h, 0, l) plane at 114 K (a), and the (h, k, 3.5) plane at 300 K (b), 114 K (c), 74 K (d), 69 K (e). Calculated TDS intensity, including only inelastic contribution and corresponding to the Cmc21 space group is shown in (f). It should be compared to the DS map in (c). The first Brillouin Zone (BZ), six shifted first BZ covering a part of the 2nd BZ (red lines), and a diffuse hexagon (green line), are schematically shown in (g). High symmetry points A, H, L, are indicated in the first BZ. A' and L' are the equivalent points out of the first BZ. Black full (open) circles show the positions of the satellites in the first (shifted first) BZ, which are L (L') points, respectively. The first BZ and a diffuse hexagon are shown in (e), while solely a diffuse hexagon position is given in (c) and (f). Light blue lines in (a) and (c) indicate line scans in DS maps. Points A' (diffuse hexagon corner) and L are scanned in c * direction in (a), while L' is scanned in the a * + b * direction in (c). The analysis of the scans is presented in Fig.4. Unit cell wave vectors are given in hexagonal settings.

Figure 3 .

 3 Figure 3. Temperature dependence of the DS intensity at the A' point. It is derived from two distinct measurements, performed on two different samples. Sample 1 (red points) is a pure BaVS3 , studied in this work. Black points are revisited data from a slightly Al-doped sample, studied previously 39 .The intensity of the second measurements is normalized in order to obtain the same intensity at a common temperature of about 110 K.

Figure 4 .

 4 Figure 4. (a) momentum HWHM (in nm -1 ) and (b) T/Im (temperature over intensity maximum) of diffuse spots at satellite positions L'-point (3, -1.5, 3.5) and L-point (-0.5, 0, -4.5), scanned, respectively, in a * + b * and c * directions. The HWHM and T/Im measured in the c * direction at the position of a diffuse hexagon corner A'-point, (-1.0, 0, -4.5), are shown for comparison. TP is for the temperature of the Peierls transition. 2kF and 1/a indicate, respectively, the CDW modulation wave vector and the inverse minimal distance between two V in the (a,b) plane.

Figure 5 .

 5 Figure 5. (a) q-dependence of the energy of the two phonons along high symmetry lines, measured at 75 K. High symmetry points are indicated for the hexagonal (orthorhombic) BZ. (b) The first BZ with the explored high symmetrylines, Γ-A-L-M-K-H-A (red arrows). Additional path is explored out of the first BZ, A'-L-H-L'-M' (blue arrows), where edges of diffuse hexagons appear. Note that the positions of the points Γ, M, K, M' are not exact, but there is an error of 5% of the reduced wave vector in the h,k-directions. Moreover, in the orthorhombic phase, at least 3 twins coexist. A general q point in experimental data becomes a superposition of nonequivalent points of the reciprocal orthorhombic unit cell. For example, the L point in the hexagonal phase becomes a superposition of T and R points in the orthorhombic phase.

Figure 6 .

 6 Figure 6. IXS scans at two satellite positions, (a) L point at (0, -0.5, 4.5), (b) L' point at (0.5, 0.5, 4.5) and (c) at the hexagon-diffuse corner, A' point at (0, -1, 4,5). They are fitted with three structures: a central peak (CP, black), a low energy phonon (LE, red) and a high energy phonon (HE, yellow).

Figure

  Figure.5a shows the phonon dispersion measured by IXS. In the energy range 0-15 meV, there are two phonons bands: the low energy phonon is indicated by red points, and the high energy phonon by black triangles. The path followed along high symmetry lines is indicated in Fig.5b. The phonon energy shows very small variations besides Γ-A, A-L and A'-L lines. Note that in A', the high energy phonon has negligible intensity. Moreover, in A, the low energy phonon has two components.DFT calculations performed on the hexagonal and orthorhombic structure are presented in Fig.6of the ref.39 . The comparison with the calculations using the hexagonal structure is not useful here. The low frequency phonon spectrum is strongly perturbed in the vicinity of the Γ point because of the instability leading to the hexagonal to orthorhombic transition. The calculations on the orthorhombic structure show that the phonons in the 0-15 meV energy range are essentially dispersionless besides when approaching the Γ point. In order to make the comparison easier, the corresponding points in the orthorhombic Cmc2 1 space group, used in the DFT calculations, are given in parentheses in Fig.5.

Figure 7 .

 7 Figure 7. Temperature evolution of the phonon (a) energy, (b) HWHM, measured at the two satellite positions. The error bars of the HWHM of the high energy phonon being too large, the data are not shown.

Figure 8 .

 8 Figure 8. IXS central pic (a) energy HWHM and (b) temperature over integrated intensity (T/Ii), measured at two satellite positions, L point at (0.0, -0.5, 4.5) (red circles), L' point at (0.5, 0.5, 4.5) (black squares), and the position of the corner of a diffuse hexagon, A' point at (0, -1, 4.5) (open black triangles). Horizontal arrow indicates the energy resolution of 1.325 meV (HWHM). The vertical arrow is for the temperature of the metal-insulator transition, as determined from the conductivity measurements. Dash-lines in (b) are for eye-guides. Data are extracted from the scans shown in Fig.6.

Figure 9 .

 9 Figure 9. (a) Calculated TDS intensity in the (h, k, 3.5) plane, corresponding to the Cmc21 space group, indexed in hexagonal settings. The green hexagon, white dotted lines and particular points A'1, L', A'2, A, indicate the portion of the momentum space where the phonon analysis was performed. (b) Reconstruction of the measured (h, k, 3.5) plane at 114 K. The green hexagon indicates the same region as in (a).

Figure 10 .

 10 Figure 10. Dispersion of low energy phonons along A'1 → A'2 (a) and A'1 → A'3 (b) directions in reciprocal space.

Figure 11 .

 11 Figure 11. Reconstruction of the (h, k, 3.5) plane at 300 K, 114 K, 74 K, and 69 K, presented in the same intensity range.


Victor Anaya
email: vanaya@cigip.es

Giuseppe Berio
email: berio@di.unito.it

Mounira Harzallah
email: mounira.harzallah@univ-nantes.fr

Patrick Heymans

Raimundas Matulevicius

Andreas L Opdahl
email: andreas.opdahl@uib.no

Hervé Panetto
email: herve.panetto@cran.uhp-nancy.fr

Maria Jose Verdecho
email: mverdecho@cigip.es

The Unified Enterprise Modelling Language -Overview and Further Work

The Unified Enterprise Modelling Language (UEML) aims to support integrated use of enterprise and IS models expressed in a variety of languages. The achieve this aim, UEML provides a hub through which different languages can be connected, thereby paving the way for connecting the models expressed in those languages. UEML offers a structured approach to describing enterprise and IS modelling constructs, a common ontology to interrelate construct descriptions at the semantic level, a correspondence analysis approach to estimate semantic construct similarity, a quality framework to aid selection of languages, a meta-meta model to organise the UEML and a set of tools to aid its use. This paper presents an overview of UEML and points to paths for further work. 

INTRODUCTION

Emerging information and communication technologies are increasingly model-driven. Unfortunately, they are often driven by models that cannot easily be interrelated because they are expressed using languages that are not interoperable. In consequence, the models can become inconsistent. Instead of producing more adaptable and integrated ICT solutions, model-driven technologies therefore run the risk of reinforcing existing interoperability problems as different information systems evolve driven by models expressed in incommensurable languages. The situation has created a need for theories, technologies and tools that allow information systems be adapted and evolve each driven by the most suitable languages, while allowing the systems and their models to be used in an integrated manner.

The Unified Enterprise Modelling Language (UEML) refers to an on-going attempt to develop theories, technologies and tools for integrated use of enterprise and IS models expressed using different languages. By this we mean keeping the existing models as they are and, in addition, establishing correspondences between them in an explicit and usable way. Useful services are consistency checking, automatic update reflection, model-to-model translation and others across modelling language boundaries. UEML would thereby act as a hub connecting different languages along with the different models expressed in those languages. UEML comprises:  a structured approach to describe enterprise and IS modelling constructs,  an evolving common ontology to describe the semantics of modelling constructs,  a correspondence analysis approach that uses the common ontology to determine semantic correspondences between constructs,  a quality framework to define and evaluate the quality of enterprise modelling languages to aid language selection for specific purposes,  a modular meta-meta model to organise the overall UEML approach and  a set of tools to aid its evolution and use.

The purpose of this paper is to present an overview of UEML and discuss paths for further work. The paper is organised as follows: Section 2 presents UEML's background and its vision. Section 3 explains how languages and constructs are described in UEML, and Section 4 shows how descriptions of constructs are tied together by a common ontology. Section 5 discusses how correspondences between languages and constructs can be established and used to support model-tomodel translation across languages. Section 6 shows how enterprise modelling languages are classified and selected in UEML according to specific goals. Section 7 presents the Anaya V., Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008). The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of the IFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08 meta-meta models that holds the UEML approach together, and Section 8 reviews the various prototype tools supporting its evolution and use. Section 9 discusses UEML in itspresent state, before Section 10 concludes the paper.

BACKGROUND

The idea of a Unified Enterprise Modelling Language first emerged during the ICEIMT'97 conference [START_REF] Goossenaerts | Formal Semantics of Enterprise Models[END_REF], with the aim of providing an underlying formal theory for enterprise modelling languages. A major motivation was the "Tower of Babel" situation that was assumed to hinder proliferation of enterprise modelling in industry [START_REF] Vernadat | UEML: Towards a Unified Enterprise Modelling Language[END_REF]. The first development version of a unified enterprise modelling was done by the UEML Thematic Network (UEML TN) (2002)(2003), funded by the EU's FP5 [START_REF] Jochem | Common representation through UEMLrequirement and approach[END_REF][START_REF] Panetto | A Unified Enterprise Modelling Language for enhanced interoperability of Enterprise Models[END_REF][START_REF] Mertins | User oriented Enterprise Modeling for Interoperability with UEML[END_REF], Berio, Anaya & Ortiz 2004). UEML development has since continued within the Interop-NoE Network of Excellence (2003Excellence ( -2007)) As UEML becomes more stable, it may be possible to extract a core set of modelling construct to use as the starting point for a new enterprise/IS modelling language, a UEML core language composed of those constructs that have proven most useful for practical, integrated model use. However, the core language scenario, should be understood as a longer term objective, beyond the scope of this paper.  Facilitating a web of languages and of models.

Whereas much research and development effort has gone into techniques and tools for integrated management of structured data (e.g., relational database theory) and of semi-structered data (e.g., XML and other web technologies), there is a lack of theory and tehcnology for integrating information resources in the form of diagrammatic models. UEML could also contribute to growing a web of languages and of models in a way that resembles the touted semantic web of semi-structured data (Berners-Lee, Hendler, Lassila 2001).

LANGUAGE AND CONSTRUCT DESCRIPTION

UEML facilitates integrated model use by making semantic correspondences between the modelling constructs of different languages clear.

Making the languages interoperable is seen as a first step towards also making the models expressed in those languages interoperable. A central part of UEML is therefore a standard, integrative and evolvable approach to describing enterprise and IS modelling constructs. By standard we mean that the approach provides a structured path to describing modelling languages, diagram types and constructs. By integrative we mean that as soon as the languages, diagram types and constructs have been described according to the approach, they have become prepared for assessment of semantic correspondences, possibly across languages. And by evolvable we mean that UEML will be able to grow and adapt by incorporation and modification of additional modelling languages and constructs without becoming overly complex and thus unmanageable.

Anaya V., Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008). The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of the IFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08 The descriptions of individual modelling constructs are particularly important, because it is this level that connects different modelling languages. Hence construct descriptions are more complex than descriptions of languages and diagram types. Specifically, in UEML, two distinct descriptions need to be made for each construct:

 Presentation (or concrete syntax), which deals with the presentation of the modelling construct as part of model diagrams or in serialised form, e.g., in an XML file.  Representation (or semantics), which accounts for which enterprise phenomena the construct is intended to represent (in particular covering reference, a central aspect of semantics). Whereas a construct can have many presentations, it can have only one representation. This paper will focus on the representation part, which has so far been most developed [START_REF] Opdahl | The UEML Approach to Modelling Construct Description[END_REF].

In UEML, semantics is described by a representation mapping of each modelling construct into a common ontology, based on earlier work by [START_REF] Opdahl | A Template for Defining Enterprise Modelling Constructs[END_REF][START_REF] Krogstie | Evaluating UML Using a Generic Quality Framework[END_REF]. The UEML approach uses separation of reference to break individual modelling constructs into their ontologically relevant atomic parts along the following six axes:

1. Which class(es) of things is the construct intended to represent? Most modelling constructs somehow represent one or more classes of things. Even when the primary purpose of a construct is to represent certain properties, states or transformations, the construct implicitly also represents a property of, state of or transformation in, one or more classes of things. (A transformation may be either an atomic even or a complex process.)

2. Which properties is the construct intended to represent? Most modelling constructs somehow represent one or more types of properties, which may either be intrinsic properties (belonging to only one thing) or relationships (properties that are mutual to several things). Some intrincis properties are laws that restrict other propertis. Even if the primary purpose of a construct is to represent classes, states or transformations, it represents classes, states or transformations that involve one or more types of property.

Which states is the construct intended to represent?

Some modelling constructs are intended to represent a more or less restricted state in one or more classes of things. The state law that restricts the state can be described in terms of the properties of those classes. Whereas most modelling constructs represent one or more properties and, at least, one or more classes, not all constructs are intended to represent a state. 4. Which transformations is the construct intended to represent? Some constructs are intended to represent a simple or complex transformation of one or more classes of things from one state to another.

The transformation law that effects the transformation can be described in terms of the states of those classes. Again, not all constructs are intended to represent a transformation. Although some constructs are apparently not intended to represent behaviour at all, other constructs represent particular states or transformations or chains of alternating states and transformations, i.e., processes.

Which instantiation levels is the construct intended

to represent? A modelling construct represents classes, properties, states and transformations at either the instance or type level or both. Anaya V., Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008). The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of the IFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08 6. Which modality (or mode) is the construct intended to represent? We usually think of enterprise models as assertions of facts about a domain, e.g., assertions that something is the case or is not the case in the enterprise. But some model elements may instead state that someone wants something to be the case, or that someone is not permitted to do something, or that someone knows something is the case¸ or that something will be the case some time in the future.

We call such statements modal (as opposed to regular) assertions, i.e., we use the term "modal" pretty much in the modal logic sense. Hence, whereas the two first axes deal with structure, the next two deal with behaviour. Together, these four axes describe the semantics of a modelling construct by describing a state of affairs, or a scene, played by several classes, properties and, perhaps, states and transformations together. The final two axes supplement the scene with information about the construct's intended use, i.e., its instantiation level and modality/mode.

The UML class diagram in Figure 1 shows the key concepts used to describe modelling languages and constructs in UEML. The upper part of the diagrams depicts modelling languages, along with their diagram types and modelling constructs. The lower part shows how each individual construct is described by a scene of interrelated classes, properties, states and transformations that the construct is intended to represent. (Construct presentation is not shown in Figure 1.)

THE COMMON ONTOLOGY

To tie modelling-construct descriptions together, UEML uses a common ontology into which the represented classes, properties, states and transformations of each construct are mapped. The common ontology thereby comes to interrelate the construct descriptions at the semantic level.

The UEML ontology is organised into four taxonomies: The classes in the ontology are organised in a conventional generalisation hierarchy. Properties, on the other hand, have their places in a precedence hierarchy, in which a property precedes another if every thing that possesses the second property must also possess the first. (For example, associated-with precedes having-content, because everything that is having-content is also associated-with that content.) There are also generalisation hierarchies of states and of transformations.

Classes, properties, states and transformations -including the state and transformation laws -all have attributes. For example, they all have unique names and there are cardinality constraints and role names on the associations between classes and properties.

The four taxonomies are interrelated. Classes are related to the properties that characterise them. Properties are related to the states they define. States are in turn entered and exited by transformations. Certain types of properties are laws that restrict other properties. State laws restrict states, whereas transformation laws effect transformations. The resulting organisation of the UEML ontology as four distinct, but interrelated taxonomies makes it possible to evolve the ontology over time without increasing complexity more than necessary. New classes, properties, states and transformations will always have a clearly identifiable location where they can be added to the appropriate taxonomy.

The UML class diagram in Figure 2 shows the key concepts of the common ontology, also based on the earlier work of [START_REF] Opdahl | A Template for Defining Enterprise Modelling Constructs[END_REF][START_REF] Krogstie | Evaluating UML Using a Generic Quality Framework[END_REF]. For every construct incorporated into UEML, each represented class, property, state and transformation is mapped into an ontology concept in the ontology. Figure 2 therefore structurally resembles the lower part of Figure 1.

The UEML ontology was first populated with a set of initial classes, properties, states and transformations derived directly from Bunge's ontological model [START_REF] Bunge | Ontology I: The Furniture of the World[END_REF][START_REF] Bunge | Ontology II: A World of Systems[END_REF] and the Bunge-Wand-Weber representation model of information systems, the so-called BWW model (Wand & Weber 1988[START_REF] Wand | On the ontological expressiveness of information systems analysis and design grammars[END_REF][START_REF] Wand | On the deep structure of information systems[END_REF]. Since then, it has evolved and grown as new constructs have been added. Currently, UEML incorporates a selection of academic and industrial modelling languages, such as ARIS, BMM, BPMN, coloured Petri nets, GRL, IDEF3, ISO/DIS 19440, KAOS, UEML 1.0 and selected diagram types from UML 2.0. In consequence, the most general concepts in the common ontology are ontologically committed, in the sense that they have grown out of Bunge's ontology and the BWW model, whereas the more specific ones have emerged through language and construct analyses.

LANGUAGE AND CONSTRUCT CORRESPONDENCES

Correspondences between any pair of constructs can be examined by comparing their mappings into the common ontology. All the modelling constructs in UEML thereby become interrelated at the most detailed level possible via the common ontology. If two modelling constructs are identical, they will map into the exact same ontology concepts. If two modelling constructs do not overlap at all, they will map into completely distinct concepts, i.e., ones that are not even closely related in their respective taxonomies. The third case is likely to be most common, where two modelling constructs map into some identical ontology concepts, some ontology concepts that are closely related and some ontology concepts that are not.

To support integrated use of models, UEML must offer ways to exploit the representation mappings to identify and manage correspondences among language constructs and among model elements. Construct correspondence refers to whether constructs refer to distinct (in several ways) or identical states of affairs in the problem domain. Three kinds of correspondences have been identified. Each of them can be precisely formulated in terms of the ontology classes, properties, states and transformations into which the constructs in the correspondence have been mapped. Of course these simple kinds of correspondences are not independent. Equal constructs will trivially contain and generalise one another. There are also complex correspondences, such as when one construct represents a state of affairs that generalises a part of the state of affairs represented by another, thus combining containment and generalisation. There are also overlapping constructs, each of which contains part, but not all, of the other. However, a complete typology of correspondences and how they combine stills needs to be worked out.

Correspondences are also characterised by different degrees of precision. For example, it is possible to only take into account how each construct is mapped into ontology concepts, ignoring how the concepts are related within the construct description. More precise correspondences can be identified by taking into account both ontology concepts and the relations between them, but ignoring the roles that the concepts may play in the relations. Finally, both the ontology concepts, the relations between them and the roles they play can be taken into account. Using different degrees of precision may be useful in order to to master complex correspondences and when dealing incomplete representation mappings and/or ontology.The work is in progress on deriving measures of correspondence between pairs of modelling constructs, providing evidence of kinds of correspondence with various degrees of precision. The measures are inspired by measures used to compare objects in the areas of classification theory and knowledge engineering [START_REF] Lin | An information-theoretic definition of similarity[END_REF][START_REF] Rodrýguez | Determining semantic similarity among entity classes from different ontologies[END_REF][START_REF] Blanchard | A treebased similarity for evaluating concept proximities in an ontology[END_REF].

Correspondence measures are also useful for validating the representation mappings and the common ontology. Correspondence measures derived automatically from the common ontology can be compared to expert estimates of the same correspondences. Deviations indicate either that the representation mapping is wrong for a construct or that the common ontology is not optimally organised. The two can also occur together, when the representation mapping is wrong because there are concepts missing from the common ontology. Another ontology problem that can be detected is missing taxonomical relations between ontology concepts, e.g., a missing generalisation relation from a sub-to a superclass. If left undetected, missing relations can lead to redundancies in the common ontology when the subclass is added again as a specialisation of the superclass. In this way, correspondence measures can also aid eliminating redundancy in the common ontology.

Correspondence measures as representative of correspondences are useful as high-level guides for model-tomodel translation and similar cross-language services. The representation mappings and common ontology provide the details for how to translate between modelling constructs belonging to different languages, as soon as the pair of modelling constructs to translate between have been decided. But it offers less help with selecting which constructs in one language to translate into which other constructs in the other one in the first place. The correspondence measures potentially aid this language-level issue by indicating, for each construct in a language, which constructs in the other language are most suitable as targets for, e.g., translation, leaving the final choice to the model manager. When the language-level construct-to-construct correspondences have been established in this way, the representation mapping and common ontology will support the detailed construct-level mappings.

LANGUAGE QUALITY FRAMEWORK

Together, the representation mappings, common ontology and correspondence measures contribute towards integrated use of models expressed in different languages. But there is also a need to select suitable languages to include in the UEML first place. For example, to quickly enrich the common ontology, it may be better to incorporate soon an almost complete and used language than a very narrow language used by specific communities. Later, when using UEML, there is a need to select suitable languages for particular purposes among the many available. For these purposes, UEML includes a language quality framework [START_REF] Anaya | Evaluating Quality of Enterprise Modelling Languages: The UEML Solution[END_REF], which aids language selection by  defining the concept of quality of a modelling language;  supporting methodical, goal-dependent evaluation of the quality of enterprise modelling languages. The current quality framework has adapted and extended SEQUAL quality framework [START_REF] Krogstie | Using a Semiotic Framework to Evaluate UML for the Development for Models of High Quality[END_REF][START_REF] Krogstie | Evaluating UML Using a Generic Quality Framework[END_REF], which provides a model of the quality of models, later extended to also account for the quality of languages. SEQUAL identifies 8 quality types for characterising what quality is: physical quality, empirical quality, syntactic quality, semantic quality, perceived semantic quality, pragmatic quality, social and organisational quality. For example, semantic quality is the correspondence between the model and the domain. SEQUAL also identifies several types of appropriateness, each indicating a language aspect that must be considered when assessing whether a language is appropriate for a particular purpose [START_REF] Krogstie | Using a Semiotic Framework to Evaluate UML for the Development for Models of High Quality[END_REF][START_REF] Krogstie | Evaluating UML Using a Generic Quality Framework[END_REF]. For example, comprehensibility appropriateness reflects the ease with which the language its model can be understood by a certain audience. In SEQUAL, each quality type is related to one or more appropriateness types and vice versa. For Anaya V., Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008). The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of the IFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08 example, domain appropriateness is used to assess physical and semantic qualities. Therefore, the different types of appropriateness provide the context to evaluate the related quality types. In addition to SEQUAL, the UEML quality framework has been inspired by two additional quality frameworks: Moody's framework ( 2003) and ISO/IEC 9126 international standard for assessing software product quality (ISO/IEC 2001). These additional frameworks have been adapted and aligned with SEQUAL's appropriateness types through a generalisation hierarchy [START_REF] Berio | Deliverable DEM2[END_REF].

The resulting appropriateness types in the UEML quality framework remain too general to allow concrete evaluations [START_REF] Anaya | Evaluating Quality of Enterprise Modelling Languages: The UEML Solution[END_REF]. Therefore, the framework also covers requirements and criteria. Requirements are collected from users (actors or experts), asking them how enterprise modelling should contribute towards enterprise integration and interoperability, based on a requirements base established in the previous UEML Thematic Network (UEML-TN 2003). are the operational, or measurable, counterparts of requirements. Each criterion can in turn be related to one or more appropriateness types, making it precise to which quality types that criterion contributes. The framework provides two complementary ways of collecting data for evaluating criteria. The language template is used to gather general and factual information about a language, such as its notations and meta models, whereas the language-evaluation questionnaire comprises both questions derived from current criteria and an associated glossary.

The framework also introduces language descriptions, covering, e.g., a language's owner and version; goal, an aggregation of criteria providing the purpose for evaluating language quality; metrics-for-goal, selected metrics relevant to a specific goal (metrics are needed to perform criteria assessment); metric evaluation, specific evaluation (for instance, a value) of a single metrics on a specific language; combined metrics evaluation, combined evaluation of several metrics evaluations for a given language and a given goal (an explicit combined metrics evaluation makes explicit how several single metrics are combined -for instance, with a weighted formula -to evaluate quality of a language wrt a given goal; additionally, it is useful because the same metrics evaluation can, if needed, be used several time).

The UML class diagram in Figure 3 shows the key concepts used to evaluate the quality of modelling languages in UEML. The associated quality evaluation method gives a clear picture of how to evaluate and select one or more enterprise modelling languages for a specific purpose. The first task is to define the goal as aggregation of criteria and then select suitable metrics for each criterion. A list of languages to be evaluated is set. The language template is used to collect factual information about each language, whereas the language-evaluation questionnaire is used to collect subjective opinions. Hence, whereas only a single filled-in language template is needed for each language, multiple filled-in questionnaires are usually needed. Once the selected criteria are assessed by using selected metrics and storing these assessments as metrics evaluations, combined metrics evaluations are calculated and stored. Finally, languages must be suitably selected based on the results stored as combined metrics evaluations. Before its use, one specific enterprise may undertake a customisation of the quality framework: This simply means to define additional requirements, appropriateness types, criteria and metrics.

META-META MODELS

The UML class diagrams of the language and construct description approach (Section 3), of the common ontology (Section 4) and of the quality framework (Section 6) are all Anaya V., Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008). The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of the IFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08 meta-meta models. They are meta-meta models because models of modelling languages are meta models and because Figure 1-3 are models of how to model modelling languages (or of how to model meta models). The UML diagrams are intended as illustrations only. For example, Figures 12do not show attributes and omit several association classes and abstract classes.

Whereas the representation mappings connect Figures 1 and2, the meta-meta model of the quality framework in Figure 3 is currently connected to Figure 1 only through the language description. The Conclusion will point out that the idea is to create a single combined, yet still modular, meta-meta model that covers all constituents of the UEML approach, the overall UEML meta-meta model.

TOOLS

UEML is supported by a set of prototype tools realised using a selection of existing technologies. There are currently five tools in the set:

 UEMLBase Repository is a Protege-OWL realisation of the representation and ontology meta-meta models of Figures 12, translated into OWL.  UEMLBase Editor is an emerging set of Eclipse GMF-based editors for browsing and updating the contents of the UEMLBase repository.  UEMLBase Manager is a Java-plugin for Protege-OWL that provides merging, reporting and other housekeeping functions for the repository.  UEMLBase Verifier is a set of Prolog rules and a Prolog rule checker that support formal verification of the concents in the UEMLBase repository, for example to check cardinality constraints and ensure that construct descriptions are concrete.  UEMLBase Correspondence Analyser uses the repository to compute similarity measures between UEMLBase constructs, paving the way for consistency checking, automatic update reflection, model-to-model translation across languages, as well as other integrated model uses. Each tool strives to be consistent with the meta-meta models presented in Section 7, although they all use more specific implementation models, such as OWL, Eclipse EMF, Java classes and Prolog facts. Hence, the meta-meta models is used to support interoperability within the UEML tool set.

DISCUSSION

The paper has presented the main constituents of the UEML approach and explained how they are related. Languages, possibly selected with the aid of the quality framework, are described using separation of reference according to the structured approach of Section 3. The descriptions of the states of affairs are then mapped into the common ontology of Section 4. It thereby becomes possible to establish correspondences between different constructs in terms of their mappings into the common ontology as in Section 5. The selection of modelling languages is guided by the quality framework of Section 6. In the long term, the most used and useful concepts in the common ontology can be used to form a core UEML language for enterprise and IS modelling. In the long tern, UEML could also contribute towards developing a web of languages and of models in a way that resembles the touted semantic web of semi-structured data (Berners-Lee, Hendler, Lassila 2001).

From an initial set of around 25 concepts taken more or less directly out of Bunge's ontology and the BWW model, the common UEML ontology has grown to comprise 110 concepts. Most of them have resulted from analyses of individual modelling constructs using separation of reference. (A few initial higher-level remain to organise and structure the four taxonomies.) As part of the Interop-NoE work, 130 constructs from the following 10 languages have been mapped into this ontology ((add references here!!!)): ARIS, BMM, BPMN, GRL, IDEF3, ISO/DIS 19440, KAOS, coloured Petri nets, UEML 1.0 and selected diagram types from UML 2.0. However, they are not all described in equal detail and none of them are yet fully validated. The languages, constructs, mappings and ontology have all been stored in the UEMLBase Repository, supported by the Editor, Manager, Verifier and Correspondence Analyser tools.

The standardised approach to language and construct description has turned out to have several advantages, in particular at the modelling construct level. The structured descriptions become complete, consistent, cohesive and, thus, more learnable and understandable. It therefore becomes easier to compare them to one another. The structured approach also offers systematic and detailed advice on how to proceed when analysing individual language constructs. It encourages highly-detailed construct description, which leads to languages that are integrated at a fine level of detail. It supports ontological analysis in terms of particular classes, properties, states and events, and not just in terms of the concepts in general.

The UEML approach has positive network externality, in the sense that incorporating an additional construct or language becomes:  more valuable the more constructs and languages that have already been incorporated, because the additional language becomes interoperable with a lrger number of other languages;

 less costly because reusing an enriched common ontology and existing representation mappings provide good reference examples and because the cost of maintaining tools and infrastructure can be shared by more UEML users.

Similar positiv network externality effects can be expected at the model level beside the langauge level discussed here.

Early experience with the construct description approach indicated that it was difficult to use because it was based on a novel, unconventional way of thinking about the semantics of modelling constructs. It was sometimes hard to find the appropriate classes, properties, states and events in the common ontology to use when describing a construct. Also, Anaya V., Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008) it was sometimes hard to determine exactly which part of a language that constitutes a modelling construct. As part of the Interop-NoE, tools and tutorials were developed that have seemingly resolved many of these problems. Also, early drafts of the common ontology have become available along with exemplary representation mappings. As a result, the first draft of several of the most recent language incorporations could be made by students with little direct supervision.

The framework for selecting and evaluating the quality of modelling languages according to specific goals also provides high benefits for users that need to decide which languages to use for practical puposes. First, it gets the voice of the customer through the consideration of the requirements of the users making them to appear in the front end of the framework. Then, these requirements are related to criteria that make them operational and applicable to the language evaluation.

CONCLUSION AND FURTHER WORK

UEML is an ambitious, long-term effort that will require several years of cooperation between academia and industry. The overall challenge for further work is to extend the theory and tools developed by the Interop-NoE network to support practical integrated use of models and languages. Although several limited paper-and-pencil trials have demonstrated the feasibility of the approach [START_REF] Berio | Deliverable DEM2[END_REF]Matulevicius, Heymans, Opdahl, 2007;Harzallah, Berio, Opdahl, 2007), detailed methods for integrated model use still need to be developed and implemented.

For UEML-supported integrated model use to be tested in large-scale, realistic settings, the common ontology and representation mappings must be verified, validated and improved. The current ontology and mappings have been contributed by several Interop-NoE research teams working in a distributed manner. The most immediate challenge is to improve the ontology and mappings in two directions. Firstly, the Editor and Verifier tools are being extended and improved. Secondly, the Correspondence Analyser tool is used to compare correspondences calculated from the common ontology and the representation mappings with correspondence estimates provided by human experts. The comparisons are used to identify weaknesses in the representation mappings. For example, when two constructs are considered similar by human experts, but not by the Correspondence Analyser, the reason might be that one or more ontology concepts have been duplicated. Accordingly, when the Analyser, but not the human experts, deem two constructs similar, the reason may be weaknesses in the generalisation hierarchies in the ontology. In this way, verification not only supports improving the representation mappings but also controls the quality of the common ontology.

As for the overall UEML approach, an obvious path for further work is to connect the meta-meta models for language and construct description and for the common ontology with the one for the quality framework. Also, the combined metameta model must be extended to account for the presentation part of language and construct description and for construct correspondences. In addition to tying together the overall approach, this work can be expected to reveal further possibilities, such as deriving quality and appropriateness metrics for languages, not only at the language level, but also at the construct level from the detailed UEML ontology and mappings.

These and other possible future developments have been organised in a UEML roadmap comprising several research directions, each detailed by specific actions (Opdahl & Berio 2006): 1. Language breadth -include more languages; 2. Ontological depth -refine the common ontology; 3. Ontological clarity -elaborate the common ontology language; 4. Presentation -extend the support for presentation issues; 5. Mathematical formality -define UEML semantics formally; 6. Tool support -develop prototype tool with GUI and validation support; 7. Model management -provide support for model management in addition to language management; 8. Validation -structural and behavioural language and model validation; 9. Dissemination -make UEML known in industry and academia and as a standard; 10. Community -establish and maintain a committed and cohesive community for managing and evolving UEML and its approach. Additional directions that deal specifically with the language quality framework are: 1. Continuing the development of the quality framework by introducing new criteria and extending the questionnaire accordingly; 2. Continuing the accommodation of existing quality frameworks by specialising appropriateness; 3. Gradually developing supporting tools based on the metameta model, starting from the current simple support for filling-in the questionnaire to complete functionality to define and evaluate metrics; 4. Launching use of the quality framework and especially by performing evaluations of languages for developing a core language. For example, more specific quality frameworks can be used to systematically introduce new appropriateness measures and to specialise existing ones. The roadmap still needs to be extended to account better for correspondence analysis.

The UEML approach may even be useful outside enterprise and IS modelling, e.g., for software modelling. Significantly, only the language quality framework is specific to enterprise modelling. The other major UEML parts might be used for a wider set of modelling domains.

Figure 3 :

 3 Figure 3: The common UEML ontology, into which all the construct descriptions are mapped.

Figure 1 :

 1 Figure 1: The main classes in the UEML representation meta-meta model.

Figure 3 :

 3 Figure 3: The UEML language quality framework.

  Anaya V.,Berio G., Harzallah M., Heymans P., Matulevicius R., Opdahl A.L., Panetto H., Verdecho M. (2008). The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of the IFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08 one can replace the other, e.g., during model-tomodel translation.  Containment occurs when the state of affairs represented by one construct has the state of affairs represented by another as a part. When one construct contains several others, the former may have to be replaced by the others during model-tomodel translation.  Generalisation occurs when one modelling construct represents a state of affairs that generalises the state of affairs represented by another. If one construct generalises another, the general construct can replace the special one in a model-to-model translation (with loss of information), but the inverse replacement is not always appropriate.

 Equality occurs when two or more constructs represent the exact same state of affairs, as explained in Section 2. If two constructs are equal,

  . The Unified Enterprise Modelling Language -Overview and Further Work. Keynote paper. Proceedings of theIFAC World Congress, 118895-11906, July 6-11, Seoul, Korea, IFAC Papersonline, ISBN 978-1-1234-7890-2/08