Pierre-Elliott Bécue

Mark Potse

Yves Coudière

Microscopic Simulation of the Cardiac Electrophysiology: A Study of the Influence of Different Gap Junctions Models

It has been suggested in the literature that a disorganisation of cardiac tissue at the cellular scale may affect the propagation of the action potential (AP) at the tissue scale, and may play a role in arrhythmia.

. We present in this article an improvement of this model, including gap junction models at any interface between cells. We then test this approach on hand-crafted two-dimensional networks of a hundred of cells, and compare it with our previous model which did not include gap junctions modelling.

Introduction

The standard bidomain or monodomain equations model a cardiac tissue at the macroscopic scale. They may be derived by homogenization of the microscopic bidomain equations, which model the propagation of the cardiac action potential (AP) on a network of individual cells. For homogenization purposes, the network is assumed to be periodic.

Yet, dysfunction or disorganization (non periodicity) of the tissue at the cellular level may affect the propagation of the AP at the tissue scale. For instance, zigzag propagation at a cellular level has been hypothesized to lead to arrhythmias [START_REF] De Bakker | Slow Conduction in the Infarcted Human Heart[END_REF]. These alterations cannot be represented by the homogenized bidomain or monodomain models.

To understand and tackle such questions, we planned to simulate the bidomain equations written at the cellular scale on a manually designed network of cells. The equations were first studied theoretically [START_REF] Colli-Franzone | Degenerate Evolution Systems Modeling the Cardiac Electric Field at Micro-and Macroscopic Level[END_REF], we proved existence of a solution [START_REF] Bécue | Theoretical and Numerical Study of Cardiac Electrophysiology Problems at the Microscopic Scale[END_REF], then we ran simulations on small handcrafted networks of individual cells [START_REF] Bécue | A Three-Dimensional Computational Model of Action Potential Propagation Through a Network of Individual Cells[END_REF].

In this paper we present an improved version of the equations presented in [START_REF] Bécue | A Three-Dimensional Computational Model of Action Potential Propagation Through a Network of Individual Cells[END_REF], that includes a proper gap junc-tions model. We briefly describe its properties, and then compare the results of both models.

2.

The bidomain model at the cell scale

We consider a set of cells indexed from 1 to N . Each cell is a connected medium Ω i . The extracellular medium is denoted by Ω 0 , and assumed to be connected. Each cell Ω i may be glued to some others. In such a case, their common interface is a gap junction, denoted by Γ ij (which is then the same as Γ ji ) where j is the index of the other cell. The interface between a cell and the extracellular medium is the cell membrane denoted by Γ 0i . An example is given Figure 1.

Figure 1. A 3 glued cells example describing the problem. The first cell is bigger because in our simulations, we want to be able to apply a stimulation on this cell without impacting the others.

For all time t > 0, the microscopic equations are -∇ • (σ i ∇u i ) = 0 for all i ∈ {0, . . . , N }. The transmission conditions between two domains Ω i and Ω j is

-σ i ∇u i • n i = σ j ∇u j • n j = c ij ∂ t v ij + F ij (v ij , w ij ).
Last, there is a no-flux boundary condition σ e ∇u e • n e = 0 on the external boundary Γ e of the system. Here, u i denotes the potential field in the cell i (or the extracellular medium 0). σ i are scalar electrical conductivities for each medium. The trans-membrane voltage across Γ ij is v ij = u i -u j . Any interface Γ ij is modeled as a capacitance c ij in parallel with an ionic current F ij . The vector w ij defined on the interface gathers the additional state variables of such an ionic current model, which is either a classic ionic model when i or j is 0, or a gap junction model otherwise. We assume that, the initial data are given only for the trans-membrane voltages v ij , as v ij (0, x) = v 0 ij (x) for x ∈ Γ ij . There exists a well posed weak formulation of these equations, where u i is searched in the space H 1 (Ω i ), with an additional gauge condition for the sake of uniqueness.

We approximate the solution with the P1-Lagrange finite element method, and use an Euler time-stepping method, implicit on the diffusion terms, and explicit on the ionic ones. Hence, for a sequence of times t n = nδt (δt > 0), we solve the discretized equations on each u i in the discrete spaces P1, with the condition that the meshes of each of the domains (cells, and the extracellular medium) match one with another.

For each time step, we have to solve a linear system of equations of the form AU n = F n where U n = U n 0 , . . . , U n N T is the vector of the degrees of freedom, and the right-hand side vector

F n = F n 0 , . . . , F n N T in-
volves the nonlinear functions F ij .

Simulations and results

We intend to compare our results with the ones we got from our previous work in [START_REF] Bécue | A Three-Dimensional Computational Model of Action Potential Propagation Through a Network of Individual Cells[END_REF]. Hence, we build two kind of problems. One type where the cells are geometrically connected by channels of intracellular material (see Figure 2), and one is with "glued" cells (as in fig. 1, or more generally fig. 3). In the former model, there is only one long cell with local reductions that model GJ channels. This is not a realistic model of gap junctions since the channels are too wide. In both figures, the "S" shape of the cell network is here for readability and also to respect the constraint that the extracellular medium has to be connected. On the latter model, each node on an interface between cells is assumed to be a gap junction. We will first test a linear gap junction model (Linear GJ), ie

F ij (v ij , w ij ) = v ij /R ij ,
where coefficients R ij are the conductances of the gap junctions. Then, we will show results with a non-linear gap junction model designed after experimentations on mouse cells [START_REF] Davidović | Microscopic Modelling of the Non-Linear Gap Junction Channels[END_REF]. In the simulations, the intracellular conductivities σ i are set to 1.7 mS cm -1 and the extracellular conductivity σ e is set to 3.0 mS cm -1 . The membrane capacitance C m is Figure 3. The top subset of a 99 glued cells network. The blue dots are spots where we measure the transmembrane voltage along time. The darkness of the cells is proportionate to their index (from 1 to 99). The whole network expands on 20 lines, the last lacking one cell compared to the others. set to 1 µF cm -2 .

3.1.

Specificities of the linear system

The symmetric matrix of the linear system is a combination of a stiffness matrix computed on all nodes and a mass matrix, computed on the nodes on any interface Γ ij . Due to the coupling conditions between the cells, this mass matrix has specific terms. It is challenging to compute these matrices efficiently in our finite-element code, CEPS.

In addition, the initial problem is a pure Neumann equation (no-flux condition on the external boundary). Hence the linear system has a unique solution defined up to a constant. Here we compute the solution perpendicular to the kernel space of the matrix A (corresponding to a gauge condition) with a conjugate gradient iterative solver, as suggested in ref [START_REF] Bochev | On the Finite Element Solution of the Pure Neumann Problem[END_REF].

Computational solver and meshes

Our model was implemented in the software code CEPS developed at Inria. It relies on the PETSc library to solve the linear systems on parallel computers. In test cases below, the problems were designed via a Python script, and meshed using the Triangle meshing software. For our simulations, the channel-version of our meshes had 34k nodes, 67.6k triangles. The glued one had 32k nodes and 63.4k triangles.

Comparison between the channel and the Linear GJ cases

We chose a resistance value R ij for the linear GJ case iteratively. Indeed, the bibliographic references always provide a resistance per unit area. Here, in two-dimensions, we do not have any precise value for the area. Furthermore, we don't really know the density of gap junctions on the interfaces. We hence chose a value that gave reasonable action potential propagation velocities, R ij = 0.015 kΩ.

For each case, we run a simulation for 420 ms, with a time step of 0.05 ms, and an output each 0.1 ms. In each case, we apply a stimulation as a current between each side of the upper-left cell. This stimulation occurs at t = 20 ms. For the Linear GJ case, the intensity of the stimulation is 2.25 × 10 -5 µA during 0.15 ms. For the channel case, the stimulation intensity is 2.70 × 10 -5 µA for the same duration. This difference results from the fact that in the channel case, the diffusion implies the need for a bigger stimulation. We show in Figures 4 and5

3.4.

Linear GJ: influence of the resistance value on the AP velocity

In the following cases, we changed the value of the resistance for the gap junctions. We used six different resistance values: R = 0.015 kΩ, R = 0.03 kΩ, R = 0.06 kΩ, R = 0.075 kΩ, R = 0.12 kΩ, R = 0.15 kΩ. The outcome is that the higher the resistance, the lower the AP velocity. We show in Figure 6 the corresponding graph. Figure 6. AP velocity as a function of the resistance of the GJCs, on the glued 99-cells problem. We see that the decrease of the velocity is not linear, but as the square root of the resistance.

3.5.

Linear GJ: influence of the GJ density on the AP velocity

We also studied how the velocity depends on the density of the interfaces along the main cell path, or similarly on the length of the cells. To this aim, we designed a singleline network of a fixed length with cells of varying length (and hence the number of cells, and hence the number of interfaces to cross). We tested this problem with both the linear Gap Junction model and the non-linear one.

We chose to represent five cases: 10 cells of 100 µm length, 20 cells of 50 µm length, 50 cells of 20 µm length, 5 cells of 200 µm length and 2 cells of 500 µm length. On Figure 7, we present the AP velocity as a function of the cell length. From this figure, we infer that the AP ve- locity depends on the density of Gap Junction the AP has to cross. This confirms the impact of the Gap Junctions on the propagation at the microscopic scale. To extract a more reliable velocity for cells with great length (200 µm or above), further testing with a longer network should be completed.

Example of non-linear GJ model

We implemented a non-linear GJ model from the results of [START_REF] Davidović | Microscopic Modelling of the Non-Linear Gap Junction Channels[END_REF], that got derived from experimentations on rat heart cells. With the same parameters as the Linear GJ simulation, but replacing the linear GJ model by the previously mentioned model, we run a simulation. Figure 8 shows the transmembrane voltages measured for this case. This Figure 8. Voltages measured on the blue dots shown in Figure 3 for the nonlinear GJ problem. The action potential takes 19 ms to cross 0.817 cm, which gives a velocity of 43 cm s -1 . result is not distinguishable from the linear problem with R = 0.015 kΩ.

3.7.

Linear GJ: example with too high resistance on the membrane between two cells

We implemented a case where the resistance between two cells (the 38 th and the 39 th ) is set to 1.5 kΩ. Figure 9 shows the transmembrane voltages measured on the blue dots in Figure 3, which shows the expected block of propagation between cells 38 and 39.

Conclusion

From these different results we were able to show the complex role of individual gap junctions on the overall averaged velocity of an action potential. The influence of the different models is to be explored further, with varying parameters, and methods, to confirm the results above. In particular, comparing simulation results with the experimental output from ref [START_REF] Davidović | Microscopic Modelling of the Non-Linear Gap Junction Channels[END_REF] could provide a decent way to assess wether our model provides a realistic propagation.

Yet, it remains many questions regarding the influence of gap junctions on the electrical activities of the heart tis-Figure 9. Voltages measured on the blue dots shown in figure 3 with R = 1.5 kΩ between the 38 th and the 39 th cells. We can see that the 39 th cell failed to depolarize and comes back to rest. sues, in particular when alterations occur. As homogenized model don't allow such studies, our model seems to provide a reasonable point to start.

Figure 2 .

 2 Figure 2. The top subset of a 100 cells network with geometrical GJCs. The blue dots are spots where we measure the transmembrane voltage along time. The whole network expands on 20 lines.

  the depolarization of the cells, measured on the blue dots shown in Figures2 and 3respectively.

Figure 4 .

 4 Figure 4. Voltages measured on the blue dots shown in Figure 2 for the channel problem. The action potential takes 20 ms to cross 0.855 cm, which gives a velocity of 42.72 cm s -1 , compatible with the literature.

Figure 5 .

 5 Figure 5. Voltages measured on the blue dots shown in Figure 3 for the linear GJ problem. The action potential takes 19 ms to cross 0.817 cm, which gives a velocity of 43 cm s -1 .

Figure 7 .

 7 Figure 7. AP velocity as function of the cell length, hence, of the density of gap junction along the main cell path.


Ali Gharaviri

Mark Potse

Rolf Krause

Angelo Auricchio

Ulrich Schotten

Effect of sodium channel blockade on the 3-dimensional substrate of atrial fibrillation: a simulation study

of Atrial Fibrillation: A simulation study 2 Background:

3 Anti-arrhythmic drugs, particularly sodium channel blocking agents, have been used as an 4 important clinical compound for cardioversion of atrial fibrillation (AF). However, several 5 experimental studies have revealed a reduction in cardioversion success rate after 6 administration of sodium channel blockers in both animal models and patients in later stages 7 of AF. The mechanisms underlying this efficacy loss are not fully understood. 8

Purpose:

9

In the present study, we investigated the loss of efficacy of Na-channel blockade to terminate 10 atrial fibrillation (AF) in structurally remodelled atria, using a novel highly detailed human 11 atrial computer model. We hypothesised that the possible explanation for the loss of efficacy 12 is that structurally remodelled atria represent a higher degree of 3-dimensionality in the AF 13 substrate. We assessed the effect of Na-channel blockade on endo-epicardial dissociation of

Results:

28 Na + -channel block lost its efficacy to terminate AF with an increase in the degree of fibrosis. 29

While in the absence of fibrosis Na+ channel block effectively reduced the number of 30 fibrillation waves, at high degrees of fibrosis the number of waves was higher than before 31 Na+ channel block. This was obviously due to the fact that a large excitable gap and an 32 increase in the degree of endo-epicardial dissociation of electrical activity did not prevent the 33 occurrence of a large number of breakthroughs (3D conduction). 34 35

Conclusion:

36

In the present study, we showed the efficacy loss of Na-channel blocker in AF termination in 37 different stages of structural remodelling. Although, Na-channel block reduced the number of 38 transmural conductions (BTs), the BT rates were still very high to support complex AF. 39

  Figure: Na-channel block effect on AF conduction pattern. A) AF survival rate. B) Number of waves. C) Excitable gap (EG). D) AF cycle length (AFCL). E) endo-epicardial dissociation (EED). F) Number of successful and unsuccessful BTs.


Ali Gharaviri

Mark Potse

Rolf Krause

Angelo Auricchio

Ulrich Schotten

Validation

Validation of a Novel Computer Model of Endo-epicardial Electrical Dissociation and Transmural Conduction during Atrial Fibrillation

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

Validation of a Novel Computer Model of Endo-epicardial Electrical Dissociation and 1

Transmural Conduction during Atrial Fibrillation 2 Ali Gharaviri, Elham Bidar, Mark Potse, Sander Verheule, Stef Zeemering, Rolf Krause, 3

Angelo Auricchio and Ulrich schotten 4

Background:

5 Recent studies demonstrated that transition from persistent to permanent atrial fibrillation 6 (AF) in goats is characterized by an increase in endomysial fibrosis, particularly in the outer 7 millimetre of the atrial wall. This phenomenon leads to loss of side to side connections in the 8 subepicardial layer as well as between the subendocardial layer and the endocardial bundle 9 network. Hence, increased fibrosis may contribute to dissociation of electrical activity 10 between these two layers (endo-epicardial dissociation of electrical activity, EED) giving rise 11 to transmural propagation adding to the overall complexity of fibrillatory conduction pattern. 12 Purpose:

13

The goal of this study was to assess the effect of epicardial fibrosis on endo-epicardial 14 dissociation and breakthroughs during AF and to validate the data against endo-epicardial 15 mapping in patients with AF. 16

Method:

17 To test this hypothesis, we developed a novel highly detailed computer model of the human 18 atria, which includes an epicardial layer with two to three layers of fibre orientations and an 19 endocardial bundle network based on a large variety of human imaging and anatomical 20 studies. To validate the model and the observed correlation between endo-epicardial 21 dissociation of electrical activity and breakthrough rates, we performed simultaneous endo-22 epicardial high-density mapping in 7 patients' right atria during cardiac surgery, 4 with 23 longstanding persistent AF (AF) and 3 without a history of AF (SR). 24

Results:

25 Both in the clinical recordings and in the simulations, the degree of EED ranged 26 approximately between 20% and 80%. In the patients, there was a trend towards more endo-27 epicardial dissociation in the patients with persistent AF. In the simulations, fibrosis clearly 28 increased the EED from 28.4±1.52% to 57±2.62% and 60.3±2.87% in control, moderately 29 fibrotic, and severely fibrotic models, respectively. Interestingly, the increase in EED activity 30 was associated with an increase in the incidence of breakthroughs both in the patient data as 31 well as in the simulations. Both in the mapping recordings and in the simulations, number of 32 breakthroughs per cycle was correlated well with and the degree of endo-epicardial 33 dissociated activity (r=0.61, p <0.05 and r = 0.72, p < 0.05 for the patient recordings and the 34 simulations respectively). 35

Conclusion:

36 Our results derived from the new computer model resembled well the basic 37 electrophysiological characteristics obtained from simultaneous endo-epicardial recordings in 38 humans. Importantly, the introduction of isolated atrial fibrosis in the epicardial layer resulted 39 in an increase in EED and breakthrough rates.

  Figure: EED and breakthroughs. A) Endo-epicardial dissociation (EED) in patient recordings. B) EED in simulation. C). Number of breakthroughs (BTs) per cycle in human recordings. G) Number of BTs per cycle in simulations.


Mathieu Chassé

Amélie Juhin

Delphine Cabaret

Steven Delhommaye

Delphine Vantelon

Georges Calas

Influence of Crystallographic Environment on Scandium K -Edge X-Ray Absorption Near-Edge Structure Spectra †

The absence of an extensive series of Sc K -edge X-ray absorption near-edge structure spectroscopy (XANES) reference spectra and the scarcity of direct structural data on Sc are major hurdles to develop our understanding of Sc chemistry. However, this first step is essential to develop new Sc-based applications and to better understand the formation of Sc deposits. Here, we present a detailed comparative study of Sc K -edge XANES spectra of three Sc-bearing compounds: a garnet (Ca 3 Sc 2 Si 3 O 12 ), an oxide (Sc 2 O 3 ) and a phosphate (ScPO 4 • 2H 2 O). Firstprinciples calculations have been performed to interpret the origin of the K pre-edge spectral features. We demonstrate the validity of our approach by reproducing satisfyingly the experimental spectra. The densities of states projected on the absorbing Sc atom and its first neighbours give the possibility to interpret the position and intensity of the pre-edge XANES features in terms of Sc local environment. The pre-edge features provide information on p-d mixing of the absorber orbitals giving clues on the centrosymmetry of the site and on the mixing of the empty 4p orbitals of the absorber with empty 3d orbitals of the neighbours via the empty p orbitals of the ligands. We also show that these features give a first estimate of the crystal-field splitting energy (ca 1.5 eV), inaccessible using other spectroscopic methods. Comparisons with K -edge spectra of other 3d 0 ions from the literature reveal the specificities of the Sc pre-edge, indicating that core-hole screening is weaker than for Ti 4+ -bearing compounds. This study provides a dataset of spectral signatures and a theoretical basis for their interpretation, a requirement for future studies on Sc chemical form in synthetic and natural systems.

Introduction

Scandium (Sc), the lightest transition element, is one of the most elusive. Often considered as part of the rare-earth elements, as both Sc and the lanthanides occur in a trivalent state, the chemical properties of Sc differ significantly from those of other rare-earth elements [START_REF] Connelly | Nomenclature of Inorganic Chemistry: IUPAC Recomendations 2005[END_REF] . This unique chemistry is exploited in high performance alloys and solid oxide fuel cells, resulting in a growing demand for this element [START_REF] Emsley | [END_REF] . A better understanding of the coordination chemistry of Sc is a prerequisite to developing innovative Sc-based applications 3 . In addition, the evaluation of future resources is of particular concern and requires an improved understanding of Sc speciation (i.e. chemical form) to trace the processes forming Sc deposits 4 .

a Institut de minéralogie, de physique des matériaux et de cosmochimie (IMPMC), Sorbonne Université, Muséum national d'Histoire naturelle, UMR CNRS 7590, IRD, F-75005 Paris, France. E-mail: mathieu.chasse@normalesup.org b SOLEIL synchrotron, L'Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette cedex, France † Electronic Supplementary Information (ESI) available: X-ray diffraction patterns, XANES spectra and demonstration of the expression of the isotropic cross section for cubic and monoclinic crystals. See DOI: 10.1039/cXCP00000x/

The determination of the chemical environment of Sc relies on a limited number of spectroscopic methods. With the exception of [START_REF] Peterson | [END_REF] Sc nuclear magnetic resonance (NMR) used to determine Sc speciation in crystals and glasses 5,6 , Sc K-edge extended X-ray absorption fine structure (EXAFS) and X-ray absorption near-edge structure (XANES) spectroscopy are the only spectroscopic methods which have provided relevant information on Sc crystal chemistry. For instance, early works focused on the site distribution of Sc in synthetic garnets 7,8 . Natural Fe oxides are major phases concentrating Sc at the Earth surface and appear to be the major Sc ore in future mines. Despite low Sc concentrations that rarely exceed a few hundred parts per millions, pioneering results have been obtained thanks to the combined chemical selectivity and high sensitivity of XANES spectroscopy 4 . X-ray absorption spectroscopy has also been used to get insights into the crystal chemistry of Sc 3+ in hydrates and aqueous solutions 3,9 . However, the absence of interpreted Sc K-edge XANES reference spectra 10 and the scarcity of direct structural data on Sc limit the identification of Sc speciation in complex systems 4 , which explains why the coordination chemistry of Sc remains one of the most challenging topics in inorganic chemistry 3 .

Except for extremely reduced environments, Sc only occurs in the trivalent state. As a consequence, Sc K-edge XANES spectra are of particular interest as the pre-edge region of 3d ions contains information on the local environment for 3d 0 configurations 11 , as the one of Sc 3+ . Three kinds of transitions can be observed: (i) local electric quadrupole (E2) transitions 1s → 3d, which probe the empty 3d states on the absorbing atom, split by the effective crystal-field splitting (10Dq) in the presence of a 1s core-hole; (ii) local electric dipole (E1) transitions 1s → 4p, where the empty p states of the absorber are hybridised with its empty 3d states. Such a p-d mixing occurs only when the site of the absorber is not centrosymmetric, such as in tetrahedral symmetry, or if the inversion symmetry is removed by atomic vibrations 11 ; (iii) non-local E1 transitions 1s → 4p where the empty 4p orbitals of the absorber are mixed with the empty 3d orbitals of the nearest metallic sites via the empty p orbitals of the ligands 12 . This paper presents a study of high-resolution Sc K-edge XANES spectra, with a special attention to the pre-edge features, for three model compounds: a garnet (Ca 3 Sc 2 Si 3 O 12 , eringaite), an oxide (Sc 2 O 3 , scandia) and a phosphate (ScPO 4 • 2H 2 O, kolbeckite) in which the local geometry around six-fold coordinated Sc (hereafter [6] Sc) is known. The acquisition of Sc K-edge XANES spectra yields a unique reference dataset, complementing previous data 4, [7][8][9] . Spectral features have been reproduced and interpreted using density functional theory (DFT). We show the validity of this approach and demonstrate its interest to interpret the origin of pre-edge XANES features for Sc-bearing compounds in terms of coordination number, local symmetry and site distortion, orbital mixing and short-to medium-range structure. As the prevailing coordination state of Sc 3+ is six-fold, these results not only offer new tools to interpret Sc X-ray absorption spectra but also give a basis to use first-principles calculations to determine Sc speciation in natural or synthetic systems, where Sc is incorporated at lower concentrations.

Experimental Methods

Characteristics of the Samples

Three pure Sc 3+ -bearing compounds were used as references for this study: 8,13 (space groups Ia 3d and Ia 3, respectively) while ScPO 4 • 2H 2 O crystallises with a monoclinic symmetry 14 (space group P2 1 /n, Fig. 1).

Ca 3 Sc 2 Si 3 O 12 , Sc 2 O 3 and ScPO 4 • 2H 2 O (
For the three phases, Sc atoms are six-fold coordinated with different symmetries ( Commercial Sigma-Aldrich Sc 3+ oxide powder -99.9 % trace rare earth metal basis was directly used.

XANES Measurement Procedures and Data Reduction

XANES spectra at the Sc K-edge were collected on the LUCIA beamline 16 at the SOLEIL synchrotron in Paris (France), operating with a storage ring current of 450 mA and an energy of 2.75 GeV. XANES spectra were collected using a Si(311) double crystal monochromator. The use of such monochromators gave high spectral resolution essential for the interpretation of the preedge spectra, with ∆E/E of ca 5 • 10 -5 . The monochromators were calibrated using the maximum intensity of the first inflection of the first derivative of Sc 2 O 3 (4492.8 eV). XANES measurements were performed in the energy range 4400 eV to 4800 eV. The energy step was (5, 0.2, 0.5 and 1) eV for energy ranges of (4400 to 4485) eV, (4485 to 4534) eV, (4534 to 4586) eV and (4586 to 4800) eV, respectively. Spectra were collected at room temperature, under vacuum, in transmission mode on pellets obtained from powdered material, diluted in a cellulose pellet and mounted on a holder between two Ultralene R films, using a photodiode. Two to five spectra were recorded for each sample with a counting time of about 10 min per spectrum. Multiple Sc K-edge XANES spectra recorded for each sample were averaged before background subtraction and normalisation using the PyMCA software 17 . The pre-edge background was removed by linear fitting of the pre-edge region and subtraction from the entire spectrum. A second linear function was used to fit the post-edge region. All spectra were normalised to the edge jump using a constant corresponding to the difference between the pre-edge and post-edge fitting functions at 4505 eV. Raw spectra can be found in the ESI † .

Theoretical Approach

First-principles calculations were performed using the Quantum Espresso plane-wave based density functional theory (DFT) suite of codes 18 . Theoretical XANES spectra are obtained in two steps: (i) the charge density is obtained via self-consistent field (SCF) calculation using the PWscf code 18 and (ii) XANES spectra are computed in a continued fraction approach using the XSpectra code 19,20 . We used the generalised gradient approximation (GGA) to the exchange-correlation functional with the Perdew-Burke-Ernzerhof (PBE) parametrisation 21 . The ionic cores were described by ultrasoft pseudopotentials 22 that have been generated according to a modified Rappe-Rabe-Kaxiras-Joannopoulos scheme 23 . Scalar relativistic effects are included in the pseudopotentials. The Sc ultrasoft pseudopotential was generated considering the 3s, 3p, 4s, 3d and 4p states as valence states, with cut-off radii of 1.30 • a 0 , 1.45 • a 0 , 1.30 • a 0 , 1.50 • a 0 and 1.45 • a 0 , respectively (a 0 is the Bohr radius and l = 2 was taken as the local part). The pseudopotential of the absorbing Sc atom (Sc * ) was generated from the Sc electronic configuration with a single 1s electron. The valence states of the Ca ultrasoft pseudopotential were the 3s, 3p, 4s, 3d and 4p states with cut-off radii of 2.30 • a 0 (l = 2 was taken as the local part). The valence states of the Si ultrasoft pseudopotential were the 3s, 3p and 3d states with cut-off radii of 1.80 • a 0 (l = 2 was taken as the local part). The valence states of the P ultrasoft pseudopotential were the 3s, 3p and 3d states with cut-off radii of 2.10 • a 0 , 2.10 • a 0 and 1.90 • a 0 . The valence states of the O ultrasoft pseudopotential were the 2s, 2p, 3s and 3p states with cut-off radii of 1.30 • a 0 , 1.35 • a 0 , 1.30 • a 0 and 1.35 • a 0 (l = 2 was taken as the local part). The valence states of the H ultrasoft pseudopotential were the 1s states with a cut-off radius of 0.90 • a 0 . Cell parameters and atomic positions used for the calculations were obtained from recently refined crystal structures of each phase 8,13,14 . Calculations were made with the 1s core hole successively located on each inequivalent Sc * of the cell. Due to the 1s core hole, the total charge of the supercell is set to +1 (full core-hole approach). Ca 3 Sc 2 Si 3 O 12 has a cubic unit cell containing eight formula units with a = 12.25 Å. Considering the large size of the cell, the interaction of the atom having a core hole with its periodic images is negligible and the unit cell was used directly to model this phase. Sc 2 O 3 also has a cubic unit cell containing twenty formula units with a = 9.84 Å, also large enough to be used directly to model this phase. ScPO 4 • 2H 2 O has a monoclinic unit cell containing four formula units with a = 5.43 Å, b = 10.20 Å and c = 8.91 Å (β = 90.50 • ). Considering the size of the unit cell, a 2 × 1 × 2 supercell was used to model this phase. The wavefunctions and charge density were expanded in plane waves and the respective cut-offs were set from convergence tests to 85 Ry and 680 Ry. Increasing these energy cut-offs to 100 Ry and 800 Ry did not significantly modify the results. The electronic occupations were fixed. The Brillouin zone was sampled according to the Monkhorst-Pack scheme 24 , using shifted 3 × 3 × 3 k-point grid, large enough to provide fully converged results. In order to interpret the pre-edge XANES features, partial electronic densities of states (DOS) were calculated on the systems having the core hole, using Lödwin projections, a shifted 3 × 3 × 3 Monkhorst-Pack kpoint grid and a Gaussian broadening of 0.5 eV.

In the XSpectra code, the reconstruction of the all-electron wavefunctions is done within the projector augmented wave method 25 .

To be able to treat large cells (more than one hundred atoms), the scheme uses a recursion method to construct a Lanczos basis and then computes the cross section as a continued fraction. For each compound, the absorption cross sections are calculated in the E1 and E2 approximations, on a 3 × 3 × 3 k-point grid, with a broadening parameter of 0.43 eV from (-10 to 5) eV relative to the Fermi level, taking into account the core-hole lifetime 26 . At higher energies, the photoelectron is fast enough to interact with the electron gas of the system and its mean free path is consequently attenuated, inducing a sharp decrease in the lifetime behavior 27 . The broadening parameter was increased accordingly up to 3 eV following an arctangent function 28 starting at 5 eV with an inflection point at 25 eV above the Fermi level. Theoretical spectra are calculated for a given orientation of the polarisation vector and, in the electric quadrupole approximation, also of the wave vector. However, the experimental spectra are measured on powdered samples and the cross section obtained corresponds to the isotropic cross section. In order to retrieve the isotropic cross section, symmetry considerations have been used 29 .

For cubic crystals, by selecting specific orientations of the polarisation and wave vectors, only one spectra needs to be calculated for the E1 approximation and two for the E2 approximation 11 . For monoclinic crystals, three spectra must be calculated in the E1 approximation and twelve in the E2 approximation. The demonstration on the number of individual theoretical spectra required to retrieve the isotropic cross sections and the corresponding orientations of the polarisation vector and the wave vector can be found in the ESI † .

In the case of Sc 2 O 3 , two individual spectra are obtained, one for each inequivalent Sc site. They are averaged considering the proportion of each site in the crystal structure and taking into account the core-level shift 30 . The calculated spectra are then aligned with the experimental ones with respect to the energy position of the main XANES peak. Experimental Sc K-edge XANES spectra (Fig. 2, in red) exhibit diverse spectral features in the main edge depending on the compound considered. In the spectrum of Ca 3 Sc 2 Si 3 O 12 , three welldefined features are observed (A, B and C), respectively located at ca 4508 eV, ca 4512 eV and in the range (4516-4520) eV. They are typical of transition metal K-edges of calcic garnet [31][32][33] . The spectrum of Sc 2 O 3 exhibits two main features, a sharp one (A) at ca 4508 eV and a broad, less intense one (B) with a maximum at ca 4517 eV. Two features are also observed in the spectrum of ScPO 7,9,10 . An additional feature (A 0 ) is visible at higher energy, ca 4503 eV, compared to the P 1 and P 2 pre-edge features.

Results and Discussion

Comparison between Theoretical and Experimental Spectra

As discussed below, it is related to the main edge and distinct from pre-edge transitions. The spectral variability observed in the studied compounds is reproduced in the calculated spectra (Fig. 2, in black). All major experimental features described previously are present. The position, shape and intensity are well reproduced for the A, B and C peaks in the main-edge region. The low energy pre-edge features (P 1 ) are shifted at higher energies in the calculated spectrum, as a result of the underestimation of the core-hole attraction on the empty d states of 3d transition metals, related to the limits of DFT 11 . However, the shape of the P 1 peaks is well reproduced, indicating our ability to explain the existence of the pre-edge features from our calculations. The relative intensity of the two P 1 features is also satisfying, the absolute intensity being only underestimated in the case of Ca 3 Sc 2 Si 3 O 12 . The P 2 and A 0 features are present with differences in terms of position and intensity.

Origin of the Pre-Edge Features

The pre-edge features contain key information on the electronic and crystallographic environments of the absorbing atom 11 . Their interpretation from calculations is not straightforward, in particular considering the complexity of the experimental spectra. To assign the transitions observed in the pre-edge spectra, the DOS for each calculated cell including the Sc 1s core hole have been projected on the absorbing Sc atoms and its first neighbours. We The P 1 feature, attributed to 1s → 3d transitions in XANES spectra of 3d transition metals, is split in two (P 11 and P 12 ) in the experi-mental spectrum of Ca 3 Sc 2 Si 3 O 12 . Our calculations show that the P 11 and P 12 features are related to E2 transitions (Fig. 3). The DOS projected on the 3d states of the absorbing Sc atom indicate that these transitions occur from 1s to 3d-t 2g and 3d-e g orbitals, respectively (Fig. 3 and Table 2). The calculated intensity is low, in particular when compared to the experimental pre-edge features. At the K-edge, the interaction between electrons and nuclear motions (i.e. vibronic coupling) can also enable E1 transitions to 3s and 3d final states 34 . Atomic vibrations are not taken into account by our theoretical approach but vibration-enabled E1 transitions may significantly enhance the intensity of the P 1 features 35 , explaining the observed discrepancy.

The A 0 feature is related to E1 transitions but its assignment is not straightforward. The DOS projected on the absorbing Sc atom indicate that these transitions are related to the 4p states of the absorbing Sc, possibly indicating non-local transitions resulting from orbital mixing with the nearest metallic neighbours, via the empty 2p states of the O ligands. The DOS have been projected on the closest Sc neighbours, located in the third cation shell. They do not explain the A 0 feature observed. By contrast, the projected DOS indicate that orbital mixing can occur with the 3p of the Ca neighbour located in the first cation shell, via the O 2p states. However, the resulting feature is split in two in the calculated spectrum while a broad but unique feature is observed experimentally. A possible explanation may be an overestimation of the core-hole lifetime in the calculations. With increasing energy, a sharp decrease of the core-hole lifetime is induced by the interaction of the photoelectron with the electron gas of the system, attenuating its mean free path 27 . The evolution of the core-hole lifetime with increasing energy along the edge is estimated empirically 28 . With a lower core-hole lifetime, the broadening of the calculated spectrum will lead to the overlap of the A 0 features, split in our calculations. Besides, in the PBE approximation used here, the energy of the Ca orbitals can be underestimated 36 . Such an energy offset may lead to the overlap of the calculated feature of highest energy with the main edge of the spectrum.

Sc 2 O 3 .
The P 1 features are reproduced in the calculations for the two inequivalent sites of the absorbing Sc atoms in Sc 2 O 3 . The contributions from sites 1 and 2 are significantly different and Sc 2 O 3 spectrum is dominated by contributions from site 2. As for Ca 3 Sc 2 Si 3 O 12 , the P 11 and P 12 features can be respectively assigned to E2 transitions, from 1s to 3d-t 2g and 3d-e g states of the absorbing Sc atom (Fig. 4 and Table 2). However, the intensity of these features is also influenced by local E1 transitions. The P 11 and, to a lesser extent, the P 12 features are enhanced by local E1 transitions to p states of the absorbing Sc located in site 2 due to 4p-3d-t 2g hybridisation. The overall experimental P 1 features are well accounted for by these calculated transitions. The P 2 features are also well reproduced and related to non-local E1 transitions. The projected DOS on the absorbing Sc atom indicate that these transitions can result from orbital mixing between the 3d-t 2g and 3d-e g orbitals of the first Sc neighbours via the empty p orbitals of the O ligands, for the P 21 and P 22 features, respectively.

As for Ca 3 Sc 2 Si 3 O 12 , the projected DOS confirm that the A 0 feature results from E1 transitions towards the empty 4p states of the absorbing Sc atom and is related to the rising edge.

ScPO 4 • 2H 2 O.
As for Sc 2 O 3 , E2 transitions are partly responsible for the P 11 and P 12 features observed in the spectrum of ScPO 4 • 2H 2 O (Fig. 5 and Table 2). These features are respectively associated to 1s → 3d-t 2g and 1s → 3d-e g local transitions. The intensity of these features is enhanced by local E1 transitions due to 4p-3d-t 2g and 4p-3d-e g hybridisation. The hybridisation is likely stronger with e g orbitals, explaining why the P 12 feature is the most intense, contrary to what would be expected from local E2 transitions. Such an hybridisation is effectively accounting for the relative intensity of the experimental P 1 features. The low intensity and broadness of experimental P 2 feature can be related to the rising edge, blurring the details of these features along with the decreasing core-hole lifetime. However, the presence of P 2 features is reproduced theoretically and explained by non-local E1 transitions. As for Sc 2 O 3 , the orbital mixing of the 3d-t 2g orbitals of the neighbouring Sc atoms with the empty 4p states of the absorbing Sc atom via the 2p states of the O ligand results in a low intensity feature, P 21 observed in the calculated spectrum but not clearly visible in the experimental one. Hybridisation with the 3d-e g orbitals of the neighbouring Sc atoms explain the presence of the P 22 feature, forming a slight feature in the experimental spectrum. The A 0 feature likely results from the orbital mixing of the 2p orbitals of the neighbouring P atoms with the empty 4p states of the absorbing Sc atom.

Insights into Electronic and Crystallographic Environments

Crystal-Field Splitting Energy.

Usually obtained by optical absorption spectroscopy, the crystalfield splitting energy, 10Dq, is only accessible through XANES spectroscopy in the case of 3d 0 ions, such as Sc. Estimates of this pa- rameter are crucial to rationalise crystal field effects on transition element ions or to serve as input values in ligand-field multiplet (LFM) analyses 37 Characteristics and Peculiarities of Scandium Pre-Edges.

The P 1 feature can always be related to E2 transitions from the 1s orbitals to 3d states localised on the absorbing Sc. However, the absolute intensity of the P 1 features and the relative intensity of the two components give information on orbital mixing or atomic vibrations. The intensity of the P 1 feature is enhanced by E1 transitions resulting from local p-d hybridisation of the absorbing atom orbitals (Figs. 4 and5). This indicates the non-centrosymmetry of the site of the absorbing Sc. Atomic vibrations can induce forbidden hybridisation, as illustrated by the discrepancies observed between the experiment and the calculation for Ca 3 Sc 2 Si 3 O 12 (Fig. 3). Local p-d hybridisation can also be highlighted by the relative intensity of the P 11 and P 12 components. When E1 transitions are absent, the P 11 component is more intense than the other one, a reversal of these intensity can be observed due to a stronger p-e g mixing compared to p-t 2g mixing (e.g. in the case of Sc 2 O 3 , Fig. 4). The P 2 features result from transitions towards empty 4p states of the absorber, orbitally mixed with the empty 3d states of the first neighbours via the empty 2p states of the O neighbours. Calculations indicate that these P 2 features are split when they result from hybridisation with 3d-t 2g and 3d-e g orbitals of the nearest metallic neighbour (e.g. in the case of Sc 2 O 3 , Fig. 4). However, the splitting of the P 2 features is not resolved experimentally when their intensity is low relative to the rising edge (e.g. in the case of ScPO 4 • 2H 2 O, Fig. 5).

The A 0 features are close in energy from the pre-edge features but their origin is distinct. As the main edge, they result form 1s → 4p transitions, with possible orbital mixing with p orbitals of the nearest neighbours.

The presence of distinct P 1 and P 2 features is not common to all 3d 0 ions. Experimental Ti K pre-edges of TiO 2 polymorphs (anatase and rutile), containing VI Ti 4+ , exhibit only three main features 38,39 . Calculations indicate however the presence of four transitions [40][41][42][43] , similar to the transitions producing the P 11 , P 12 , P 21 and P 22 features observed in Sc pre-edges (Fig. 4). The E2 transition towards the 3d-e g orbitals of the absorbing Ti is superimposed to the E1 transition towards the p orbitals of the absorbing Ti mixed with the 3d-t 2g of Ti neighbours. The core-hole attraction of 3d Sc orbitals is more important indicating a lesser core-hole screening for Sc compounds than for Ti ones. This effect is related to the nature of the chemical bonding between metal and oxygen, which may be more covalent for Sc compounds. By contrast, the presence of higher energy features, such as the A 0 feature of Ca 3 Sc 2 Si 3 O 12 , related to neighbouring alkaline-earth metals, is also identified for Ti-bearing compounds, such as CaTiO 3 39 and SrTiO 3 [START_REF] Cabaret | AIP Conference Proceedings[END_REF] . Further comparison with six-fold coordinated 3d 0 ions will require additional studies. In V 5+and Cr 6+ -bearing compounds, the metallic ion is most often tetrahedral or five-fold coordinated, prohibiting comparative analyses [START_REF] Peterson | [END_REF]46 . Calcium K-edge XANES spectra have also been published 47 . As for Sc spectra, P 1 and P 2 features seem to be present. However, the spectral resolution in the pre-edge region is limited, concealing the splitting of these features and calculated spectra are required for unequivocal assignment of the features.

In some cases, the interpretation of pre-edge features at the Sc K-edge is limited by the broadening resulting from the fast decaying core-hole state. Complementary information could be gained from resonant inelastic X-ray spectroscopy (RIXS).

In the case of 1s3p RIXS at the Sc pre-K-edge, the incident photon is absorbed, promoting a 1s electron to the first empty 3d state of Sc. Then, the 1s core-hole is filled by a 3p electron accompanied by the resonant emission of a Kα photon. The resulting high energy resolution fluorescence detection-X-ray absorption spectrum (HERFD-XAS) shows an apparent broadening which is reduced with respect to conventional XANES as it results from both 1s and 3d core-hole lifetimes 48 . In addition, as it is a two-photon resonant process, it is possible to probe both empty and occupied electronic states of the absorbing atom giving additional information on the local electronic environment of Sc, in particular when combined to LFM calculations.

Conclusions

The influence of the crystallographic environment of Sc on the spectral features observed in Sc K-edge XANES spectra has been studied through the experimental and theoretical characterisation of three compounds of known structure. This study provides a new set of Sc K-edge XANES spectra, of high interest to determine Sc speciation in natural or synthetic systems 4,10 . These successful calculations provide a theoretical basis for further first-principles calculations of Sc K-edge XANES spectra on poorly characterised systems. We demonstrate that such an approach is critical to interpret the observed pre-edge features. Combining experimental and calculated spectra gives access to an estimate of the crystalfield splitting energy, needed for ligand-field multiplet analysis of Sc XANES spectra 37 . Such an analysis also gives access to the local environment of Sc, informing on the degree of symmetry of the absorbing atom via the intensity of the main pre-edge features and on orbital mixing with the neighbouring atoms via low intensity spectral features present on the rising edge.

Fig. 1

 1 Fig. 1 Polyhedral representation of the unit cell of the compounds under study: (a) Ca 3 Sc 2 Si 3 O 12 ; (b) Sc 2 O 3 ; (c) ScPO 4 • 2H 2 O. Scandium is sixfold coordinated with C 3i (Ca 3 Sc 2 Si 3 O 4 and site 1 of Sc 2 O 3 ), C 2 (site 2 of Sc 2 O 3 ) and C 1 (ScPO 4 • 2H 2 O) point-group symmetry, respectively.

Fig. 2

 2 Fig. 2 Comparison between experimental (red) and calculated (black) normalised Sc K -edge XANES spectra for (a) Ca 3 Sc 2 Si 3 O 12 ; (b) Sc 2 O 3 and (c) ScPO 4 • 2H 2 O. Calculated spectra for the two inequivalent Sc sites of Sc 2 O 3 (site 1 in blue and site 2 in green) are shown separately, the theoretical spectrum (red) corresponds to the sum of these two spectra weighted by the relative proportion of each site (eight sites 1 for twentyfour sites 2 in the unit cell).

Fig.Ca 3

 3 Fig. Experimental and calculated pre-edge regions of Sc K -edge XANES spectra of Ca 3 Sc 2 Si 3 O 12 . The calculated E1 and E2 contributions have been separated. The corresponding local p and d DOS projected on the absorbing Sc (Sc * ), averaged on the six 1st O neighbours and on the six 1st Ca neighbours of Sc * are shown. The vertical line (E F ) corresponds to the Fermi level. The energy scale of the projected DOS has been shifted in order to match with experimental XANES spectra.

Fig. 4

 4 Fig. 4 Experimental and calculated pre-edge regions of Sc K -edge XANES spectra of Sc 2 O 3 . The calculated E1 and E2 contributions have been separated for the two inequivalent Sc sites. The corresponding local p and d DOS projected on the absorbing Sc (Sc * ), averaged on the six 1st O neighbours of Sc * and on the twelve 1st Sc neighbours of Sc * are shown. The vertical line (E F ) corresponds to the Fermi level. The energy scale of the projected DOS has been shifted in order to match with experimental XANES spectra.

Table

Table

of the Samples Ca 3 Sc 2 Si 3 O 12 . This

  compound was prepared following the experimental method of Bettinelli et al. (2011) 15 without addition of 99.99 % Eu 2 O 3 .

). In Ca 3 Sc 2 Si 3 O 12 , all Sc atoms are crystallographically equivalent in the 16a Wyckoff position with C 3i symmetry, corresponding to distorted octahedral sites, with an inversion centre and a three-fold symmetry axis. In Sc 2 O 3 , there are two crystallographically inequivalent Sc atoms in the 8b and 24d Wyckoff positions, with C 3i and C 2 symmetry, respectively. The site with C 2 symmetry corresponds to a distorted trigonal prism with a two-fold symmetry axis. In ScPO 4 • 2H 2 O, all Sc atoms are crystallographically equivalent in the 4e Wyckoff position, with no symmetry (C 1 ), corresponding to a highly distorted octahedron.

Preparation

Sc 2 O 3 .

Table 1

 1 List of the compounds under study along with their crystallographic parameters and the geometric properties of the six-fold coordinated Sc sites (cryst. sys.: crystallographic system, pos.: position, abs.: absorbing, ref.: references). Sc 2 Si 3 O 12 and #0004189 for ScPO 4 • 2H 2 O).

	Formula	Cryst. sys.	Space	Cell parameters	Z	Wyckoff pos.	Point group	Mean quadratic	Bond angle	Ref.
			group			(abs. atom)	symmetry	elongation	variance
	Ca 3 Sc 2 Si 3 O 12	Cubic	Ia 3d	a = 12.2500 Å	6	16a	C 3i	1	0	8
	Sc 2 O 3 , site 1	Cubic	Ia 3	a = 9.8431 Å	12	8b	C 3i	1	180.5	13
	site 2					24d	C 2	0.999	234.5
	ScPO 4 • 2H 2 O	Monoclinic	P2 1 /n	a = 5.4258 Å	4	4e	C 1	1.001	19.1	14
				b = 10.2027 Å
				c = 10.2027 Å
				β = 90.502 •
	ScPO 4 • 2H 2 O. This compound was precipitated by adding 1 g
	of Na 2 HPO 4 • 12H 2 O to 10 mL of 0.3 mol • L -1 ScCl 3 . The solid
	product was centrifuged, washed with distilled water and dried at
	45 • C for 24 h, giving 0.3 g of ScPO 4 • 2H 2 O.
	Characterisation of the Samples
	The crystal structures of the mineral phases have been char-
	acterised by X-ray diffraction (XRD) and the diffraction pat-
	terns are given in the ESI † . The data were collected in Bragg-
	Brentano geometry with a PANalytical X'Pert PRO MPD diffrac-
	tometer equipped with an X'Celerator detector. Cobalt Kα radia-
	tion (λ Kα1 = 1.78897 Å, λ Kα2 = 1.79285 Å) was used. Data were
	recorded over the 3 • 2θ to 90 • 2θ range, with 0.017 • 2θ steps and
	a total counting time of 6 h. The nature of the phases obtained was
	confirmed using the PDF-2 2016 database from the International
	Centre for Diffraction Data (ICDD, reference #00-005-0629 for
	Sc 2 O 3 ) and the American Mineralogist Crystal Structure Database
	(AMCSD, references #0010338 for Ca 3

  4 • 2H 2 O, the most intense (A) being centred at ca 4509 eV, the second one (B) is broader with a maximum at ca 4519 eV. A split pre-edge feature (P 1 ) is observed for each spectrum in the range (4491-4495) eV. A second group of pre-edge features (P 2 ) is observed for the Sc 2 O 3 and ScPO 4 • 2H 2 O spectra in the range (4495-4501) eV. The energy of the P 1 pre-edge feature (centred at ca 4493 eV) is steady and consistent with previous works

  distinguish the pand d-DOS of the absorbing Sc atom, the p-DOS of the first O neighbours, the pand d-DOS of the first Ca neighbours for Ca 3 Sc 2 Si 3 O 12 , the d-DOS of the first Sc neighbours for Sc 2 O 3 and ScPO 4 • 2H 2 O and the p-DOS of the first P neighbours for ScPO

4

• 2H 2 O. For each type of neighbouring element, the projected DOS are averaged over all neighbours belonging to the same shell. In this discussion, we assume that the six-fold coordinated site of Sc is close to an ideal ScO 6 octahedron, with O h point group symmetry. This consideration involves a crystal-field splitting of Sc 3d orbitals into t 2g and e g orbitals.

Table 2

 2 Origin of the experimental pre-edge (P) and A 0 features deduced from XANES calculations and DOS projections (neighb.: neighbour, Sc

	Compound	Feature Assignment
	Ca 3 Sc 2 Si 3 O 12	P 11	E2: 1s → 3d-t 2g of Sc*
		P 12	E2: 1s → 3d-e g of Sc*
		A 0	E1: 1s → 4p mixed with 3p of Ca neighb.
	Sc 2 O 3	P 11	Site 2, E1: 1s → 4p-3d-t 2g of Sc*
			+ Site 1 and 2, E2: 1s → 3d-t 2g of Sc*
		P 12	Site 2, E1: 1s → 4p-3d-t 2g of Sc*
			+ Site 1 and 2, E2: 1s → 3d-e g of Sc*
		P 21	Site 1 and 2, E1: 1s → 4p mixed with 3d-t 2g
			of Sc neighb.
		P 22	Site 1 and 2, E1: 1s → 4p mixed with 3d-e g
			of Sc neighb.
		A 0	E1: 1s → 4p
	ScPO 4 • 2H 2 O P 11	E1: 1s → 4p-t 2g of Sc*
			+ E2: 1s → 3d-t 2g of Sc*
		P 12	E1: 1s → 4p-e g of Sc*
			+ E2: 1s → 3d-e g of Sc*
		P 21	E1: 1s → 4p mixed with 3d-t 2g of Sc neighb.
		P 22	E1: 1s → 4p mixed with 3d-e g of Sc neighb.
		A 0	E1: 1s → 4p mixed with 2p of P neighb.

* : absorbing Sc).

  . The projection of the DOS demonstrated that the P 11 and P 12 features observed in the spectra of the compounds under study are related to 1s → 3d-t 2g and 1s → 3d-e g transitions. The energy difference between these two features is expected to give a good approximation of 10Dq, despite the presence of the 1s core hole. Experimental spectra indicate 10Dq values of 1.8 eV, 1.3 eV and 1.5 eV for Ca 3 Sc 2 Si 3 O 12 , Sc 2 O 3 and ScPO 4 • 2H 2 O, respectively. This value is in agreement with the value of ca 1.5 eV found by XANES spectroscopy for six-fold coordinated Sc 3+ complexes in hydrated salts9 . The crystal-field splitting energies retrieved from calculated spectra is slightly overestimated, with respective values of 1.9 eV, 1.4 eV and 1.6 eV for Ca 3 Sc 2 Si 3 O 12 , Sc 2 O 3 and ScPO 4 • 2H 2 O. The values estimated from the peak positions in the experimental Experimental and calculated pre-edge regions of Sc K -edge XANES spectra of ScPO 4 • 2H 2 O. The calculated E1 and E2 contributions have been separated. The corresponding local p and d DOS projected on the absorbing Sc (Sc * ), averaged on the six 1st O neighbours, on the four 1st P neighbours and on the twelve 1st Sc neighbours of Sc * are shown. The vertical line (E F ) corresponds to the Fermi level. The energy scale of the projected DOS has been shifted in order to match with experimental XANES spectra.spectra must be considered with care as these features are not always simply related to E2 transitions towards 3d orbitals. They can also result from E1 transitions through p-t 2g or p-e g mixing or because of atomic vibrations which can slightly modify the energy of the features. The presence of two sites, as is the case for Sc 2 O 3 , also has an influence. The projected DOS give another estimate of 10Dq with respective values of 1.9 eV, 1.8 eV, 1.5 eV and 1.7 eV for Ca 3 Sc 2 Si 3 O 12 , the sites 1 and 2 of Sc 2 O 3 and ScPO 4 • 2H 2 O. The comparison with 10Dq values obtained from calculated spectra indicate that, when orbital mixing allows E1 transitions, the 10Dq value obtained from the spectra tends to be underestimated. A rough estimation of 1.5 eV for the 10Dq value in the presence of a core hole seems consistent with experimental spectra and can be used in a first step but needs to be later refined by calculations. The effect of the core hole on 10Dq can also be estimated from the position of the 3d-t 2g and 3d-e g maxima in the DOS projected on neighbouring Sc atoms (i.e. with no 1s core hole). This crystalfield splitting energy is respectively estimated to be 2.1 eV, 2.6 eV, 2.6 eV and 2.0 eV for Ca 3 Sc 2 Si 3 O 12 , Sc 2 O 3 and ScPO 4 • 2H 2 O. The presence of a core hole therefore leads to a significant underestimation of 10Dq, up to ca 1 eV.

	Fig. 5

Acknowledgements

We thank Marco Bettinelli and Roberta Oberti for providing reference samples. We acknowledge SOLEIL for provision of synchrotron radiation facilities and we would like to thank the staff of the LUCIA beamline (Proposal No. 20150692). Computer facilities were provided by GENCI-IDRIS (Projects Nos. A0020906863 and A0040906863). Mathieu Chassé has been supported by scholarships from the École normale supérieure (ENS) and Macquarie University. This work was funded by the Institut universitaire de France (IUF).

Conflicts of interest

There are no conflicts to declare.


D Armentano

J-M Azaïs

David Ginsbourger
email: ginsbourger@idiap.ch

J R León

Conditions for the finiteness of the moments of the volume of level sets

Let X(t) be a Gaussian random field R d → R. Using the notion of (d -1)-integral geometric measures, we establish a relation between (a) the volume of the level set (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.

Introduction

Let X(t) be a centered, stationary, Gaussian random field

X : Ω × R d → R,
with continuous sample paths. By a scaling argument, and without loss of generality, we may assume that X(t) is centered with variance 1. On the other hand, for a given u ∈ R, let us consider the level set restricted to some compact set K ⊂ R d Cu,K := {t ∈ K : X(t) = u}.

If the sample paths of X(t) are almost surely (a.s.) differentiable and if a. s. there exist no point t such that X(t) = u, ∇X(t) = 0 (where ∇X(t) is X's gradient), then by the implicit function theorem, Cu,K is almost surely a manifold and its (d -1)-volume is well defined and coincides with its (d -1)-Hausdorff measure, namely, H d-1 (Cu,K ). Under some nondegeneracy hypothesis, the Kac-Rice formula (KRF) Azaïs-Wschebor [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]) gives an expression for the moments of this measure. If we consider the expectation, the compactness of the set K and the KRF imply that the first moment is finite, so we already have a sufficient condition of finiteness (but as we will see, the latter is not necessary). For higher moments the KRF provides a multiple integral, the integrand of which is degenerate on the diagonal so the study of finiteness is not straightforward.

When d = 1, Cu,K is a.s. a set of points and its measure is just the number of points. We have several result on finiteness of moments, see Sections 3 and 4. They all use at some stage the Intermediate Value Theorem. Unfortunately these methods are completely inoperative in higher dimensions. Here, we appeal to integral geometry in order to establish dimension-independent necessary and sufficient conditions of almost sure finiteness of level set volumes that boil down to one-dimensional results.

In Section 2 we recall the definition of the (d -1)-dimensional integralgeometric measure, which is defined as the integral of the number of points over a family of lines.

Our three main results follow

• In Section 3 we establish the equivalence between (a) the finiteness of the expectation of the (d-1)-dimensional integral-geometric measure of the level set and (b) the finiteness of the second spectral moment matrix. This result gives a simpler presentation and shorter proof of the results of Wschebor [START_REF] Wschebor | Surfaces Aleatoires[END_REF] which uses De Gorgi perimeters.

• In Section 4 we give sufficient conditions for finiteness of the second moment (Theorem 2) using the Geman condition (See [START_REF] Donald | On the Variance of the Number of Zeros of a Stationary Gaussian Process[END_REF]).

• In the same section, we prove finiteness all moments (Theorem 5), under some conditions, when the sample paths are smooth.

2 Integral geometric measure, Crofton formula

Let B be a Borel set in R d . Following Morgan [START_REF] Morgan | Geometric measure theory[END_REF] (and also Federer [START_REF] Federer | Geometric measure theory[END_REF]) we define the (d -1)-integral geometric measure of B by

I d-1 (B) := c d-1 v∈S d-1 y∈v ⊥ # {B ∩ v,y } dH d-1 (y) dS d-1 (v) (1)
where S d-1 is the unit sphere in R d with its induced Riemannian measure, and v,y is the affine linear space {y + tv : t ∈ R}. The constant can be easily computed, using the Crofton formula below and considering the particular case of the sphere, yielding,

c d-1 = Γ d+1 2 2π (d-1)/2 .
The integrand in (1) is measurable (see for example Morgan [6, page 13]), and since it is non-negative, the integral is always well defined, finite or infinite.

In particular, if B is (d -1)-rectifiable, then Crofton's formula [START_REF] Morgan | Geometric measure theory[END_REF] p. 31 yields

H d-1 (B) = I d-1 (B), (2)
where H d-1 is the (d -1)-Hausdorff measure.

3 Characterisation for the finiteness of the expected volume of the level set.

The spectral measure F of X(•) is a symmetric measure with mass one: it is a probability measure. Let Λ2 be the second spectral moment matrix defined by

(Λ2)ij := R d λiλjdF (λ).
This matrix may be finite or infinite, infinite meaning by convention that at least one entry is infinite. When Λ2 is finite, it is easy to prove that X(•) is differentiable in quadratic mean. If in addition the sample paths are almost surely differentiable (which is a little stronger) and if a.s. there exist no point t such that X(t) = u, ∇X(t) = 0, we have:

• the level set Cu,K is almost surely a submanifold of codimension 1,
and its Riemannian volume can be defined and coincides with its (d -1)-Hausdorff measure, namely, H d-1 (Cu,K );

• the KRF (see Adler-Taylor [START_REF] Adler | Random fields and geometry[END_REF] or Azaïs-Wschebor [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]) implies that

E(H d-1 (Cu,K )) = E(I d-1 (Cu,K )) = L d (K)E( X (0) ) e -u 2 /2 √ 2π = L d (K)F(Λ2)e -u 2 /2 , (3)
where I d-1 (Cu,K ) is the the (d -1)-dimensional integral-geometric measure defined above, L d is the Lebesgue measure on R d and

F(Λ2) := 1 (2π) (d+1)/2 z∈R d (z Λ2z) 1/2 e -z 2 /2 dL d (z). (4)
The second equality in (3) is the true Kac-Rice formula, the third is due to classical integration.

We need to extend the definition of F(Λ2) by setting it to +∞ when Λ2 is infinite.

So we consider the following relation:

E(I d-1 (Cu,K )) = L d (K)F(Λ2)e -u 2 /2 . ( 5
)
Note that its terms on both hand sides are now always well defined, finite or infinite.

The goal of this section is to prove that in a broad sense this formula is always true :

• Whenever Λ2 is finite, of course the RHS of (3) is finite, but also the LHS is finite also and equality holds true.

• If Λ2 is infinite then both sides of (3) are infinite.

Such a kind of property is known since the work of Cramér-Leadbetter [START_REF] Cramér | Stationary and related stochastic processes[END_REF] for d = 1 and from the work of Wschebor [START_REF] Wschebor | Surfaces Aleatoires[END_REF] for d > 1. Our proof uses Cramer-Leadbetter's result and generalised Crofton's formula. We first recall a result due to Cramér-Leadbetter, main result of Section 10.3 of [START_REF] Cramér | Stationary and related stochastic processes[END_REF]. The expected number of crossings Nu([0, T ] of a stationary processes with any level u on an interval [0, T ] is finite if and only if λ2 < ∞, where λ2 denotes the second spectral moment. In case λ2 is finite we have furthermore

E(Nu([0, T ]) = T π √ λ2e -u 2 /2 .
This result is based on polygonal approximation and intermediate values theorem, so it heavily relies on one-dimensional settings.

We now turn to our first main result.

Theorem 1. Let X(t) be a centered, stationary random field X : R d → R, with continuous sample paths. Then, we have equivalence between:

• E(I d-1 (Cu,K )) < ∞,
• The second spectral moment matrix Λ2 is finite.

In such a case we have

E(I d-1 (Cu,K )) = L d (K)F(Λ2)e -u 2 /2
Proof. Since X is almost surely continuous, then Cu,K is a Borel set on R d a.s., and therefore its integral geometric measure is well defined. By Fubini theorem we get that

E(I d-1 (Cu,K )) = c d-1 v∈S d-1 y∈v ⊥ E(# {Cu,K ∩ v,y }) dH d-1 (y) dS d-1 (v)
As a matter of fact, because of stationarity of the process and by Cramér-Leadbetter applied to the process t → X(y + tv), we get

E(# {Cu,K ∩ v,y }) = H1(K ∩ v,y ) v Λ2v 1 π e -u 2 /2 .
Then,

E(I d-1 (Cu,K )) is equal to e -u 2 /2 c d-1 π • v∈S d-1 v Λ2v y∈v ⊥ H1(K ∩ v,y ) dH d-1 (y) dS d-1 (v)
and by Fubini, we obtain that

E(I d-1 (Cu,K )) = L d (K)e -u 2 /2 c d-1 π S d-1 v Λ2v dS d-1 (v). ( 6
)
Integrating in polar coordinates the expression F(Λ2), given in (4), we obtain

F(Λ2) = 1 (2π) (d+1)/2 +∞ 0 ρ d e -ρ 2 /2 dρ v∈S d-1 (v Λ2v) 1/2 dS d-1 (v).
Furthermore, making the change of variable u = ρ 2 /2, it is straighforward to conclude that

F(Λ2) = c d-1 π S d-1 v Λ2v dS d-1 (v), (7)
and therefore from ( 6) yields

E(I d-1 (Cu,K )) = L d (K)e -u 2 /2 F(Λ2). (8)
We consider the two following cases.

• When Λ2 is finite the integral on the RHS of ( 8) is finite and therefore we get the desired result in this case.

• When Λ2 is infinite, this means that this matrix has at least one infinite element. In such a case we define the linear subspace

G(Λ2) := {v ∈ R d : v Λ2v < +∞}.
We prove that G(Λ2) is of dimension strictly smaller than d. Let v1, . . . , v d 0 be a maximal set of linearly independent vectors of G(Λ2).

Then by standard linear algebra:

the space span(v1, . . . , v d 0 ) generated by v1, . . . , v d 0 is in G(Λ2). This implies that d0 < d, for every v / ∈ span(v1, . . . , v d 0 ): v Λ2v = +∞ (unless v1, . . . , v d 0 is not maximal) , this implies that G(Λ2) = span(v1, . . . , v d 0 ).

In conclusion the integrand in ( 8) is almost everywhere infinite so the integral is infinite and by consequence the expectation of the integral geometric measure is infinite.

Finitness of k-moments of the volume of the level set

Using Formula (1) it is possible to obtain sufficient conditions under which the random variable I d-1 (B) has finite moments. To illustratethis we will first consider the second moment. Thus we have the following Theorem 2. Let assume that • The second spectral moment matrix Λ2 is non-degenerate.

• There exists δ > 0 such that the spectral mesure F satisfies

R d ||λ|| 2+δ dF (λ) < ∞.
Then we have

E(I d-1 (Cu,K )) 2 < ∞. Remark 3.
Let us point out that under the assumption that R d ||λ|| 2+δ dF (λ) is finite, the Kolmogorov-Chentsov criterion implies that the field X has a.s. C 1 sample paths. Thus the following equality takes place 2 . Moreover, the Riemannian volume of Cu,K can be defined and coincides with its (d -1)-Hausdorff measure.

E(I d-1 (Cu,K )) 2 = E(H d-1 (Cu,K ))
Proof. Without loss of generality we assume that δ < 2. Let r be the covariance function of X, and let us first consider the field restricted to the line y,v : y + tv, t ∈ R: Xy,v(t) = X(y + tv). Its covariance function is given by rv(t) = E[X(y + tv)X(y)] = r(tv).

Note that because of stationarity it does not depend on y.

It is sufficient to prove the assertion of the theorem for a set K being a centred ball Ba with sufficiently small diameter a. In that case note that the integral in the right-hand side of ( 1) is finite since for |y| > a the integrand vanishes. Since the second spectral moment matrix is finite and non degenerate

R d λ, v 2 dF (λ),
is bounded below and above. On the other hand, using a monotone convergence argument, as b tends to infinity

R d \B b λ, v 2 dF (λ) ≤ R d \B b ||λ|| 2 dF (λ) → 0. ( 9
)
So it is easy to conclude that for b sufficiently large, for any v ∈ S d-1

B b λ, v 2 dF (λ) > 1/2 R d λ, v 2 dF (λ). (10)
In the rest of the paper C will denote some unimportant constant, its value may change from an occurence to another.

Applying the Jensen inequality (with respect to the integral) yields

E(I d-1 (Cu,K )) 2 ≤ C c 2 d-1 v∈S d-1 y∈v ⊥ E(#{Cu,K ∩ y,v }) 2 dH d-1 (y) dS d-1 (v).
As already remarked, the integral is over a bounded domain and it is sufficient to prove that the integrand is uniformly bounded. Remark also that B b ∩ y,v is always a centred interval with length 2c less that 2a. Consequently,

E(#{Cu,K ∩ y,v }) 2 ≤ E(#{Cu,K ∩ 0,v }) 2 .
It remains to prove that

E(#{Cu,K ∩ 0,v }) 2
is uniformly bounded.

In fact, because of the Rolle theorem, if Uu is the number of upcrossings of the level u on the line 0,v , then

#{Cu,K ∩ 0,v } ≤ 2Uu + 1.
So it is sufficient to bound the second moment of Uu and even, because we have proved in the previous section that the first moment is uniformly bounded, it is sufficient to bound the second factorial moment. Since the variance has been assumed to be 1 and Λ2 is non-degenerate, the Kac-Rice formula applies and yields

E(Uu(Uu -1)) = a -a a -a E X + (s)X + (t) X(s) = X(t) = u 1 2π 1 1 -r 2 v (s -t) dsdt ≤ C 2a 0 (2a -τ )E X + (0)X + (τ ) X(0) = X(τ ) = u 1 1 -r 2 v (τ ) dτ,
where X stands for X0,v. By a standard regression formula, see for example [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] page 99,

E(X (0) X(0) = X(τ ) = u) = -E(X (τ ) X(0) = X(τ ) = u) = -r v (τ )u 1 + rv(τ ) .
Also,

σ 2 v (τ ) : = Var(X (0)|X(0) = X(τ ) = u) = Var(X (τ )|X(0) = X(τ ) = u) = λ2,v(1 -rv(τ )) -r 2 v (τ ) 1 -r 2 v (τ )
.

Set θv(τ ) := rv(τ ) -1 + λ2,vτ 2 /2 using the inequality z + t + ≤ (z + t) 2 /4 and the fact that θv(τ ), θ v (τ ), θ v (τ ) are non-negative we get

E(Uu(Uu -1)) ≤ Ca 2a 0 2λ2,vτ θ v (τ ) 1 -r 2 v (τ ) -3/2 .
Now, there exists a constant C0 such that 0 < w < 1 implies that 1 -cos(w) ≥ C0w 2 .

This implies in turn that for τ < 1/b where b has been defined in (10)

1 -rv(τ ) = +∞ 0 1 -cos(λτ ) dFv(λ) ≥ 1/τ 0 1 -cos(λτ )dFv(λ) ≥ C0τ 2 1/τ 0 λ 2 dFv(λ) ≥ 1 2 C0τ 2 +∞ 0 λ 2 dFv(λ) ≥ Cτ 2 ,
where Fv is the spectral measure along the line 0,v (for convenience it is on (0, +∞)). The penultimate equality uses (10), the last inequality is due to the fact that Λ2 is non-degenerate.

On the other hand it is direct to prove that 1 -rv(τ ) ≤ λ2,vτ 2 and, by compactness, the quantity λ2,v is bounded as a function of v giving that 1 + rv(τ ) ≥ 1 as soon as the radius a of the ball is sufficiently small. This yields

1 -r 2 v (τ ) ≥ Cτ 2 . As a consequence E(Uu(Uu -1)) ≤ Cλ2,va 2a 0 2 θ v (τ ) τ 2 dτ. (11)
The integrand in (11) can be bounded because

∞ 0 λ 2+δ dFv(λ) ≤ I(δ) := R d ||λ|| 2+δ dF (λ) < ∞.
We have

θ v (τ ) τ 2 = τ -2 ∞ 0 (τ λ2 -λ sin(λτ ))dFv(λ). Define R(u) := (u -sin(u)).
Its behaviour at zero and at infinity implies that for every δ, 0 < δ < 2, there exist a constant C δ such that

0 ≤ R(u) ≤ C δ u 1+δ .
This implies that

θ v (τ ) τ 2 ≤ τ -2 ∞ 0 λ|R(λτ )|dFv(λ) ≤ C δ τ -2 ∞ 0 λ(λτ ) 1+δ dFv(λ) ≤ C δ τ δ-1 ∞ 0 λ 2+δ dFv(λ),
Implying the convergence of the integral in (11), uniformely in v.

Next, we consider a Gaussian field having C ∞ sample paths. This is for instance the case of Gaussian random trigonometric polynomials in several variables or the random plane wave model [START_REF] Zelditch | Real and complex zeros of Riemannian random waves[END_REF]. A result of Nualart & Wschebor, quoted as Theorem 3.6 in the book [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], can be used for obtaining that all the moments of the random variable I d-1 (Cu,K ) are finite. The background result is the following: Proposition 4. Consider a Gaussian process χ, R → R satisfying Var(χ(t)) > κ for all t ∈ I a compact interval of R and some κ > 0. Then for all u ∈ R, and m, p ∈ N such that p > 2m, it holds

E[(Nu) m ] ≤ Cp,m 1 + C + E X (p+1) ∞ ( 12
)
where Nu is the number of points t ∈ I such that χ(t) = u, Cp,m is a constant depending only on p, m and the length of the interval I, and C is a bound for the density of χ(t).

Let us assume now that the field X has C ∞ sample paths and we assume that the variance is bounded below. As in the proof on Corollary 2 the process Xy,v(t) = X(y + tv) is a real process but now with C ∞ trajectories. Chose p = 2m + 1 then from Proposition 4, for every m,

E(# {Cu,K ∩ v,y }) m < Cp,m 1 + C + E X (2m+2) y,v ∞ .
It is an easy consequence of the Borel-Sudakov-Tsirelson inequality that E X (2m+2) y,v ∞ is finite. An argument of continuity shows that it is uniformly bounded. A further application of Jensen's inequality gives our third main result Theorem 5. Let X(t) a Gaussian random field R d → R with C ∞ sample paths and with variance bounded below. Then for every integer m and every compact set K,

E(I d-1 (Cu,K )) m < ∞.

Acknowledgements: This work has been partially supported by the French National Research Agency (ANR) through project PEPITO (no ANR-14-CE23-0011). David Ginsbourger would like to thank Andrew Stuart for a stimulating discussion in Cambridge that has incented the present collaboration, and in turn to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme "Uncertainty quantification for complex systems: theory and methodologies" (supported by EPSRC grant no EP/K032208/1).