E Bévillon

J P Colombier

V Recoules

R Stoian

Free electron properties of metals under ultrafast laser-induced electron-phonon nonequilibrium: a first-principles study

Keywords: numbers: 81.40.-z, 79.20.Ds, 72.15.Lh

The electronic behavior of various solid metals (Al, Ni, Cu, Au, Ti and W) under ultrashort laser irradiation is investigated by means of density functional theory. Successive stages of extreme nonequilibrium on picosecond timescale impact the excited material properties in terms of optical coupling and transport characteristics. As these are generally modelled based on the free electron classical theory, the free electron number is a key parameter. However, this parameter remains unclearly defined and dependencies on the electronic temperature are not considered. Here, from first-principles calculations, density of states are obtained with respect to electronic temperatures varying from 10 -2 K to 10 5 K within a cold lattice. Based on the concept of localized or delocalized electronic states, temperature dependent free electron numbers are evaluated for a series of metals covering a large range of electronic configurations. With the increase of the electronic temperature we observe strong adjustments of the electronic structures of transition metals. These are related to variations of electronic occupation in localized d-bands, via change in electronic screening and electron-ion effective potential. The electronic temperature dependence of nonequilibrium density of states has consequences on electronic chemical potentials, free electron numbers, electronic heat capacities and electronic pressures. Thus electronic thermodynamic properties are computed and discussed, serving as a base to derive energetic and transport properties allowing the description of excitation and relaxation phenomena caused by rapid laser action.

I. INTRODUCTION

The dynamics of laser-excited materials is an area of intense research as diagnostics of laser-matter experiments can be discussed by back-tracking the transient properties of the irradiated material. Particularly, the primary phenomena of transient electronic excitation and energy transport are of utmost importance. Irradiating a metal by a short laser pulse (∼ 100 fs) can lead to a significant rise of the electronic temperature with respect to the ionic lattice as the energy of the laser pulse can be deposited before the material system starts dissipating energy by thermal or mechanical ways. The electronic excitation can affect both electronic and structural properties of the solid, impacting optical coupling, transport and phase transitions. The confinement of the absorbed energy at solid density pushes the matter into an extreme nonequilibrium state and new thermodynamic regimes are triggered. The interplay between the ultrafast excitation and the material response still requires a comprehensive theoretical description for highly excited solid materials including in particular the excitation-dependent band structure evolution as this influence the response to laser action. 1 Recent advances in studying processes induced by short laser pulses have revealed the determinant role of primary excitation events. Their accurate comprehension is necessary to correctly describe ultrafast structural dynamics, 2,3 phase transitions, 4,5 nanostructure formation, 6 ablation dynamics, 7,8 or strong shock propagation. 9 In such nonequilibrium conditions, con-duction electrons participating to energy exchange are expected to evolve in time, depending on the excitation degree. 10 They largely determine the material transient properties and transformation paths. In this context, they are a crucial parameter required to describe complex ultrafast phenomena involving relaxation of excited states. Particularly, pure electronic effects (population and band distribution) determining transient coefficients before structural transitions set in are of interest and we will follow excitation influence in the form of nonequilibrium electronic temperature.

At the very beginning of the irradiation process, excited electrons are unhomogeneously distributed within the electronic band structure of the materials. By collisions and energy transfer, they fastly reach a Fermi-Dirac distribution. The electronic subsystem is thermalized and the concept of electronic temperature (T e ) can be applied. Mueller et al. have recently shown that the electron subsystem thermalizes within a characteristic time τ in the range of tens of femtoseconds for T e larger than 10 4 K. 11 Then, a first relaxation channel is the energy transfer between electronic and vibrational excitations. This is commonly described by a two temperature model (TTM) [START_REF] Kaganov | Relaxation between electrons and crystalline lattices[END_REF]13 based on the assumption that the occupation of electronic and phonons states can be separately described by two effective temperatures, the electronic T e and the ionic temperature T i . In standard approach, energy transfer between electrons and ions can be modeled by the product of the electron-phonon coupling parameter γ and the temperature difference. This relaxation occurs in the picosecond timescale. 14 Thus, there exists a period of time where the ionic temperature remains low while electrons fastly reach a thermalized state of high electronic temperature. In this case, strong alteration of electronic properties preceeds structural transformation, with consequences on the efficiency of energy deposition. Such nonequilibrium states can be modelled in the framework of first-principles calculations, by interrogating the electronic influence at various degrees of electronic heating, while disregarding in a first approximation, the ionic temperature effects.

Experimentally, excited solids in steady-state cannot be created and thermal nonequilibrium data in these particular conditions are difficult to be determined from integrated or time-resolved measurements. [15][16][17] In such complex conditions, simple estimations and models are used to access the behavior of intrinsic material properties. A strong need then exists to perform multiscale calculations both in space and in time, capable of replicating the observed behaviors and to predict material response under excitation. Most of the macroscopic models and behavior laws are based on the picture of free electrons commonly used to describe metals or even dielectric dynamics under laser irradiation. 1,3,13,18 Ab initio microscopic calculations can supply macroscopic approaches (optical, thermal, hydrodynamical or mechanical) with implicit dependencies on material properties and electronic band structures. Since the density of free electrons n e is not an observable variable in a quantum mechanical perspective, its dynamics in thermally excited solids remains poorly explored whereas the transport parameters and measurable dynamics are commonly depicted and fitted by laws depending on n e .

It has already been shown in previous works that electronic structures determine the thermodynamic functions and scattering rates of the heated electron subsystem. 19 This work analyses and extends the largely used electronic thermodynamic properties derived from the free electron gas model by interrogating the evolution of the free carriers. This model, based on the assumption of free and non-interacting electrons, works satisfactory in case of simple metals (Na, Mg, Al...). However it cannot encompass the complexity observed for transition metals, where d-electrons with a higher degree of localization than sp-electrons can still participate to optical processes. This indicates a potentially important role of electron confinement within more or less diffuse orbitals. Moreover, it has already been shown that the increase of the electronic temperature strongly affects the shape of the d-band. 5 Under such strong modifications of the electronic systems, it is important to extract from calculations an effective free electron number per atom N e classically defining n e = N e n i . This effective parameter can have importance whenever experimental optical or thermal properties are derived and used to extract other parameters such as temperature, stress or conductivity from a nonequilibrium solid. The objective of the investigations presented here is to quantify the effects of ther-mal activation energy ∼ k B T e around Fermi energy on N e consistently with a rigorously calculated band-structure accounting for Fermi smearing and d-band shifting within the range of 0.01 ≤ T e ≤ 10 5 K.

We report results from a systematic study on DOS energy broadening performed on a free electron like metal (Al) and on transition metals (Ni, Cu, Au, W and Ti), some of them with noble character. Section II is devoted to the calculation and procedure details. In section III, DOS dependence on the electronic temperature is discussed. We focus on the observable energetic shift and narrowing of the d-band and the implications on the chemical potential. Finally, in order to obtain the electron density relevant to light absorption, heat flux, or mechanical stress induced by electronic heating, an estimation of the number of free electron per atom, based on delocalized states considerations, is calculated at all electronic temperatures and discussed for all metals in Section IV. Concluding remarks on the effects of n e evolution on energetic and transport parameters, especially on electronic pressure and electronic heat capacity are made in section V.

II. CALCULATIONS DETAILS

Calculations were done in the framework of the density functional theory (DFT), 20,21 by using the Abinit package 22 which is based on a plane-waves description of the electronic wave functions. Projector augmentedwaves atomic data [23][24][25] (PAW) are used to model nucleus and core electrons. The generalized gradient approximation (GGA) in the form parameterized by Perdew, Burke and Ernzerhof 26 or the local density approximation (LDA) functional developed by Perdew and Wang 27 are considered for the exchange and correlation functional. Semicore electronic states are included in PAW atomic data of Ti and W as they significantly improve the description of material properties. The Brillouin zone was meshed with Monkhorst-Pack method, 28 with a 30 × 30 × 30 k-point grid. From the studied metals, only Ni is expected to have ferromagnetic properties, but our calculations showed that magnetic properties vanish above T e = 3 × 10 3 K, thus, all calculations were done without using spin-polarized methods. To ensure high accuracy of calculations, lattice parameters were relaxed up to the point where stress goes beyond 10 -4 eV/ Å, with a cutoff energy of 40 Ha.

Al, Ni, Cu, Au, Ti, W cristallize in different phases depending on the environment conditions. Here, we focus on cristal structures adopted by metals at ambiant conditions, namely: face-centered cubic (FCC, space group F m 3m, 225) structure for Al, Ni, Cu and Au; hexagonal close-packed (HCP, space group P 6 3 /mmc, 194) structure for Ti; and body-centered cubic (BCC, space group Im 3m, 229) structure in the case of W. Once structures are chosen, the accuracy of PAW atomic data is tested through calculations at T i = T e = 0K conditions of lat- tice parameters and bulk moduli using Birch-Murnaghan equation of states. A good agreement is found between our calculated values and experimental data, that are provided in Table I. Some differences are noticeable between computed and experimental bulk moduli, [31][32][33] especially when zero-point phonon effects are not taken into account. 31 This confirms the reliability of the used PAW atomic data. The theoretical lattice parameters computed at this step are then used to calculate T e dependent DOS.

To model laser irradiation effects, we consider timescales where the electrons are thermalized and their distribution can be described by electronic temperatures. Calculations were done with T e ranging from 10 -2 K to 10 5 K while T i remains equal to 0K. A number of 40 bands per atom is used to ensure a maximum occupation below 10 -4 electrons of the highest energy band at 10 5 K. T e dependent DFT calculations are performed following the generalization of the Hohenberg and Kohn theorem on many-body systems to the grand canonical ensemble as proposed by Mermin. 29 The finite electronic temperature is taken into account by considering a Fermi-Dirac distribution function applied to the Kohn-Sham eigenstates, ensuring a single thermalized state of electrons during the self-consistent field cycle. This involves a T e dependent electronic density and an electronic entropy part in the free energy potential with implicit and explicit dependencies. 22,30 Equilibrium electronic density at a finite electronic temperature is determined by minimizing the free energy, the variational functional here, resulting in a T e dependent electronic structure.

III. Te EFFECT ON DENSITY OF STATES

The following discussion is based on the precise determination of the electronic density of states of all the considered metals. Calculations are performed at twelve different electronic temperatures, from 10 -2 to 10 5 K. [START_REF]Tabulated data of Te dependent DOS and electronic properties are[END_REF] The DOS and associated Fermi-Dirac electronic distribution functions of the discussed metals are shown in Fig. 1 at the electronic temperatures of 10 -2 , 10 4 and 5 × 10 4 K. Here, for simplicity, the beginning of the valence band of the DOS was set at 0 eV for each T e . According to this representation, the number of valence electrons N v e can be expressed as:

N v e = ∞ 0 g(ε, T e )f (ε, µ, T e ) dε, (1)
where g(ε) is the DOS and

f (ε, µ, T e ) is the Fermi-Dirac distribution (f (ε, µ, T e ) = {exp[(ε -µ(T e ))/(k b T e )] + 1} -1
). If we first focus the discussion on the DOS obtained at T e =0K, we can notice that they are similar to previous works. 5,19,[START_REF] Papaconstantopoulos | Handbook of the Band Structure of Elemental solids[END_REF] For Al, the DOS adopts the shape of square root function of the energy, characteristic for a free electron like metal. With transition metals, the d-band appears with a typical d-block having a much higher density. This d-block is filled or almost filled in case of Ni, Cu and Au, while the filling is roughly 1/3 and 1/2 in the case of Ti and W as showed by the location of the Fermi energy. Generally speaking, the d-bands are narrow in the case of Ni, Cu and Au since almost all d-states are filled, which leads to a weak and non-directional character of the d-bonding. [START_REF] Ponomareva | [END_REF] At the opposite, they are much more expanded with the presence of pseudo-gaps in the case of Ti and W, exhibiting a stronger and more directional d-bonding. [START_REF] Ponomareva | [END_REF]

A. Shift and shrinking of the d-block

Density of states exhibit different dynamics when the electronic temperature increases. In the case of Al for example, the DOS is almost insensitive to T e . This constant behavior of the electronic structure was already noticed in Ref. [5]. On an other hand, transition metals exhibit more complex DOS due to the presence of d-bands. Metals with d-block fully or almost fully occupied by electrons (Ni, Cu and Au) exhibit a shrinking and a strong shift of the d-block toward lower energies when T e increases. On the contrary, metals with partially filled d-block (Ti and W) display an expansion and a shift toward higher energies of their d-block when T e is increased. In order to quantify these phenomena, we show in Fig. 2 The d-block modifications observed here for Au were already reported in Ref. [5], with an explanation of the phenomenon based on changes of the electronic screening. When an electronic temperature is applied, the depopulation of 5d-block leads to a decrease of the electronic screening which makes the effective electron-ion potential more attractive. 5 The consequence is a global shift of the electronic states toward lower energies. A similar effect likely occurs in the case of Ni and Cu even if the depopulation is now concerning 3d-electrons instead of 5d as in Au. Considering the shift toward higher energies and extension of the d-block in case of Ti and W, one should expect an increase of the screening effect with the augmentation of T e for these two metals. To validate this assumption, the change of the number of d-electrons ∆N d and the changes of Hartree energies ∆E Ha as a marker of the changes of electronic localization have been evaluated and are presented in Fig. 3. We specify that the concept of electronic localization refers to a certain degree of spatial concentration of the charge density. The relevance of these observations are discussed below.

Except the case of Al, all considered metals have electronic configurations in the form of (n-1)d x ns y . Since the main quantum number is one of the dominant parameter characterizing the diffuse nature of an orbital, 37 (n-1)d-orbitals are more localized and overlap less than nsp-ones. Consequently, the resulting d-band is confined energetically with higher density of state. Electrons occupying this band are also experiencing spatial localization. Thus, one can expect that changes in the electronic screening mainly come from changes of the electronic occupation of this d-band. To gather deeper in- Semicore electronic states also undergo an electronic depopulation at T e above 5 × 10 4 K. For example, at 10 5 K, the depopulation reaches 0.3 electrons for 3p-semicore electronic states of Ti and 0.4 electrons for 4f -semicore electronic states of W. Consequently, the total number of d-electrons increases in agreement with a strengthening of the electronic screening at least up to 5 × 10 4 K, where the depopulation of semicore states seems to moderate the effect, as we can see on Fig. 2 with a decrease of shifts and width changes. Finally, the behavior of Al is particular with an increase of N d and no effects on the density of states. This is related to the occupation of a high energy d-band that does not contribute to the electronic localization. However, DOS angular-momentum projection methods suffer from some drawbacks. Firstly, they are performed on spheres centered on atoms and there is always some intersphere informations lost during the process. Consequently, some electronic states and some electrons are not accounted for. Secondly, projections on angularmomentum do not allow distinctions between 3d, 4d or 5d bands. Then, the number of d-electrons computed can be the summation of electrons from localized (n-1)d band with nd or even (n+1)d delocalized bands. Thus, the variation of d-electron numbers has some uncertainties, but still provides information about trends. In order to get a more precise perspective of how electronic screening is affected by ∆N d , we rely on Hartree energy. The changes of Hartree energies with respect to T e are plotted in Fig. 3b. We recall that this energetic term is the Coulomb repulsive self-energy of the electronic density:

E Ha = 1 2 n(r)n(r ′ )
|r-r ′ | drdr ′ . It corresponds to the global electron-electron interaction. In a perfectly homogeneous spatial distribution, this quantity would reach a minimum value, however, constrained by the electronic structure, electrons are not homogeneously distributed. Specifically, semicore states and (n-1)d-bands strongly concentrate the charge density, which generates an electronic screening of the nucleus. Variations of the electronic occupation of these electronic states with T e impact Hartree energies, which can be related to a change of electronic screening. Thus, the evolution of ∆E Ha with T e is an indicator for the gain or the loss of electronic localization. This will be mainly the consequence of N d changes due to the spatial confinement of the (n-1)d-band we discussed above. The evolution of ∆E Ha also reflects the change of electronic screening of the ions. On Fig. 3, we can notice a good correlation between ∆N d and ∆E Ha with T e at least up to temperatures of 5 × 10 4 K. When the number of d-electrons decreases there is a loss of electronic localization. This leads to a decrease of E Ha which signals a decrease of electronic screening. The opposite phenomenon occurs when the N d increases, with a gain of electronic localization leading to an increase of E Ha correlated to an increase of electronic screening. Finally, ∆E Ha of Al remains roughly equal to zero, while the electronic occupation of high energy bands increases. It confirms that high energy bands, including d-bands, are sufficiently delocalized to be considered as uneffective on electronic localization.

From E Ha and N d changes, some unexpected behaviors are also observed. Firstly, ∆N d is correlated to ∆E Ha in the case of Cu and Ni. The stronger is the decrease of ∆N d , the stronger is the decrease of ∆E Ha . However, this rule is no longer available when considering Au. This is due to the fact that d-electrons belong to the 5d-orbitals in the case of Au whereas they belong to 3d-orbitals in the case of Cu and Ni. Associated to more diffuse orbitals, d-electrons of Au are already less localized than those of Cu and Ni, and it leads to a lower loss of localization when d-band is depopulated with the increase of T e . In other words, ∆E Ha is lowered in the specific case of Au since its d-electrons are already less localized. Secondly, ∆E Ha is not correlated to ∆N d in case of Ti and W at electronic temperature higher than 5 × 10 4 K. Whereas the number of d-electrons still increases, the Hartree energy decreases, with a particularly strong decrease for W. This is attributed to the electronic depopulations of semicore electronic states that occur at high T e as discussed above.

The strong evolution of Hartree energy for W originates from the significant change of electronic screening generated by the depopulation of highly localized 4f -electrons. We assume here that semicore electrons are fully thermalized with valence electrons even if semicore thermalization timescales are difficult to estimate. Being low in energies, these semicore states are not directly excited by laser irradiation and thermalization time is dependent on electron-electron collision frequency. 11,38 At 10 5 K, Fisher et al. 38 estimate about 1-10 sucessfull impact probabilities during irradiation timescale, for binding energies of semicore electronic states similar to the ones considered here. This suggests a short thermalization time but accurate description is beyond the scope of the present work.

Generally speaking, we note a relatively good agreement between ∆N d and ∆E Ha on one side, and ∆ε d and ∆W d on the other side. For Ni, Cu and Au, the decrease of ∆N d leads to a loss of electronic localization that generates a decrease of ∆E Ha . The effect can be related to the decrease of the electronic screening and thus to the increase of the global electron-ion effective potential that shifts electronics states of these metals toward lower energies. This shift applies non-homogeneously on the d-block since bottom d-block is less affected by depopulation and thus by changes of electronic screening than the top part, as can be seen on Fermi-Dirac distributions on Fig. 1(b-d). As a result the d-block is shrinked, and ∆ε d and ∆W d decrease. Shifts also apply to electronic states of higher energies, as shown for Cu on Fig. 4(a,b). For Ti and W, the increase of ∆N d leads to an increase of ∆E Ha . The corresponding gain in electronic localization produces a stronger electronic screening. As a result, the electron-ion effective potential is less attractive and bands are shifted toward higher energies. It also applies inhomogeneously to the d-block with its bottom states less affected by changes of electronic population than its top states, as can be seen on Fermi-Dirac distributions on Fig. 1(e,f). As a consequence, the d-block extends and is shifted toward higher energies, and ∆ε d and ∆W d increase. Higher energy electronic states are also affected by this increase of electronic screening, as we can note with the shift of other bands toward higher energies for Ti on Fig. 4(c,d). This discussion synthesizes T e effects on DOS for a range of representative metals, with various possible impacts on electronic properties.

B. Electronic distributions

As discussed above, the electronic distribution is ensured by the Fermi-Dirac function characterized by the electronic chemical potential µ(T e ). As already showed by Lin et al., 19 the electronic chemical potential exhibits strong variations, depending on the material studied. More precisely, these variations are related to the assymmetric distribution of the density of electronic states from both sides of the Fermi energy. For this reason, the electronic chemical potential moves toward higher energies in the case of Ni, Cu and Au, while it is displaced toward lower energies in the case of Al, Ti and W. In the present calculations, the T e dependence of the DOS produces important shifts of the d-block. Since the d-block concentrates electronic states, µ(T e ) is also strongly affected by these shifts. In Fig. 5, the relative changes of the electronic chemical potential is shown for all studied materials.

One important observation has to be made here. Previous works 19 reported evolutions of T e dependent properties computed from DOS performed at T e = 0K. In the present case, this approach takes into consideration the relaxion of DOS with T e , and changes in the electronic structure are impacting the electronic chemical potential. As a general note, one can observe that trends are similar between the temperature evolution of µ derived from T e dependent DOS and µ originating from T e = 0K DOS. The agreement is very good in the case of Al, that we can attribute to weak changes of its electronic structure with T e . However, in case of Ni, Cu and Au, the increase of µ is lowered in T e dependent DOS situation, a difference that is due to the shift of the d-block with T e . Since the d-block is the main electron reservoir, when it is shifted toward lower energies, the electronic chemical potential accounts for this evolution and its displacement tends to follow. In the case of Cu for example, at 5 × 10 4 K, µ from T e dependent DOS is decreased by 2 eV compared to µ from T e = 0K DOS. For W and Ti, the decrease of µ is also lowered in T e dependent DOS. This is attributed to the shift of the d-block toward higher energies, which is accommodated by µ. This indicates the importance of considering T e effects on band structure while regarding the evolution of electronic populations, with direct consequences on the determination of free electron numbers.

IV. FREE ELECTRON NUMBERS

The number of free electrons is a quantity difficult to define since the quality of being "free" is elusive. In metals, electrons are implicitly considered as free if they belong to orbitals having the highest main quantum number in the atomic electronic configuration (EC). [START_REF] Ashcroft | Solid State Physics[END_REF] For example, EC of Al and Cu are respectively 3s 2 3p 1 and 3d 10 4s 1 , and their corresponding number of free electrons are 3 and 1, respectively. The highest main quantum number is chosen as it characterizes the diffuse and overlapping nature of the orbitals. Hence, electrons belonging to those orbitals are assumed to be free of moving in a large space and by extension in the whole metal, with trajectories limited by collisions and with parabolic dispersion laws. However, the use of atomic electronic configurations induces a degree of incertitude when applied to condensed phases. In addition, N e determined from EC does not allow any change while T e increases. As a consequence, and motivated by the necessity of giving a certain evaluation of the free electron quantity, we computed the number of free electrons directly from the electronic structures of metals.

In order to improve the determination of N e , we have to distinguish electrons belonging to localized states (assumed to be non-free) from those belonging to delocalized states (considered as free). For this, it is important to determine which are the localized states in the density of states. d-orbitals from electronic configuration of transition metals defined by (n-1)d x ns y are less diffuse than sp-orbitals due to lower main quatum number. Consequently, they overlap less and thus interact less than sporbitals. The resulting d-bands produce a characteristic d-block of very high density of electronic states. At the opposite, sp-bands are delocalized and generate sp-bands of low density with a square root distribution, similar to Al (see Fig. 1). The difference of density between localized d-block and delocalized sp-bands is large and it is then easy to distinguish them in the DOS. Using a method to remove localized states from the DOS, one can compute free electron numbers according to the previous description, i.e. electrons occupying delocalized electronic states only. Accordingly, the number of free electrons is given by the integration of the DOS weighted by the Fermi-Dirac distribution:

N e = ∞ 0 g deloc (ε, T e )f (ε, µ, T e ) dε, (2)
with g deloc being the delocalized part of the density of states only.

To remove localized states from the DOS, several methods can be used. One of the most simple implies to fit the density of states to a curve having a square root shape, as this will artificially remove the high density of states of the d-block. [START_REF] Loboda | [END_REF] The square root shape is chosen since it is the distribution of the density of electronic states adopted by a free electron like metal, as observed for Al on Fig. 1a. Considering the electronic structure of dband metals, square root shape is thus a criterium for indentifying delocalized states. Here, in order to keep all DOS subtleties, this square root fit is only used to replace the d-block, resulting in a DOS of delocalized states, that can be written as:

g deloc (ε) = g(ε) -[g(ε) -α √ ε] dblock , (3)
where α √ ε is the fit of the DOS. The correction is only applied to the energy range containing the d-block. Fig. 6 exemplifies the whole process in the case of the DOS of Cu, indicating the fit results at 5 × 10 4 K. By proceeding that way, a small part of d-electrons also contributes to the free electron number. We assume that a small part of the d-band can be considered as delocalized too. The analysis of Hartree energies (Fig. 3b) already showed that some d-electrons are less localized than others (case of Au 5d-electrons versus case of 3d-electrons of Ni and Cu). Moreover, the d-band is not restricted to the very localized d-block, as shown in Fig. 4. Part of it easily matches a square root distribution of electronic states, indicating non-negligible overlaps and interactions between d-components, in a similar way to what is observed with sp-components. Finally, as previously discussed, d-bonds are mainly weak and non-directional, 36 thus it would not be surprizing that, despite a strong localization character, part of d-electrons have an ability to be mobile. Once they are only made of delocalized states, the DOS can now be integrated and N e can be deduced. The number of free electrons per atom for all considered metals is presented in Fig. 7. It shows strong variations as T e increases, exept in the case of Al, where this number remains constant with N e = 3.0 free electrons per atom. This is an expected free electron behavior since excited electrons are leaving delocalized states to reach other delocalized states. For Ni, Cu and Au, the localized d-block can be considered as a reservoir of non-free electrons susceptible to be depopulated with T e , depending on the relative location of the Fermi energy. Then, nonfree electrons from the localized d-block reach delocalized states and become free, which leads to an increase of N e with T e . In the case of Ti and W, the partially occupied d-block plays an ambivalent role. At low T e the bottom part of the d-block is filled of non-free electrons while the top part is empty but consists of localized states that can potentially trap excited electrons. As a consequence, N e remains constant or slightly decreases at low electronic temperatures. However, at temperatures above 10 4 K, N e increases as in the case of Ni, Cu and Au, by populating delocalized states of higher energy.

The typical values of N e deduced only from electronic configurations of isolated atoms are independent on the electronic temperature. They are provided in Table II alongside with the N e obtained by the present approach at 0K. As already mentioned, we obtained a good agreement in case of Al since the electronic structure is only made of delocalized states and thus N e remains constant and equal to the number of electrons from EC. However, differences can be large with respect to other metals, especially since electronic structure of condensed phases allows the transfer of electrons between bands. They also come from the fact that we considered the d-band as partially delocalized. We observed that the free elec- tron numbers calculated here approach values from other estimations. 10,[START_REF] Ebeling | Thermophysical Properties of Hot Dense Plasmas[END_REF] In case of Au, the present N e is higher than the value typically used. This is attributed to different Fermi energies, almost twice higher in our case (10.2 eV) than the value generally considered in the litterature (5.5 eV). [START_REF] Ashcroft | Solid State Physics[END_REF] It should also be noted that prior hypothesis concerning the charge density led to compensating corrections on other electron parameters related to classical models (Drude), via e.g. the effective mass used for Au. 42

V. ENERGETIC AND TRANSPORT PARAMETERS

In order to investigate to what level the band structure dependence on the excitation degree (via T e ) affects macroscopic transient characteristics and physical quantities, we evaluated thermodynamic properties of electrons when T e differs from T i . Such excitation character corresponds to ultrashort pulse laser irradiation of metal free surfaces, where the spatio-temporal evolution of the electron distribution and its return to equilibrium is calculated by a Boltzmann formalism 18 or by the two temperature model (electronic thermalization assumed). We recall that the TTM describes the energy evolution of electrons and ions subsytems using two diffusion equations coupled by an electron-phonon transfer term. 13 The electronic thermal energy gain is connected to the temperature by the electron specific heat C e . An accurate evaluation of this property is crucial in laser-matter interaction simulations since it provides a correct estimate of the rise of temperature in the electron system. TTM enables to take into account nonequilibrium effects on the kinetics of the material when it is included in classical molecular dynamics simulations or in two temperatures hydrodynamics approaches. 6,[START_REF] Inogamov | AIP Conf. Proc[END_REF][START_REF] Lin | Computational study of the generation of crystal defects in a bcc metal target irradiated by short laser pulses[END_REF] The electronic contribution to P e results directly from the electron heating, depending thus on T e but also on the density of states. Consequently, swift matter dynamics start especially due to the pressure gradient generated in the electron system. The key parameters to correctly reproduce the ultrafast nonequilibrium evolution of the material are then based on an electronic equation of states, connecting electronic specific heat C e and pressure P e with free electron density n e and temperature. C e and P e are generally derived from the thermodynamic properties of an ideal Fermi gas. [45][46][47][48] This work allows to insert the subtle effects of DOS modifications allowing more accurate perspectives. Transport properties, i.e. electronic thermal and electrical conductivity and electron-phonon coupling strength are also important to complete the kinetic equations but they are beyond the scope of this paper and will be only briefly mentioned. We will focus here on the influence of band structure evaluation on electronic thermodynamic properties.

The electron specific heat of metals can be derived with respect to the electronic temperature by C e = ∂E/∂T e , where E is the internal energy of the electron system. The evolution of the specific heat under electronic excitation is shown in Fig. 8 where several typical behaviors are observed. For Al, C e rapidly saturates to the lowest value of all other considered metals. The rest of metals remains far from saturation and reaches much higher values than Al. For W and Ti, C e exhibits a first leveling from 10 4 K to 4 × 10 4 K and restarts to increase at higher temperatures. At low and intermediate T e a relatively good agreement is found with those obtained from g(ε) evaluated at 0K, where [∂g(ε)/∂T e ] V is neglected. 19 For transition metals, increasing discrepancies appear at higher temperatures, mainly due to shifts of the d-block that are highly affected by T e increase. As expected, the temperature dependence of C e is linear at low electronic temperature and tends to saturate for high T e toward the non-degenerate limit 3/2n el k b , where n el includes both free electrons and part of d-electrons. As already mentioned by Lin et al., 19 thermal excitation from the dband results in a positive deviation of C e from the linear temperature dependence. Similar results are obtained with our temperature dependent calculations, with an additional deviation resulting from excitation of semicore electrons in case of Ti and W and high T e .

The electronic pressure P e is determined by the derivative of the electronic free energy F with respect to volume as P e = -∂F/∂V = -∂E/∂V +T e ∂S/∂V , where S is the entropy of the system. The last term corresponding to the entropy contribution to the pressure has been shown to be largely dominant in this range of T e . 30 P e evolution for the different metals is plotted in Fig. 9 and shows that P e increases rapidly as T 2 e for low excitation then scales as T e for higher temperatures. At 2.5 × 10 4 K, the electronic pressure is in the order of tens of GPa, and exceeds 100 GPa at 5 × 10 4 K for all metals except Al. Finally, at 10 5 K more than 300 GPa are reached in case of Ni, Cu, Au and W while it approaches a level of 200 GPa in case of Al and Ti. The fast increase of the electronic pressure is of interest since it is likely impacting the stability or the properties of materials and their evolution upon excitation, notably the initial steps of the thermodynamic trajectories. This strong increase of P e with T e comes from the occupation of the high energy states by the electrons, which is governed by the electronic structure and by entropic changes with the electronic temperature.

In order to exhibit band structure effects and to test our free electron approach, we renormalized P e with respect to the free electron gas pressure limit n e k b T e . The ratio is plotted in the inset of Fig. 9. At low electronic temperatures, degeneracy and band structure effects are dominating and the curves are far from the value of unity, that would characterize an ideal non-degenerated electron gas behavior. However, at higher electronic temperatures, curves tend to saturate at the value of 1.0, which indicates that the free electron numbers we have derived are consistently characterizing the electronic pressure of the system. This asymptotic behavior is not achieved using constant values of n e given by Ref. [39]. On an other hand, the effect of the entropy contribution on the renormalized P e is enhanced. The entropy contribution reflects the electronic disorder centered around the Fermi level. It comes from a compromise between the number of available electronic states and the number of electrons allowed to fill these states. In this context, when the Fermi level is within the d-block, the entropy effect is stronger, as for Ti and W, than when it is located somewhere else in the DOS (as in the case of Ni, Cu, Au). Al shows the lowest renormalized values due to lowest density of states.

Based on these results, an equation of states of the electron system can be constructed, including the interactions with a static array of positive ions but neglecting interactions with phonons. To predict laser light absorption, subsequent material heating and eventual phase transitions, the transport model should take into account the observed changes in band structure. In a first approach, transport properties can be estimated based on classical formulas but including the evolution of n e with T e . In this way, k e conductivity can be estimated roughly based on the relation derived from the kinetic equation between the electron heat capacity and k e . 49 If τ e is the relaxation time between collisions and v e the average electron velocity, then the mean free path l e = v e τ e allows to make the standard connection which writes k e (T e ) = 1 3 C e (T e )l e v e . Concerning the laser absorption, insights from the classical formalism of an electron in an optical electric field, i.e. the Drude-Lorentz model, can also be obtained from the calculated properties. Whereas the energy levels and their own occupation define the response of the material to applied optical fields, it is possible to describe the dielectric function by intraband and interband contributions. The optical properties are related to the structure and the electronic configuration of the material and the interband is highly sensitive to the DOS derived in section III. On the other hand, the intraband part (Drude) only depends on the number of free carriers per atom and on an effective momentum scattering time τ as σ = n e e 2 τ /m(1 -iωτ ), where ω is the laser angular frequency. In this way, it becomes possible to make reasonable deductions about excitation when optical property changes are measured. 50 These insights could become more accurate if we consider informations based on electronic structure calculations, with higher computational costs in this case. 51 Simpler models based on n e and interband transition from filled to empty states could be useful as a complementary approach, with nevertheless a lower accuracy.

VI. CONCLUSION

In the present study, T e dependent density functional calculations were performed on a representative range of metals: Al, Ni, Cu, Au, Ti and W, with simple and transition character. Electronic temperatures from 10 -2 to 10 5 K were used to evaluate electronic properties in nonequilibrium conditions.

In a first step DOS modifications with T e are discussed. It is shown that almost all bands are affected by energy shifts, but the most affected states are those involving the localized part of the d-band which characterizes the transition metals. Shifts towards lower energies and shrinking are observed for filled or almost filled d-block metals, illustrated by Ni, Cu and Au. Shifts towards higher energies and extensions for partially filled d-block, with Ti and W as examples. All these modifications are explained by evolutions of electron-ion effective potential that result from variation of the electronic screening generated by changes of the electronic occupation of the localized dblock. This was validated by two consistent approaches, the evolution of the number of d-electrons and the modification of the Hartree energies with T e .

Changes of the DOS with electronic temperatures impact electronic properties, like the electronic chemical potential and the electronic heat capacities that are discussed and compared to previous calculations performed at T e = 0K. A good agreement is obtained at low and intermediate temperatures, while an increasing discrepancy is observed when shifts within the electronic structures become stronger at high temperatures. The concept of electronic pressures is also addressed, with pressures rapidly reaching high values, of tens or hundreds of GPa, questioning material stability as T e increases.

Free electron numbers, dependent on the electronic temperature, are also computed from DOS. They are defined as belonging to delocalized states characterized by density of states having a square root energy dependence. As expected, for a free electron like metal as Al, this number remains constant. However, N e always increases for transition metals, with specific behaviors depending of the filling degree of the d-block. Ni, Cu and Au exhibit a N e that rapidly increases with T e while a small lag is observed for Ti and W, with an increase at higher temperatures. At high temperatures, these free electron numbers are found consistent with P e in an ideal gas limit.

Finally transport properties were addressed via the evolution of the electronic heat capacity and the electronic pressure with T e . The electronic heat capacity links the quantity of absorbed energy to the electronic temperature, giving an access to the number of free electrons. On the other hand, the rapidly growing electronic pressure may impact phase stabilities. Both are crucial properties, since significant deviations of the thermophysical properties of metals from the commonly used approximations may have important practical implications in ultrashort laser material processing applications.

VII. AKNOWLEDGMENTS

We acknowledge Marc Torrent for providing efficient PAW atomic data. This work was supported by the ANR project DYLIPSS (ANR-12-IS04-0002-01) and by the LABEX MANUTECH-SISE (ANR-10-LABX-0075) of the Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Part of the numerical calculations has been performed using resources from GENCI, project gen7041.

  FIG. 1. (color online). Electronic density of states (solid lines), associated Fermi-Dirac distribution functions (dotted lines) and corresponding electronic chemical potential (dashed lines) for all studied metals. Data for the following electronic temperatures are shown: 10 -2 K (black), 10 4 K (red) and 5 × 10 4 K (blue) curves.[START_REF]Tabulated data of Te dependent DOS and electronic properties are[END_REF]

FIG. 2 .

 2 FIG. 2. (color online).Relative changes of the d-block center (a) and width (b) with the electronic temperature, for all studied transition metals.

FIG. 3 .

 3 FIG. 3. (color online). Changes of the number of d-electrons (a) and changes of the Hartree energy (b) with respect to the electronic temperature.

FIG. 4 .

 4 FIG. 4. (color online). Evolution of the total DOS and spdfcomponents of the DOS for Cu at 0K (a) and 5×10 4 K (b), and for Ti at 0K (c) and 5 × 10 4 K (d). Dashed lines correspond to Fermi levels.

FIG. 5 .

 5 FIG. 5. (color online). Electronic chemical potential evolution with the electronic temperature for all studied metals.

FIG. 6 .

 6 FIG.6. (color online). The blue curve is the density of delocalized electronic states for Cu at 5 × 10 4 K. The empty red dots show the square root fit of the DOS and the dotted black curve represents the removed part of the initial DOS, constituted of the non-contributing localized d-block. The hatched part refers to the free electrons and is computed from the integration over the Fermi-Dirac distribution corresponding to Eq. (2).

FIG. 7 .

 7 FIG. 7. (color online). Evolution of the number of free electrons per atom (Ne) with the increase of Te, for the considered metals.

FIG. 8 .

 8 FIG.8. (color online). Evolution of the electronic heat capacity with respect to the electronic temperature for the studied metals.

FIG. 9 .

 9 FIG. 9. (color online). Evolution of the electronic pressure with the electronic temperature for all the studied elements. Inset: renormalization of Pe with respect to the free electron gas pressure limit nek b Te.

TABLE I .

 I Electronic structure of atoms and theoretical, experimental and relative error (%) of lattice parameters ( Å) and bulk moduli (GPa) of metal phases at ambiant conditions.

	Elt	XC Functionals	Elec. Struc.	Chem. Struc.	l th	lexp	Rel. Err.	B th	Bexp	Rel. Err.
	Al	GGA	3s 2 3p 1	FCC	4.04	4.05	-0.4	79	81	-2
	Ni	GGA	3d 8 4s 2	FCC	3.51	3.52	-0.3	192	191	1
	Cu	GGA	3d 10 4s 1	FCC	3.64	3.61	0.6	142	133	6
	Au	LDA	5d 10 6s 1	FCC	4.05	4.08	-0.7	195	167	14
	Ti	GGA	3s 2 3p 6 4s 2 3d 2	HCP (a,b) HCP (c)	2.93 4.66	2.95 4.69	-0.6 -0.6	112	114	2
	W	GGA	5s 2 5p 6 4f 14 5d 4 6s 2	BCC	3.18	3.17	0.7	295	296	-5

TABLE II .

 II Number of free electrons, from Ref.[39] and from this work obtained at Te = 0K.

	Ne	Al	Ni	Cu	Au	Ti	W
	Ref. [39]	3	2	1	1	2	2
	This work	3.0	1.4	1.9	2.4	1.4	2.2


Gabriel Vasile
email: gabriel.vasile@gipsa-lab.grenoble-inp.fr

Adrian Tudoroiu

Frédéric Gottardi
email: frederic.gottardi@edf.fr

Joël Gailhard

Alexandre Girard
email: alexandre.girard@edf.fr

Guy D'urso

UNCONSTRAINED NONLINEAR OPTIMIZATION OF A DISTRIBUTED SWE MODEL USING MODIS AND IN SITU MEASUREMENTS OVER THE FRENCH ALPS

Keywords: Snow, optimization, MODIS

In this paper we propose the optimization of the snow submodel of MORDOR using MODIS and in situ measurements for the case study of the Serre-Ponçon reservoir (one of the largest artificial lakes in Western Europe) on the Durance River in the French Alps. We consider the problem of optimizing the snow model as an unconstrained nonlinear optimization problem.

INTRODUCTION

Snowpack estimation in mountainous regions is important for hydroelectric power supply since it allows to anticipate the water resources available during spring snowmelt and thus to optimize filling reservoirs. To achieve this estimate of the snowmelt contribution, EDF (the main French hydropower company) uses both hydrological modeling (MORDOR model [START_REF] Garçon | Prévision opérationnelle des apports de la Durance à Serre-Ponçon à l'aide du modèle MORDOR[END_REF]) and local in situ measurements. In this paper we propose the optimization of the snow submodel of MORDOR using MODIS and in situ measurements for the case study of the Serre-Ponçon reservoir (one of the largest artificial lakes in Western Europe) on the Durance River in the French Alps. The area of interest is delimited by the green rectangle from Fig. 1-(b). The water resources located inside the watershed limit in red are collected to the Serre-Ponçon reservoir located Fig. 1-(a).

Using 36 meteorological stations operated by EDF in the area of interest, the MORDOR forecasting model currently provides the snow water equivalent (SWE) used to evaluate the potential intakes to the reservoir and to optimize the operation of the Serre-Ponçon dam. Over one year, the snow water equivalent is analyzed in two phases: accumulation (SWE slowly increasing) and melting (SWE decreasing fast). Using 36 meteorological stations operated by EDF in the area of interest, the MORDOR forecasting model currently provides the snow water equivalent (SWE) used to evaluate the potential intakes to the reservoir and to optimize the operation of the Serre-Ponçon dam. Over one year, the snow water equivalent is analyzed in two phases: accumulation (SWE slowly increasing) and melting (SWE decreasing fast).

DISTRIBUTED SWE MODEL OPTIMIZATION

After kriging [START_REF] Goovaerts | Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall[END_REF] the 36 in situ measures (temperature and precipitation), this physical model is anticipating the amount of water that is contained in the snow pack over one season [START_REF] Gottardi | Quantitative estimation of precipitation over the French mountainous areas using snow measurements and weather patterns approach[END_REF]. The proposed optimization is carried out by comparing the MORDOR model with the SWE data measured on the terrain by remotely controlled cosmic-ray snow gauges (4 stations in the basin) [START_REF] Paquet | Operation feedback and prospects of EDF cosmic-ray snow sensors[END_REF], and also with binary snow maps derived from MODIS images [START_REF] Klein | Development and validation of a snow albedo algorithm for the MODIS instrument[END_REF]. We shall consider the problem of optimizing the snow model as an unconstrained nonlinear optimization problem. The MORDOR model will be described as a nonlinear function of two parameters that needs to be optimized: k f (the melting coefficient) and c p (the accumulation coefficient). Different optimization scenarios are studied:

1. according to the snow phase: accumulation or melting; 2. according to the parameter to be optimized: c p or k f ; 3. according to the MSE minimization method:

Nelder-Mead, pattern search, genetic algorithms and simulated annealing.

To evaluate the difference between the data provided by the snow model and the data collected from the measurement stations, we use the standard Mean Squared Error (MSE) measure. Let g(c p , k f , t) be the SWE from MORDOR and let h(t) be the measured SWE over one year. Then we can write the following equation:

c p , k f ! " # $ = f MSE g(c p , k f , t), h(t) { } , c p _ init , k f _ init ! " # $ ( ) . (1)
The proposed optimization f starts at the point

MODIS MAP OPTIMIZATION OF THE SNOW MODEL

To compare the snow model SWE data with the data measured by MODIS (Fig. 2), the MORDOR output data must be firstly binarized. One way to do this is by choosing a specific threshold value T: all SWE values less than T are labeled to "0" and the values greater than T to "1". Unfortunately this method does not allow conventional optimization since the function f is not continuous. To obtain this, we propose to apply continuous binarization by using the tangent hyperbolic function:

bin = tanh h s σ b ! " # $ % & +1 2 , ( 2
)
where σ b is a parameter depending of the incertitude of our maps and h s is the height of the snow.

To find the best binary fit of the MORDOR data, we must find the best value for σ b by optimizing the snow model with the data from the MODIS maps and also with data from the in situ measurements: where m is data from MODIS map. Using this new value for the binarization function we can optimize the snow model as in the previous section:

σ b [ ] = f MSE bin g(c p , k f , t),σ b ! " # $ ⋅ g(c p , k f , t), m(t)⋅ h(t) ( ) ,σ init ( ) (3)
c p , k f ! " # $ = f MSE bin g(c p , k f , t),σ b ! " # $ ⋅ g(c p , k f , t), m(t)⋅ h(t) ( ) , c p _ init , k f _ init ( ) (4)

EXPERIMENTAL RESULTS

In this section, we present some results derived for only one station (6415). In the proposed experiment, the results from four stations were considered: station number 6415, 6460, 6470 and 6510. The obtained experimental results, illustrated in Fig. 3, indicate a good matching with respect to in situ SWE measures on each ground station. Finally, qualitative performance assessment is proposed in table 1.

CONCLUSIONS

The optimization of the snow sub-model of MORDOR using MODIS and in situ measurements was performed for the case study of the Serre-Ponçon reservoir (one of the largest artificial lakes in Western Europe) on the Durance River in the French Alps. The problem of optimizing the snow model was treated as an unconstrained nonlinear optimization problem. Several optimization strategies were investigated.

Fig. 1 :

 1 (a) the Serre-Ponçon dam, France, (b) the area of interest.

  x 0 =[c p_init , k f_init ] and finds the global minimum x=[c p , k f ] for the MSE. Optimization strategies tested in this paper are: • fmin search -Nelder-Mead simplex function minimization [6]; • pattern search -constructs a sequence of iterations that converge to an optimal solution [7]; • genetic algorithm -based on a natural selection process that mimics biological evolution [8]; • simulated annealing -probabilistic search algorithm that seeks to improve the current minim by slowly reducing the extent of the search [9]. We can split the optimization problem in more phases (intervals): 1. first interval -phase of snow accumulation and melting (the whole year [D s , D e ]), 2. second interval -the phase of snow accumulation [D s , D m ], 3. third interval -the phase of snow melting [D m , D e ]. To automatically detect D s (start day), D m (middle day) and D e (end day), we use gradient-oriented thresholding of the measured data.

Fig. 2 :

 2 MODIS map for (a) 03/07/2000 and (b) 03/16/2000.

Fig. 3 :

 3 Fig. 3: (a), D s , D m and D e computed for one station, (b) binary data from real measures compared to binary data from MODIS maps, (c) Optimization of the c p and the k f coefficients of the MORDOR model for the snow accumulation phase.

ACKNOWLEDGEMENT

This work was supported by the French National R&D Program TSEN-X PEPS UJF-CNRS 2011.


Jean Paul Fitoussi Subprime mortgages meltdown and the behaviour of Central Banks*

How near of a financial crisis and thus of an economic crisis tout court, are we? The answer to this question depends a lot on the behaviour of Central Bankers. To understand why, a brief description of the present situation is in order.

At the outset there is the crisis of the subprime mortgage market in the US, which is rather small in size. The contagion to other compartments of the credit market, and eventually to the stock exchange arose for two reasons. The first is securitization of private debt which by itself is an intelligent way of spreading the risks so as to allow access to credit by a larger number of borrowers. It significantly reduces the amount of credit rationing in the economy. The structured finance markets allow thus the issuance of asset backed securities (ABS) and in particular asset backed commercial paper (ABCP). Usually most of ABCP are diversified, relying on a rather wide mix of assets, among which one can find in small proportion residential mortgage loans. Usually also, liquidity, in the case of an inability to roll (to pay CP on time) due to a market disruption is provided by bank liquidity facilities. Canada was a special case where the liquidity facilities are not as strong; the banks refused to pay, but market participants worked out rescue plans among themselves that will allow these conduits to be unwound over time. Some programs however are single sellers extendible ABCP, backed by mortgage collateral. But many of these programmes are insulated from market value declines by market values swap, which will ultimately protect investors; yet the banks have to buy the CP or the long-term assets that the conduits own, and the amounts are huge -ABCP programmes deleveraged by an average of €10 billion per day for the last three weeks of August. Other CP programmes lack structural protection and the investors may not get all their money or have to wait a long time before they do. Now one can understand why the loss of confidence in the latter propagates to all segments of the ABCP market, creating a liquidity crisis. The main responsibility for this contagion, and the second, but the true reason for the crisis, is borne by the credit rating agencies which did not properly distinguish between the different kinds of ABCP programs, because they were blindly believing in their sophisticated models. "Once the market lost confidence in rating agency models following the subprime debacle, the CP didn't look so safe." (Richard Robb) 1 Hence the stepping in of Central Banks and especially the ECB was a proper reaction but it was obviously not sufficient. If confidence is lost because the market participants become conscious of their imperfect knowledge, the players would unlikely lend to each other. That arose even between banks themselves increasing the opacity of the risk structure of their assets. What begun as a liquidity crisis may well degenerate in a solvency one that could be stopped only by a decrease of the rate of interest. Apparently the President of the FED understood this point as testifies his decision of cutting the refinancing rate. The gesture was more important than its content, because of the expectations it led to.

In Europe things are moving more slowly in a context which requires exactly the reverse: a crystal clear discourse and a quick decision by the Lender of Last Resort. Only transparency and action can take care of a confidence crisis. At the contrary, the discourse of the President of the ECB is all but clear, as it sticks to its pre-crisis wording of the possibility of an interest rate hike, adding some obscure considerations about the consequences of the financial market turmoil in a way which is not increasing our understanding. What seems to be clear from his message is that the crisis helped to return to a normal pricing of risks. What was abnormal according to his view was the flattening of the interest rate curve in the pre-crisis period.

There may even be some truth in this statement, but it does not help to cure a confidence crisis.

What may explain this European specificity is a convergence of two ex-ante contradictories lines of thought. The first seeing any alleviation of "the pain" of the market as a reward to speculators, the second stemming from the strong believers in general equilibrium economic theory according to which the variation in asset prices is just reflecting an exogenous shock, "a reduced appetite for risk". Whatever their widely different roots the two lines of thought converges towards the conclusion that nothing has to be done, as if such passivity would not arm the "real economy". And indeed the question is phrased in those terms: will the subprime crisis affect the real economy? Of course, it will: how can one imagine, even for a second, that a decrease of asset prices (especially housing prices) and an increase in the average rate of interest (through an increase in the risk premium) would not have negative consequences on growth and employment? As Chairman Ben S. Bernanke put it: "It is not the responsibility of the Federal Reserve -nor would it be appropriate -to protect lenders and investors from the consequences of their financial decisions. But developments in financial markets can have broad economic effects felt by many outside the markets, and the Federal Reserve must take those effects into account when determining policy"2 . It is only recently on August 27 in Budapest, that President Trichet corrected its preceding statements -"What I said was before the market turbulence" -letting us know that the ECB is paying close attention to the liquidity crisis and may not tighten on Sep 6.

The ECB understands its mission as one of anchoring inflation expectations around its inflation goal. It does not seem to fully understand that it has also the duty of managing expectations about the real economy. Now what has been termed colourfully "a reduced appetite for risk" may well reflect a radical shift in expectations about the real economy. In this context, raising interest rate would not be a sign of independence from the ECB, but the sign that it is misunderstanding its role. This episode reflects also the obvious, but not widely recognised fact that we are leaving in an imperfect knowledge world, one where uncertainty is pervasive. In this word, the informations for a perfectly rational calculus are just missing, and the framework of dominant economic theory can't be applied. (See on these questions the remarkable book by Roman Frydman and Michael Goldberg: Imperfect Knowledge economics: Exchange Rates and Risks). This applies in particular to financial markets whose role is supposed to coordinate future intertemporal plans relative to saving and investment. In such a world, notwithstanding moral hazard problems, the main task of a Central Banker is to re-establish confidence by making known by all means that he will not leave the real economy to be affected by the sudden disruption of financial markets.

To summarize: 1.

It was wise for the ECB to provide the market with liquidities 2.

It was not appropriate to speak to the market in such an obscure way 3.

The normal reaction of the ECB should to reconsider its former discourse of "strong vigilance" (August 2) and not to tighten at its 6september meeting. If the real economy in Europe starts to slow, there will be plenty of time to decrease the rate of interest. Monetary policy has to take into account the different shocks to the economy among which a financial shock can be the gravest, as it is generally reflecting a shift in expectations about the real economy. It is the only policy instrument which can be applied timely taking into account new information stemming from the very functioning of markets.

This is a quote from a mail sent to me by Richard Robb, Columbia University.

Remarks at the Federal Bank of Kansas City's Economic Symposium, August

31, 2007: "Housing, Housing Finance, and Monetary Policy".


The ECB' monetary policy strategy and structural reforms

Jean-Paul Fitoussi

There is a mounting discontent in Europe about the conduct of macroeconomic policy in the euro area and especially its policy mix, in the broad sense of the term -the combination of monetary policy, fiscal policy and structural reforms. If the proof of the pudding is in the eating, the weak economic performances of the euro area since 2001 is a testimony that some thing has gone wrong with the reaction of public policies to the financial, confidence and geopolitical shocks which have hit the world economy. In the three years period -2001-2003 -the average growth rate of the zone would be of about 1%, and the forecast for 2004 is not that of a bright rebound, but of a soft recovery (1.8%). Of course, one can argue that we should not expect from economic policies more than they can deliver. They act at the margin, and they can't prevent much of the consequences of, say, the explosion of a financial bubble which will go its way through the private sector, policy reactions notwithstanding.

The actors in the policy debate held different views which may be expressed in a sketchy way as follows. The monetary policy authorities think that the responsibilities lie with the governments (bad management of fiscal policies of which high budget deficits are a consequences), and the resistance of societies to structural reforms. The governments think that they lie with the central bank (delayed and weak responses to shocks) and also with societies (the difficulty to implement reforms). The average citizen think that all actors have a share of responsibilities: fiscal and monetary authorities by not reacting enough, and the private sector by opposing reforms and/or by trying to escape the burden of adjustment to shocks.

In a nutshell each actor reproaches to the others to be insufficiently flexible, if by flexibility we mean the capacity to adjust quickly to new informations. And all of them are rights, but in a different degree. In a former briefing paper1, I tried to show how the stability pact should be reinterpreted to allow for a greater degree of "flexibility" of fiscal authorities. Here, I will focus on structural reforms and the new ECB's monetary strategy.

Structural reforms

The plea for structural reforms is twofold: they will allow the economy to operate at a higher degree of activity; they will increase the self regulatory functioning of a market economy, reducing the scope and the need for macroeconomic policies. The reference model describes an economy with perfect competition and rational expectations. In such a model full employment is always a characteristic of the temporary equilibrium, and rational expectations lead to the ineffectiveness of monetary policy. Hence the central bank should focus on price stability and can achieve it at no costs if it is credible. Departures from perfect competition will increase the equilibrium rate of unemployment, without giving to the central bank the opportunity to act to reduce it, but with mounting pressures from governments and societies towards the bank so as it acts in a more expansionary manner. If the bank succumbs to this pressures, the result would be an acceleration of inflation. The reasons is simply that markets imperfections lead to an increase in the non accelerating inflation rate of unemployment (NAIRU).

This framework lead to emphasize the role of institutions in economic performances, especially labour market institutions. These institutions lead to diverse rigidities which explain the departures from the reference model and bad economic outcomes. This vision has two major (and related) flaws: The first is that it is based on a simplistic application of the welfare theorems, by which a perfectly competitive market will always reach the most efficient price/quantity allocation. It is simplistic because the step from the theoretical result to the policy prescription is wider than one could think, and has to be taken cautiously (as was done by the founders of general equilibrium theory). First, even assuming that market forces were able to attain the maximum efficiency, there would still exist a problem of equity in the distribution of the resources. A democratic society may have a legitimate taste for redistribution and for the implementation of a costly system of safety nets; in this case the strict optimality notion delivered by the free market ideology may not coincide with a broader notion of social welfare, as the democratic process may lead the society to trade some of the efficiency in favour of higher social cohesion. Furthermore, it may be argued2 [START_REF] Fitoussi | La règle et le choix[END_REF]) that democracy and political adhesion of the population to the economic government of a society can actually enhance efficiency, guaranteeing the flexibility, transparency and consensus that would be missing when ruling according to the strict application of a doctrine. In this sense, the political process and the market would be complements rather than substitutes. This complementarity is even more striking when we realize that the efficiency of the market outcome strongly depends on a number of assumptions that are rarely observed in the real world, from perfect competition to complete markets and information. The long and widely studied list of "market failures" gives a scope for public intervention that may not be overlooked only because we also observe "government failures". In particular, a number of institutional rigidities that we observe in the European labour markets can be explained by the different bargaining power of workers and entrepreneurs. In its Wealth of Nations Smith had already highlighted the problems that this asymmetry could cause. The norms on labour protection can then be seen as a legitimate outcome of the democratic process, aimed at reestablishing some fairness in the bargaining process. The following paragraphs will tackle, more in detail, the debate on institutions and labour market performance.

The influence of institutions, especially labour market institutions, on labour market performances has been a long debated question at least since the twenties. I remember a paper by Jacques Rueff published in 1931, whose title was self explaining: "Unemployment insurance cause of persistent unemployment", in which the author was trying to explain the increase in unemployment in the twenties in the United Kingdom. Simple correlations convinced him that there was no other cause of the evolution of unemployment in that country. Of course we have done a lot of technical progress in dealing with data since then, but that the question is still debated today shows that almost a century of hard and rigorous research has been unable to settle the debate.

According to most studies of which the one realized by Stephen Nickell and associates (2003) is representative "the equilibrium level of unemployment is affected first by any variables which influences the ease with which unemployed individuals can be matched to available job vacancies, and second, by any variable which tend to raise wages in a direct fashion despite excess supply in the labour market". These variables includes the unemployment benefit system, the real interest rate, employment protection, active labour market policy, union structures, the extent of coordination in wage bargaining, labour taxes etc..(terms of trade, shifts in trend productivity growth). But it is fair to say that until now, there is no convincing evidence that labour market institutions are responsible for the high level of unemployment in continental Europe or the disappointing macroeconomic performances of Europe during the nineties. What is striking is the weak, to say the least, explanatory power of the institutional variables, especially those considered as being the more important, namely the benefit replacement rate and employment protection. That the latter may have ambiguous effects has long been recognized in the literature: the fact that firms are more cautious about hiring, because of strong labor protection may increase the efficiency of the matching process. But what has not been recognized is that the same may be said for the workers. The fact that unemployment benefit allows the unemployed to search for a job better suited to their skills and expectations, may also increase the efficiency of the matching process. Certainly labor productivity could be greater if the worker has the feeling that his job is corresponding better to his desire [START_REF] Jp | Comments on Nickell, Nunziata, Ochel and Quintini[END_REF]. There is thus a hiatus between usual recommendations and the weaknesses of the evidence to support them. At best empirical studies are able to explain 3 See [START_REF] Jp | Roots of the recent recoveries: labor reforms sound public finance or private sector forces?[END_REF].: "Roots of the recent recoveries: labor reforms sound public finance or private sector forces?, Brookings papers on economic activity 4 See [START_REF] Krugman | Competitiveness: a dangerous obsession[END_REF] on this point : "Competitiveness: a dangerous obsession", Foreign Affairs. 73(2): 28-44. 5 See [START_REF] Cohen | French Unemployment: translatlantic perspective[END_REF] discuss the case of employment protection. "French Unemployment: translatlantic perspective", Economic Policy 25:267-91. 6 See [START_REF] Fitoussi Jp | Réduction du chômage: les réussites en Europe[END_REF] : "Réduction du chômage: les réussites en Europe", Conseil d'Analyse Economique 23, La Documentation Française ;and Freeman (2000) : "Single peaked vs diversified capitalism: the relation between economic institutions and outcomes, NBER Working Paper 7556. second order of importance effects of institutions on unemployment.

Economic outcomes are more easily explained by the big shocks that OECD countries have suffered: changing trend in productivity growth, the oil shocks, the important increase in the real rate of interest. Besides, structural reforms in the countries which implemented them, do not appear to have played an important role either3. This is why a new empirical strategy has been developed which consists to study the interaction between shocks and institutions (Cf. e.g. [START_REF] Blanchard | The role of shocks and institutions in the use of European unemployment : the aggregate evidence[END_REF]Wolfers, 2000, or Fitoussi andPhelps, 2001). Institutions which may be innocuous in normal time can matter in case of shocks, because they prevent adaptation to these shocks. There are two reasons why the interaction strategy seems more appealing. First unemployment rates in the OECD countries had risen roughly in unison from the mid-seventies to the mid 1980s. Thus all the favoured candidates to explain the phenomenon are OECD-wide shocks. Second labour-market institutions may have mainly played a role in propagating these shocks rather than originating the shocks4. The welfare state had its origins well before the rise in unemployment at the beginning of the 1950s when Europe was enjoying low unemployment and generally good labour-market performance.

Moreover, a better understanding of the early shocks that drove unemployment to new heights in previous decades in Europe can possibly help save some institutions that, by making up for failures in insurance markets and training, may be worth keeping. Finally, the institutions reflect a social contract that arises from a democratic process. In this process there are winners and losers so structural reform is unlikely to lead to a Pareto-improving outcome, even one supported by a majority of the electorate5. And former research showed that it is hard to explain change in unemployment by change in institutions.

We come thus to the second flaw of the 'structural reform argument': The assumption that the free market paradigm is always superior to any other institutional arrangement, an assumption that is not supported by the data, nor by common sense. Two recent studies independently conducted on the subject6, reached the same conclusion out of a sample of 19 OECD countries. In market democracies, the institutional structure is not a powerful factor in explaining economic performances. Capitalism is sufficiently robust to accommodate rather different institutional settings. That conclusion is at odds with the common wisdom according to which the diversity of institutional structures plays a 7 As Solow remarks at the end of Keynes Lectures: "If pure unadulterated labour-market reform is unlikely to create a substantial increase in employment, then the main reason for doing it is anticipated gain in productive efficiency, however large that may be. But if we respect the wage earner's desire for job security, and it seems at least as respectable as anyone's desire for fast cars or fat-free desserts, then an improvement in productive efficiency gained that way is not a Paretoimprovement. More labor market flexibility may still be worth having -and I think it is -but then the losers have a claim in equity to some compensation. The trick is to find a form of compensation that does not cancel the initial gain in labour-market flexibility." determinant role in explaining both unemployment and growth. And indeed empirical evidence in favour of such a thesis appears at best very weak. If we had in each decade followed common wisdom we would have recommended to every countries in the world to follow the French institutional model in the 60s, the Japanese one in the seventies, the German one in the eighties, and the US one in the 90s. The nationality of the model of the present decade (the years 2000s) is still unknown7.

The diversity of the institutional framework in OECD countries shows that institutions are the outcome of a political process anchored in the specific history, culture and anthropology of the country, rather than a way to increase efficiency. If for example, the labour contract which emerged from the world war II was open ended and of indefinite duration despite the cultural diversity of the countries concerned, it may be just because after a war, the solidarity between social groups had to be reassessed. My second remark is thus that there is a strong presumption that institutions are endogenous. Structural measures are usually taken to cope with a problem, in a sense which is not necessarily the one economists would recommend. Rising unemployment in Europe in a period where stabilisation policies were aimed at disinflation and at maintaining monetary parities, has lead to structural activism to alleviate the pain of the unemployed. There is for example in France a clear causal relation between restrictive policies, mounting unemployment and the 1997 law on the reduction of working time -the 35 hours week. And in the US, the increase in the generosity of employment benefits has followed the collapse of financial markets and September 11. Some active labour market policy have also the (unintended?) effect of putting the blame almost uniquely on the side of the workers, who are considered as inclined to adopt a passive behaviour when confronted with unemployment or an egoistic behaviour when they are insiders. What is striking is that in Europe the advocates of orthodox macroeconomic policies ( the stability pact and the single objective for monetary policy) are recommending incredibly interventionist structural policies which in a way are more limitative of the freedom of people that any kind of classical demand policy.

The ECB's monetary policy strategy

They are two readings of the press release about the new ECB's monetary policy strategy. The first, sympathetic, will focus on what seems to be a radical departure from the initial strategy taking into account the most voiced criticisms about this strategy:

a point target of 2% for the inflation rate, i.e an objective equal to the upper n bound of the preceding definition: "the governing council agreed that in the pursuit of price stability it will aim to maintain inflation rates close to 2% over the medium term." Note that this change represents a real progress in two directions with respect to the past. A point target is an unambiguous and symmetric definition; and second the target has been chosen at a higher level that the implicit target that one could deduct from the reference growth rate of the money supply used by the ECB (1.5%). Furthermore the press release lays out the reasons why it is so: "This clarification underlines the ECB's commitment to provide a sufficient safety margin to guard against the risks of deflation. It also addresses the issue of the possible presence of a measurement bias in the HICP and the implications of inflation differentials within the euro area". Even if the three arguments have not the same weight, they all effectively imply an increase in the inflation target. The first and the third are linked through the Balassa-Samuelson effect -the convergence of price levels between countries with different levels of development which implies the persistence of inflation differentials -but also because the exchange rate parities at the outset of the launching of the euro were characterized for some countries by an overvaluation and for others by an undervaluation of their currencies. If the inflation objective is too low, it will thus impose on the more developed countries an inflation rate close to zero or even a risk of deflation if it happens that the currency of these countries have been overvaluated. Germany is obviously a case in point. The second argument, the possibility of a measurement bias in the HICP, is quite distinct as it has better to be clarified the soonest possible. It should require as that has been done in the US the establishment of an independent committee to review the question and to suggest new methods of measurement.

The two-pillar monetary-policy strategy is de facto evolving towards a one pillar n n n n strategy, through the downgrading of the first pillar (the money supply M3). As was underlined in my last briefing paper, the ECB has been following until now, at least in theory, the strategy called of the "two pillars", heritage of the compromises that underlie its creation; on one side, it was supposed to target money growth ( the "reference value" is 4.5% per year8); on the other hand, it had to take into account a composite indicator of inflationary pressures. In practice, the money growth objective was not followed, and the ECB simply focussed on the second pillar. But this ambiguity between the strategy the Bank was supposed to follow and the one it actually implemented, was a nuisance for the transparency of its acts, the effectiveness of its communication strategy, and finally damaged the ECB in its quest for credibility. The flaw with the first pillar has been extensively analysed in the many reports by the ECB watcher group and practically by all economists who have evaluated the strategy. The change is thus welcome. "The Governing Council wishes to clarify communication on the cross-checking of information in coming to its unified judgement on the risks to price stability. To this end, the introductory statement … will start with the economic analysis to identify short to medium-term risks to price stability …. The Governing Council also decided to no longer conduct a review of the reference value on an annual basis." All that amounts to say that the money supply will play almost no role in the decisions for setting the interest rate, although it may be taken into consideration as one of the numerous elements allowing to assess the risks to price stability. The fact that monetary analysis may be useful to assess medium to long term trend in inflation is not really under dispute, nor is it essential to fix short term interest rates.

One may conclude from this first, sympathetic reading, of the press communiqué that in changing its strategy in this way, the ECB has proven its capacity to ear well grounded criticisms and thus that it is much more pragmatic that one would like us to believe. If this interpretation is correct, then the new strategy would allow for a more "flexible" monetary policy, i.e. both more reactive and more proactive. There is though a second reading of the press release, pointing to its contradictions (see e.g. Paul de Grauwe: "The central bank that has missed the point", The Financial Times, [START_REF] De | The central bank that has missed the point[END_REF], and to its omissions. There is in effect a contradiction in the press communiqué as far as the definition of price stability is concerned. The Governing Council begin by confirming the definition which it announced in 1998, namely "price stability is defined as a year-on-year increase in the Harmonised Index of Consumer Prices (HICP) for the euro area of below 2%." And it is after this confirmation that the new definition that I considered as a point target of 2% followed. How can we reconcile this two definitions, the first being asymmetric, the second symmetric. For Paul de Grauwe, "instead of creating clarity, the ECB has managed to create confusion about its true intention." That is unfortunate, as a pragmatic move may be interpreted as an incapacity to decide about such fundamental a point as a final objective of economic policy. How the markets could anticipate monetary policy if it is not sure about the objective the ECB is pursuing? The Committee of economic and monetary affairs of the European parliament should ask for more clarity in its next hearing of the president of the ECB. The omissions concern mainly the other ingredients of an inflation targeting procedure, as has been repeatedly emphasized in a number of briefing papers, especially those of Svensson. According to Christopher Sims (2003), "economists should recognize that they have a history of proposing simple "nominal anchor" prescriptions for monetary policy that have eventually proved not to be very useful." So we should not consider inflation targeting as The deus ex machina which always and everywhere will lead to a correct monetary policy. It is an improvement because it enhances goal and model transparency which is by itself desirable both for efficiency and democratic reasons. The other omission concerns the ways the ECB would participate to the pursuit of other objectives of economic policy according to its mandate. After all almost nobody expect that the inflation rate could be significantly higher than 2% in the two years to come, that is we are in a situation were the participation of the bank to the other objectives of the government will not prejudice its primary objective. So according to its statute it has to act. But these clarifications can't perhaps be made in a press release, and we should await the first implementation of the new strategy and the reading of future monthly bulletins.

Concluding remarks and summary

In the policy debate, each actor is reproaching to the others a lack of "flexibility". The stability pact reduce the reactivity of fiscal authorities, as has been emphasized in my December 2002 briefing paper. In this briefing paper I focus on the ECB and structural reforms.

The "new" ECB's monetary policy strategy represents a real progress. First the objective for price stability becomes a point target of 2% for the inflation rate, i.e. it is symmetric and higher than the implicit target of the former strategy. Second the two-pillar monetary strategy is de facto evolving towards a one pillar strategy, through the downgrading of the first pillar (the money supply). One may only regret that the press communiqué contains also the old definition of price stability witch can lead to some confusions. If one disregard the contradiction in the press release regarding the inflation objective the new strategy could allow for a more flexible monetary policy, that is both more reactive and more proactive.

For several years the ECB has been advocating structural reforms implying that they would have a decisive impact on price levels and on growth potential, through an increase in flexibility. I developed the argument along two lines. The first is that the theoretical argument in favour of structural reforms, notably in what concerns the labour market, is weak. It relies on too a simplistic application of the welfare theorems, and above all it does not take into account the necessary interaction between the democratic process and the market process which is the characteristic of a modern capitalistic society. The second is that the long debated question of whether labour market institutions affect growth performances and unemployment is empirically unsettled. The evidence is often weak, and even when institutions seem to explain some features of unemployment, their effect is of a second order of importance. Furthermore, capitalism appears from different empirical analyses as a flexible system, capable to accommodate different institutions at different moments of time. This goes against the tendency to invoke a single recipe (structural reforms, reduced role of the government) for each country at every time.

The above considerations lead to a strong conclusion: It does not exist a single superior institutional arrangement, and more specifically, it is impossible to advocate always more flexibility in the name of a technical argument. The role of the government in the economy, the balance between efficiency and equity considerations, the institutional setting that is given to a society are fundamentally political choices; and as such should never be entrusted to a technical body.

We should thus better think twice before recommending to governments to change such essential thing as the social contract on the basis of weak, uncertain and even contradictory evidences. That does not mean that structural reforms are not needed, but that they are much more complex to design and to implement -if one wants to avoid adverse effects -that the slogan "be more flexible" would want us to believe.

See Fitoussi

2002a 2 I did so at length in, La règle et le choix, Seuil 2002.

This may be arrived at by summing potential output growth, estimated at 2.5%, the target inflation rate, say 1.75%, and a trend shift in velocity of about 0.5% per year.