Samir F Matar
email: matar@icmcb-bordeaux.cnrs.fr

Thomas Fickenscher

Birgit Gerke

Oliver Niehaus

Ute Ch

Adel F Rodewald

Naïm Al Alam

Rainer Ouaini

Pöttgen
email: pottgen@uni-muenster.de

Rodewald

Adel F Al Alam

Naïm Ouaini

Rainer Pöttgen

Chemical bonding in RFe6Ge4 (R = Li, Sc, Zr) and LuTi6Sn4 with rhombohedral LiFe6Ge4 type structure

Keywords:

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

Introduction

Kagomé networks of transition metal atoms are important basic building units in diverse intermetallic structure types. Prominent examples are the hexagonal Laves phases [1,2] and the huge family of CaCu5 phases [START_REF] Villars | Pearson's Crystal Data: Crystal Structure Database for Inorganic Compounds (release 2013/14[END_REF][START_REF] Parthé | TYPIX-Standardized Data and Crystal Chemical Characterization of Inorganic Structure Types, Gmelin Handbook of Inorganic and Organometallic Chemistry[END_REF] and related superstructures [START_REF] Pöttgen | [END_REF].

Such triangular systems are interesting model compounds for the study of frustration, especially for magnetically ordering phases [START_REF] Blundell | Magnetism in Condensed Matter[END_REF].

Stacking of Kagomé networks has also been observed in the rare structure type LiFe6Ge4 [7,8]. The few representatives are listed in Table 1. The transition metal atoms form the Kagomé networks and they are separated by hexagonal bipyramids of tetrel atoms around the cations (Li, Sc, Zr, or a rare earth element). So far only few studies on magnetic properties have been reported for the RETi6Sn4 series (RE = Gd-Tm) [12]. These compounds are paramagnets down to 3 K.

The family of LiFe6Ge4 type compounds is well suited for a systematic study of chemical bonding as a function of the cation and the valence electron count per formula unit, similar to the series of LiCo6P4 [16] and U4Re7Si6 type compounds [17]. Herein we report on single crystal X-ray diffraction data of ScFe6Ge4 (so far only powder X-ray diffraction diffraction data were reported [9,10]) and a comparative study of the chemical bonding of RFe6Ge4 (R = Li, Sc, Zr) and LuTi6Sn4 [12] with the remarkable motif of titanium Kagomé networks.

Experimental

Synthesis

Starting materials for the preparation of the ScFe6Ge4 sample were pieces of scandium chips (Smart Elements), iron granules (Alfa Aesar) and germanium granules (Chempur), all with stated purities > 99.9 %. Pieces of the three elements were mixed in the ideal 1 : 6 : 4 atomic ratio and arc-melted [18] under an argon pressure of ca. 800 mbar. The argon was purified over titanium sponge (900 K), silica gel and molecular sieves. The product button was re-melted several times to ensure homogeneity. The total weight-loss after several meltings was smaller than 0.5 %. The brittle ScFe6Ge4 sample has metallic luster and is stable in air over weeks.

The arc-melted sample was essentially polycrystalline. For the growth of small single crystals for structure refinement, a small piece of polycrystalline ScFe6Ge4 was sealed in an evacuated silica tube and annealed in a muffle furnace at 970 K for three weeks.

EDX data

Semiquantitative EDX analyses of the ScFe6Ge4 crystals investigated on the diffractometer were carried out by use of a Zeiss EVO  MA10 scanning electron microscope in variable pressure mode with scandium, iron and germanium as standards.

The experimentally observed composition of 101 at.-% Sc : 561 at.-% Fe : 341 at.-% Ge was close to the ideal one (9.1 : 54.5 : 36.4). No impurity elements were observed.

X-Ray diffraction

The ScFe6Ge4 sample was characterized through a Guinier powder diffractogram using CuK 1 radiation and -quartz (a = 491.30 and c = 540.46 pm) as an internal standard. The Guinier camera was equipped with an imaging plate technique (Fuji Film, BAS-READER 1800). The lattice parameters (Table 1) were obtained from a least-squares refinement. The experimental pattern was compared to a calculated one [19] to ensure correct indexing.

Small single crystals of ScFe6Ge4 were isolated from the crushed annealed sample.

They were glued to thin quartz fibers and investigated on a Buerger precession camera (white Mo radiation, Fuji-film imaging plate) in order to check their quality for intensity data collection. The data set was collected at room temperature by using a STADIVARI (Mo microfocus source and a Pilatus 100 K Detector with a hybrid-pixel-sensor) diffractometer. A spherical absorption correction was applied to the data set. Details on the crystallographic data are given in Table 2.

Computational details

An appropriate framework for the study of the electronic structure and the qualitative bonding behavior is the quantum density functional theory DFT [20,21].

Among the many computational methods built within the DFT we used two of them in a complementary manner. The Vienna ab initio simulation package (VASP) code [22][23][24] allows geometry optimization and cohesive energy calculations; it was also used to predict the electronic properties of MgFe6Ge4 starting from those determined herein for ScFe6Ge4. For this we used the projector augmented wave (PAW) method [23,25], with the generalized gradient approximation (GGA) scheme following Perdew, Burke and Ernzerhof (PBE) [26]. For Li, Mg, Sc and Zr the semi-core states were included for the building of the potentials. This enables for a better chemical picture in the solid state, ex. Li with 3 electrons (1s 2 , 2s 1 ), i.e. with a larger basis including core electrons, provides a charge of Li +0.82 in LiFe6Ge4 instead of fully ionized Li +1 if only 2s 1 were considered (cf. section 3.3.1). In fact the larger the basis set the better the results so that one avoids excessively ionic behavior far from reality. For instance this is the case of the ionic metal hydride MgH2 where, if Mg 3s 2 states are only considered, then we have Mg 2+ , H -1 from the calculations, which doesn't reproduce reality. Including semi-core Mg 2p 6 in the construction of the pseudo-potentials is then needed to get H -0.82 , closer to reality. Preliminary calculations with local density approximation LDA [27] led to largely underestimated volumes versus the experiment; the LDA being known as overbinding. The conjugate-gradient algorithm [START_REF] Press | Numerical Recipes[END_REF] is used in this computational scheme to relax the atoms. The tetrahedron method with Blöchl corrections [START_REF] Blöchl | [END_REF] as well as a Methfessel-Paxton [30] scheme was applied for both geometry relaxation and total energy calculations. Brillouin-zone (BZ) integrals were approximated using the special k-point sampling. The optimization of the structural parameters was performed until the forces on the atoms were less than 0.02 eV/Å and all stress components less than 0.003 eV/Å 3 . The calculations are converged at an energy cut-off of 500 eV for the planewave basis set with respect to the k-point integration up to 11  11  11 (kx, ky, kz) for best convergence and relaxation to zero strains. The calculations are scalar relativistic and assume spin degenerate total spins. Then all-electron calculations with the GGA were carried out for a full description of the electronic structure and the chemical bonding, using the augmented spherical wave (ASW) method devised by Williams, Kübler and Gelatt in 1979 [31] as a linearized method close to the LMTO (Linearized Muffin Tin Orbitals) method. It has benefited from continuous developments leading to full potential FP-ASW with implementation of chemical bonding according to different schemes (cf. text book by V. Eyert [START_REF] Eyert | The Augmented Spherical Wave Method -A Comprehensive Treatment[END_REF] and therein references). The ASW method uses a minimal basis set for the valence states with the outermost shells representing one of each kind: the valence states and the matrix elements are constructed using partial waves up to lmax+1 = 4 for Lu, lmax+1 = 3 for Fe, Sc and Zr and lmax+1 = 2 for Li, Mg and Ge. Self-consistency is achieved when charge transfers and energy changes between two successive cycles were below 10 -8 and 10 -6 eV, respectively. BZ integrations were performed using the linear tetrahedron method within the irreducible wedge. Besides the site projected density of states, we discuss qualitatively the pair interactions based on the overlap population analysis with the crystal orbital overlap population (COOP) [33].

In the plots, positive, negative, and zero COOP indicate bonding, anti-bonding, and non-bonding interactions, respectively.

Magnetic susceptibility measurements

The magnetic measurements were carried out on a Quantum Design Physical Property Measurement System (PPMS) using the VSM (Vibrating Sample Magnetometer) and the VSM-Oven option. In order to achieve temperatures above 350 K the latter one is equipped with a heater stick, additionally. For the VSM measurement in the low-temperature area a 15.661 mg piece of the crushed sample was fixed to the sample holder rod, while a 15.316 mg piece was attached to the heater stick by using an alumina based cement. For all measurements above 300 K, the sample was at first heated up to 600 K to ensure a virgin sample. Due to the heating limits of the low-temperature option [Tmax (VSM) ≈ 400 K] this is not accessible with our equipment.

Magnetic investigations were performed in the temperature range of 3 to 600 K with magnetic flux densities up to 80 kOe.

57 Fe Mössbauer spectroscopy

A 57 Co/Rh source was available for the 57 Fe Mössbauer spectroscopic investigations. 40 mg of ScFe6Ge4 were placed in a thin-walled PVC container at an optimized thickness of about 6.4 mg Fe/cm². The measurements were performed by using the usual transmission geometry at ambient temperature. The spectrum was fitted by using the Normos-90 program system [START_REF] Brand | Normos Mössbauer fitting Program[END_REF].

Results and discussions

Structure refinement

The ScFe6Ge4 data set showed a trigonal lattice and no further systematic extinctions. The centrosymmetric space group R3 m was found to be correct in agreement with previous studies on the stannides RETi6Sn4 [12]. The atomic positions of LuTi6Sn4 [12] were taken as starting values and the structure was refined with anisotropic displacement parameters for all atoms with Shelxl-97 (full-matrix leastsquares on Fo 2 ) [START_REF] Sheldrick | [END_REF]. In order to check for deviations from the ideal composition, the occupancy parameters were refined in separate series of least-squares cycles. All sites were fully occupied within three standard deviations. The final difference Fourier synthesis revealed no significant residual densities. The atomic coordinates and interatomic distances are listed in Tables 3 and4.

Further details of the structure refinement may be obtained from the Fachinformationszentrum Karlsruhe, D-76344 Eggenstein-Leopoldshafen (Germany), by quoting the Registry No. CSD-428504.

Crystal Chemistry

A view of the ScFe6Ge4 structure approximately along the [110] direction is presented in Fig. 1, emphasizing the Kagomé networks formed by the iron atoms as well as the Sc@Ge8 hexagonal bipyramids. The Fe-Fe distances within the Kagomé networks of 248 and 260 pm are slightly longer than twice the covalent radius of 232 pm [START_REF] Emsley | The Elements[END_REF], but close to the Fe-Fe distance of 248 pm in bcc iron [START_REF] Donohue | The Structures of the Elements[END_REF], indicating substantial Fe-Fe bonding interactions.

The Kagomé networks are stacked in AABBCC sequence. Each pair of Kagomé networks forms a kind of sandwich for the Sc@Ge8 hexagonal bipyramids. The apices of the bipyramids penetrate the hexagons of the Kagomé networks. The Sc@Ge8 hexagonal bipyramids share six common edges in the ab plane. Although this is a purely geometrical description of the ScFe6Ge4 structure, it allows a complete description with only two basic building units.

The Sc-Ge bond lengths within the Sc@Ge8 hexagonal bipyramids cover a broader range. The two apices are at the shorter Sc-Ge distances of 265 pm, close to the sum of the covalent radii of 266 pm [START_REF] Emsley | The Elements[END_REF]. The six germanium atoms building the ground plane of the bipyramid are at the much longer distance of 293 pm, indicating an asymmetrical bonding pattern.

Both of the two crystallographically independent germanium sites have nine, respectively six nearest iron neighbors with Fe-Ge distances ranging from 249 to 261 pm, similar to the binary iron germanides FeGe2 (254 pm) [START_REF] Wallbaum | [END_REF] and Fe6Ge5 (238-270) [39]. These Fe-Ge bonds interconnect the Kagomé networks and the Sc@Ge8 hexagonal bipyramids.

These ternary iron germanides exist with lithium, scandium, and zirconium at the 3a Wyckoff site. In going from the lithium to the zirconium representative, one observes an increase of the a and c lattice parameters (Table 1), a consequence of the different covalent radii of 123, 144, and 145 pm for lithium, scandium, and zirconium, respectively. This difference in size mainly influences the bonding to the two shorter Li-Ge, Sc-Ge, and Zr-Ge neighbors. The influence of the different valence electron count (65, 67, and 68 for LiFe6Ge4, ScFe6Ge4, and ZrFe6Ge4, respectively) is discussed below on the basis of ab initio electronic structure calculations. Keeping the size and the diagonal relationship between lithium and magnesium (covalent radius of 136 pm) in mind, one might also speculate about a magnesium-containing representative. However, so far, our synthesis efforts gave no hint for such a compound.

Apart from the germanides the isostructural series of RETi6Sn4 stannides exists [12][13][START_REF] Fartushna | XI th International Conference on Crystal Chemistry of Intermetallic Compounds[END_REF][START_REF] Ma | [END_REF]. The much larger tin atoms along with the rare earth atoms force a drastic increase of the a and c lattice parameters (Table 1). These stannides contain the remarkable motif of a titanium Kagomé network with a distinctly lower valence electron count of 43. In the following section we evaluate the electronic structures of the germanides LiFe6Ge4, ScFe6Ge4, and ZrFe6Ge4 along with the stannide LuTi6Sn4. The negative values for all four ternaries point to cohesive compounds. Particularly the hypothetic MgFe6Ge4 should be able of synthesis. However the trends of cohesive energies are not homogeneously changing along the series, i.e. with the regular increase of the number of electrons from A = Li, Mg, Sc and Zr one can notice that the most stable compound is ScFe6Ge4. Regarding LuTi6Sn4, the cohesive energy is large and amounts to -4.76 eV/cell leading to assign it the largest stability.

Electron transfer via charge values are obtained from the analysis of the charge density CHGCAR issued from VASP calculations using the AIM (atoms in molecules theory) approach [40]. Typically in a heteroatomic chemical bond, the charge density reaches a minimum between atoms and this is a natural region to separate them from each other. Such an analysis does not constitute a tool for evaluating absolute 2. The energy reference along the x axis is with respect to the Fermi level EF. For the sake of completeness the DOS of hypothetic MgFe6Ge4 are included. LuTi6Sn4 (Fig. 2 a) shows different features from the AFe6Ge4 series (Figs. 2 b-e). This is particularly exhibited by strongly localized Lu 4f within the valence band (VB) at ~-5 eV due to their filling up to saturation with 14 electrons on the one hand and with the broader Ti PDOS versus Fe PDOS which cross the Fermi level at much higher magnitude on the other hand. This also arises from the low filling of the Ti 3d states versus the larger filling of Fe 3d states. The localization of the latter and the large Fe d-PDOS magnitude at EF are a strong indication for a magnetic instability of the Fe compounds in the presently considered spin degenerate NSP configuration in the Stoner theory of band ferromagnetism [START_REF] Mohn | Magnetism in the solid state -An introduction, springer series[END_REF]. This mean field Stoner theory can be applied to address the tendency for spin polarization. This is shortly presented: The total energy of a spin system results from the exchange and kinetic energies. Referring the total energy to the non-magnetic state, this is expressed as: E = constant{1-In(EF)}. In this expression, I (eV) is the Stoner integral, which is calculated and tabulated for the metals by Janak [START_REF] Janak | [END_REF] and n(EF) (1/eV) is the PDOS value for a given state -mainly d-at the Fermi level in the non-magnetic state. If the unit-less Stoner product In(EF) is larger than 1, E is lowered and the system stabilizes in a magnetically ordered configuration.

Then the product In(EF) provides a criterion for the stability of the spin system.

Considering the NSP results for LuTi6Sn4 and AFe6Ge4, the PDOS at EF, n(EF), amount to n Ti (EF) = 1.69 eV -1 on the one hand and n(E F) LiFe6Ge4= 4.5 eV -1 , n(E F) ScFe6Ge4= 4.8 eV -1 , n(E F) MgFe6Ge4= 4.9 eV -1 and n(E F) ZrFe6Ge4= 4.7 eV -1 on the other hand. From ref. [START_REF] Janak | [END_REF] the Stoner integral values are I{Ti} = 0.34 eV and I{Fe} = 0.46 eV resulting into I Fe .n Fe (EF) = 2.08, 2.26, 2.22 and 2.17 for AFe6Ge4 with A = Li, Mg, Sc and Zr respectively. On the other side I Ti .n Ti (EF) = 0.58. Then the Stoner criterion 1-In(EF) is negative for the AFe6Ge4 family, not for LuTi6Sn4.

Energy lowering upon the onset of magnetization, i.e. intra-band spin-polarization should occur when spin polarization is allowed. Subsequent spin polarized (SP) calculations with equal initial  and  spin populations, were carried out. At selfconsistency a finite magnetization can be identified within an implicit long range ferromagnetic order. Nevertheless if magnetic exchange energy is not sufficient, zero local moments can result, so that the calculations are not biased initially. This also depends on the BZ mesh precision, i.e. the calculations are usually carried out in steps of increasing precision until no more changes are observed in the variational energy and the magnetic moments. Subsequent spin polarization (SP) led to a non magnetic solution for LuTi6Sn4 (as also determined experimentally [12]) and to a large energy stabilization E(SP-NSP) ~ -2 eV within the AFe6Ge4 series thanks to magnetic exchange. This confirms the Stoner criterion trends discussed above and finite moments could be identified on Fe with small negative moments on A and Ge arising from the mixing of the valence states of the respective basis sets through the chemical bond: eV. In spite of the small energy magnitude for the latter, the results are comforted by the property studies of ScFe6Ge4 which lead to a ferromagnetic ground state (vide infra).

The site and spin projected DOS are shown in Fig. 3 for all five 1:6:4 compounds.

Contrary to the AFe6Ge4 series (Fig. 3 b-e) which clearly show an energy shift between ↑ and ↓ spin PDOS with the former towards lower energy within the VB and the latter at higher energy whence their labeling respectively as majority and minority spins, there is no shift between ↑ and ↓ spin PDOS in LuTi6Sn4. (Fig. 3a). This reflects the magnetic results discussed above and shows that the Fermi level crosses a small magnitude Fe ↑ PDOS and a large ↓ spin PDOS. Then spin polarization has led to an overall lowering of the density of states at EF which goes along with the energy stabilization and underlines different bonding characteristics between the NSP and SP configurations.

In the NSP and SP DOS panels, the VB is characterized below ~-2 eV by the itinerant states of the respective constituents showing similar PDOS shapes. This indicates the mixing between the states accounted for in the basis set. Also within the transition metal Kagomé substructure Ti-Ti and Fe-Fe bonding is expected. These features are made explicit with the analysis of the chemical bonding using the COOP approach, exemplarily for LuTi6Sn4 and ScFe6Ge4. Focusing on the interactions between chemical species within the substructures of different kinds, Fig. 5 explicates them within NSP ScFe6Ge4. Clearly the weakest bonding occurs between Sc and Fe which are isolated from each other by the Sc@Ge8 hexagonal bipyramids. The COOP run similarly to the DOS from the energy standpoint whereby the bonding occurs with Fe d states at (anti-bonding) and slightly below (bonding) EF. On the opposite, Fe-Ge interactions are of bonding nature and occur at lower energy within the VB, i.e. at the Ge p-PDOS. This is similar to the Ti-Sn interactions in LuTi6Sn4 shown in panel (d) which exhibit bonding behavior throughout the VB and keep a bonding behavior a few eV above EF. In spite of the Sc@Ge8 polyhedra Sc-Ge bonding is of lower magnitude (due to the small multiplicity of 1:2 on each Ge site with slightly larger in-plane Sc-Ge2 COOP magnitudes versus Sc-Ge1).

Magnetic properties of ScFe6Ge4

At first the low-temperature area (3-300 K) of ScFe6Ge4 was investigated by performing a ZFC (zero-field-cooled) measurement at an applied field of 10 kOe (not shown here). Due to a significant magnetic moment per formula unit and the absence of any Curie-Weiss like behavior, we decided to perform measurements above 300 K. The top panel of Fig. 6 presents the temperature dependence of the magnetic and inverse magnetic susceptibility (χ and χ -1 data) of ScFe6Ge4 in the range of 300 to 600 K measured at 10 kOe. Due to limitations of our equipment we were not able to obtain reproducible results above 600 K. Within the investigated temperature area and in accordance to the theoretical calculations (vide supra) a ferromagnetic ordering can be clearly observed slightly below 500 K. Above the magnetic ordering an initial linear temperature dependency of the reciprocal susceptibility can be observed. Unfortunately, the linear range is not sufficient to execute a reliable fit by using the Curie-Weiss law.

In order to obtain more precise information about this cooperative phenomenon, a low-field measurement was performed in a zero-field-and field-cooled mode (ZFC/FC) which is shown in the inset of the top panel of Fig. 6. A significant bifurcation between the ZFC-and the FC-curves is visible which is typical for a ferromagnet and the Curie temperature could be determined to be TC = 491(3) K.

The bottom panel in Fig. 6 displays the magnetization isotherms of ScFe6Ge4 measured at 3, 300, 400, 500 and 600 K. A linear field dependency of the magnetization isotherms above the ordering temperature (500 and 600 K) is expected due to the expected paramagnetic character of the compound. However, at low magnetic fields a steep increase is observed which is most likely due to elementary α-Fe that exhibits a ferromagnetic ordering at TC = 1041 K [START_REF] Holleman | Lehrbuch der Anorganischen Chemie[END_REF]. Nevertheless, the intrinsic ferromagnetic character of ScFe6Ge4 is undoubtedly due to significant steeper increase at low magnetic fields of the 400 K isotherm in comparison to the 500 K one. The magnetic moment at 3 saturation magnetizations of Fe 2+ or Fe 3+ .

The discrepancy between calculated and measured moments can be assigned to the fact that the calculations account for a perfect defect void structure, while the measured sample is polycrystalline with a distribution of domain sizes. Ultrafine particles with size below single (mono)-magnetic domain might even be super-paramagnetic.

57 Fe Mössbauer of ScFe6Ge4

The 57 Fe spectrum of ScFe6Ge4 is presented in Fig. 7 together with a transmission integral fit. It could be well reproduced by a single signal with an isomer shift of 0.22 (1) and an experimental line width of 0.29(1) mm•s -1 ranging in the usual isomer shift region of metallic iron [START_REF] Van Der Kraan | [END_REF] and intermetallic scandium iron germanides [45]. No second resonance as expected for an α-Fe impurity (δ = -0.1 mm•s -1 , BHf = 33 T) could be observed. A weak quadrupole splitting (ΔEQ = 0.34(1) mm•s -1 ) is a consequence of the non-cubic site symmetry of the iron atom. Since the measured spectrum was recorded well below the ferromagnetic ordering temperature a clear Zeeman splitting came visible with a magnetic hyperfine field of 19.1(1) T. In order to check the magnetic stability of ScFe6Ge4 at room temperature a second spectrum was recorded at 5 K (not shown here). It lead to the same hyperfine field strength, no increase could be detected.

Conclusions

Comparative studies with DFT of the electronic and magnetic structures as well as the bonding behavior were carried out for the AFe6Ge4 family and comparisons with LuTi6Sn4 were established. Whereas similar electronic structure trends were observed within the AFe6Ge4 family (A = Li, Mg, Sc, Zr) for cohesive energies and the small charge transfer within the Fe substructure, larger electron transfer from the Ti substructure is found. The Fe Kagomé substructure is found magnetically unstable in a spin degenerate configuration and develops finite magnetization with M(Fe) ~ 1.7 B with a spin dependent bonding mainly driven by differentiated COOP(↑) and COOP(↓). 57 Fe Mössbauer spectroscopic measurements confirm the metallic character of the Fe atoms predicted by VASP calculations. The ferromagnetic nature of ScFe6Ge4 is reflected in the magnetic data (TC = 491(3) K) as well as in a magnetic hyperfine field splitting of 19.1 T.

3. 3 .

 3 Electronic structure and chemical bonding 3.3.1. Geometry optimization, cohesive energies and relative charge transfer From Table 5 the geometry optimized crystal structure results of ScFe6Ge4 are found in relatively good agreement with the experiment. The results were then used to predict the values for the as yet non-synthesized germanide MgFe6Ge4. This helps drawing trends of cohesive energies in the AFe6Ge4 (A = Li, Mg, Sc and Zr) series. The cohesive energies can be obtained from the difference between the total energy at self consistent convergence on the one hand and those of the atomic constituents in their ground state structures on the other hand: E(LiFe6Ge4) = -69.83 eV/cell; E(MgFe6Ge4) = -68.63 eV/cell; E(ScFe6Ge4) = -75.11 eV/cell; E(ZrFe6Ge4) = -76.93 eV/cell. E(Li) = -1.92 eV/at.; E(Mg) = -1.39 eV/at.; E(Sc) = -6.18 eV/at.; E(Zr) = -8.53 eV/at.; E(Fe) = -7.82 eV/at.; E(Ge) = -4.48 eV/at.. The resulting cohesive energies are then: Ecoh.(LiFe6Ge4) = -3.07 eV/cell; Ecoh. (MgFe6Ge4) = -2.40 eV/cell; Ecoh. (ScFe6Ge4) = -4.09 eV/cell; Ecoh. (ZrFe6Ge4) = -3.56 eV/cell.

  LiFe6Ge4: M(Li) = -0.04 B; M(Fe) = 1.64 B; Maverage(Ge) = -0.1 B. MgFe6Ge4: M(Mg) = -0.04 B; M(Fe) = 1.79 B; Maverage(Ge) = -0.1 B. ScFe6Ge4: M(Sc) = -0.05 B; M(Fe) = 1.78 B; Maverage(Ge) = -0.1 B. ZrFe6Ge4: M(Zr) = -0.02 B; M(Fe) = 1.82 B; Maverage(Ge) = -0.05 B. The calculated SP configuration implicitly applies for a ferromagnetic ground state. Also antiferromagnetic configurations were obtained by doubling the unit cell along c and by considering Fe as UP SPINS in the first and as DOWN SPINS in the second one (AF1) on one hand and by considering Fe as half UP and half DOWN in the first subcell and half DOWN and half UP in the second cell (AF2), on the other hand. Both AF configurations led to a raise of the energy with respect to SP-Ferro configuration although with different magnitudes, i.e. E(SP-AF1) ~ -0.51 eV; E(SP-AF2) ~ -0.22

  Figs.[START_REF] Villars | Pearson's Crystal Data: Crystal Structure Database for Inorganic Compounds (release 2013/14[END_REF] are followed in the COOP and lower magnitude antibonding COOP's are observed at EF. The ↑ spin COOP are mainly bonding with some anti-bonding contribution at ~-2 eV arising from Fe-Fe COOP whereas ↓ spin COOP are all of bonding nature with very small negative COOP at the top of the VB. Bonding is then spin-dependent whereby it is larger for minority spins thus bringing stabilization to this compound.

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

 12345 Fig. 1. Crystal structure of ScFe6Ge4. The iron Kagomé networks and the edge-sharing Sc@Ge8 hexagonal bipyramids are emphasized.

Fig. 6 .

 6 Fig. 6. Magnetic properties of ScFe6Ge4: (top) Temperature dependence of the magnetic susceptibility  and its reciprocal  -1 in the range of 300 to 600 K measured with a magnetic field strength of 10 kOe. The inset shows the magnetic susceptibility in zerofield-(ZFC) and field-cooled (FC) mode at 100 Oe; (bottom) Magnetization isotherms at 3, 300, 400, 500 and 600 K.

Fig. 7 .

 7 Fig. 7. Experimental (data points) and simulated (continuous lines) 57 Fe Mössbauer spectra at ambient temperature.

Table 1

 1 Lattice parameters of LiFe 6 Ge 4 type compounds.

	Compound	a / pm	c / pm	V / nm 3	reference
	LiFe 6 Ge 4	504.5	1966	0.4334	[7, 8]
	ScFe 6 Ge 4	507.9(3)	2000.9(1)	0.4470	this work
	ScFe 6 Ge 4	506.6	2001.3	0.4448	[9, 10]
	ZrFe 6 Ge 4	507.3(2)	2009.9(9)	0.4480	[11]
	YTi 6 Sn 4	578.2(1)	2273.0(5)	0.6582	[12]
	GdTi 6 Sn 4	578.6	2279.8	0.6610	[15]
	GdTi 6 Sn 4	578.1(2)	2276.1(6)	0.6587	[12]
	TbTi 6 Sn 4	577.7(1)	2274.8(2)	0.6575	[12]
	DyTi 6 Sn 4	577.2(1)	2268.6(4)	0.6545	[13, 14]
	DyTi 6 Sn 4	577.6(1)	2272.1(4)	0.6564	[12]
	HoTi 6 Sn 4	577.7(1)	2267.1(4)	0.6553	[12]
	ErTi 6 Sn 4	577.1(1)	2266.0(3)	0.6536	[12]
	TmTi 6 Sn 4	576.8(1)	2263.7(3)	0.6522	[12]
	LuTi 6 Sn 4	577.0(2)	2260.4(5)	0.6517	[12]

Table 3

 3 Atomic coordinates and anisotropic displacement parameters (pm 2 ) for ScFe 6 Ge 4 . The anisotropic displacement factor exponent takes the form: -2π 2 [(ha * ) 2 U 11 + ... + 2hka * b * U 12 ]. U eq is defined as one third of the trace of the orthogonalized U ij tensor.

	Atom	site	x	y	z	U 11 = U 22	U 33 U 13 = U 23 U 12	U eq
	Sc	3a	0	0	0	185(6)	152(10)	0	93(3) 174(5)
	Fe	18h 0.49633(8) -x	0.10182(3)	171(3)	180(4)	7(1)	90(3) 172(3)
	Ge1	6c	0	0	0.13226(5)	184(3)	152(4)	0	92(2) 173(3)
	Ge2	6c	0	0	0.33256(5)	181(3)	151(5)	0	91(2) 171(3)

Table 4

 4 Interatomic distances (pm) in the structure of RFe 6 Ge 4 (R = Li, Sc, Zr) and LuTi 6 Sn 4 . All distances of the first coordination shells are listed. Standard deviations are all less or equal 0.2 pm.

	LiFe6Ge4		ScFe6Ge4		ZrFe6Ge4		LuTi6Sn4
	Li:	2 Ge1 247.7	Sc:	2 Ge1 264.6	Zr:	2 Ge1 267.5	Lu: 2 Sn1 286.2
		6 Ge2 291.3		6 Ge2 293.2		6 Ge2 292.9		6 Sn2 333.1
		12 Fe	323.5		12 Fe	325.6		12 Fe	326.7		12 Ti	374.7
	Fe:	1 Ge2 248.9	Fe:	2 Fe	248.4	Fe:	1 Ge1 241.6	Ti:	2 Ti	281.8
		1 Ge2 250.0		1 Ge1 248.8		2 Fe	243.0		1 Sn1 284.8
		1 Ge1 251.6		1 Ge2 250.4		1 Ge2 248.5		1 Sn2 291.0
		2 Fe	252.3		1 Ge2 251.7		1 Ge2 256.8		1 Sn2 291.8
		2 Fe	252.3		2 Fe	259.5		2 Ge1 261.1		2 Sn1 292.3
		2 Ge1 256.3		2 Ge1 261.2		2 Fe	264.3		2 Ti	295.2
		2 Fe	289.6		2 Fe	296.5		2 Fe	300.0		2 Ti	319.8
		2 Li	323.5		2 Sc	325.6		2 Zr	326.7		2 Lu	374.7
	Ge1: 1 Li	247.7	Ge1: 3 Fe	248.8	Ge1: 3 Fe	241.6	Sn1: 3 Ti	284.8
		3 Fe	251.6		6 Fe	261.2		6 Fe	261.1		1 Lu	286.2
		6 Fe	256.3		1 Sc	264.6		1 Zr	267.5		6 Ti	292.3
	Ge2: 3 Fe	248.9	Ge2: 3 Fe	250.4	Ge2: 3 Fe	248.5	Sn2: 3 Ti	291.0
		3 Fe	250.0		3 Fe	251.7		3 Fe	256.8		3 Ti	291.8
		3 Li	291.3		3 Sc	293.2		3 Zr	292.9		3 Lu	333.1
		3 Ge2 291.3		3 Ge2 293.2		3 Ge2 292.9		3 Sn2 333.1

Table 5

 5 Geometry optimized lattice parameters and atomic coordinates of ScFe 6 Ge 4 and predicted MgFe 6 Ge 4

	Atom	site	x	y	z
	ScFe 6 Ge 4 (a = 507 and c = 2008 pm)
	Sc	3a	0	0	0
	Fe	18h	0.496	-x	0.102
	Ge1	6c	0	0	0.132
	Ge2	6c	0	0	0.333
	MgFe 6 Ge 4 (a = 509 and c = 2010 pm)
	Mg	3a	0	0	0
	Fe	18h	0.481	-x	0.100
	Ge1	6c	0	0	0.133
	Ge2	6c	0	0	0.331

Acknowledgments

This work was financially supported by the Deutsche Forschungsgemeinschaft. Computational facilities provided by the MCIA-University of Bordeaux and support from Conseil Régional d'Aquitaine are acknowledged. B.G. and O.N. are indebted to the Fonds der Chemischen Industrie and the NRW Forschungsschule Molecules and Materials -A Common Design Principle for PhD fellowships.

ionizations but allows establishing trends between similar chemical compounds. For the compounds under consideration the average charge values (Q) are:

In all four compounds Fe keeps a metallic behavior with very small negative magnitude of charge whereas the electron transfer is mainly within the A@Ge8 entities. Whereas monovalent Li shows the expected charge transfer magnitude, close but smaller than 1 as discussed in the computational details above, one should notice the large magnitude of electron transfer from Mg to Ge in hypothetic MgFe6Ge4 contrary to the Sc and Zr compounds. Also it can be noted that the charge transfer trends do not follow the course of the electronegativity values: (Li) = 0.98, (Mg) = 1.31, (Sc) = 1.36; (Zr) = 1.33; (Fe) = 1.83 and (Ge) = 2.01. This could be assigned to the particular feature of the Kagomé Fe substructure between the A@Ge8 units. Particularly the divalent, trivalent and tetravalent characters of Mg, Sc and Zr, respectively, do not appear as it would show in more ionic compounds as oxides for instance; this is also valid for the most electronegative element of the series, Ge, and most likely arises from the covalent bonding character. Different charge transfers are observed for LuTi6Sn4 with Q(Lu) = +1.36; Q(Ti) = +1.14 and Q(Sn) = -2.05. This is likely connected with the more electropositive character of titanium as compared to iron (see the course of the electronegativities (Ti) = 1.54 and (Fe) = 1.83).

All-electron calculations and bonding properties

Using the experimental data of Tables 1 and2 we carried out detailed calculations of the electronic structure and chemical bonding with the scalar relativistic ASW method.

The calculations assume firstly non-spin-polarized (NSP) configurations. At self consistent convergence of the energies and charges, small charge transfers similar to the above trends were obtained. The site projected density of states PDOS are shown in Fig.


Leander Schietgat

Bertrand Cuissart

Alban Lepailleur

Kurt De Grave

Bruno Crémilleux

Ronan Bureau

Jan Ramon

Comparing chemical fingerprints for ecotoxicology

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The evolution of data mining techniques provides methods for efficiently finding relations between chemicals and toxicological endpoints in large datasets. A fundamental part of any data mining study is how to encode the information. Molecular fingerprints [START_REF] Todeschini | Molecular Descriptors for Chemoinformatics[END_REF], where structural features are represented by a Boolean array, are a standard and computationally efficient representation of chemical compounds. They have been successfully used in molecular similarity search [START_REF] Willett | Similarity-based virtual screening using 2D fingerprints[END_REF][START_REF] Geppert | Advances in 2D fingerprint similarity searching[END_REF], and thus provide a relevant description of molecular structures. We focus on the use of 2D fingerprints in predictive toxicology and we compare different algorithms to construct them. These algorithms can be divided into four classes: (1) dictionary-based, (2) path-based, (3) radial-based, and (4) atom pair-based. As an alternative to the above approaches, we also considered a maximum common substructure (MCS) algorithm to describe the molecules. We trained predictive toxicology models from each of the chemical descriptions with several machine learning algorithms widely used in chemoinformatics [START_REF] Varnek | Machine learning methods for property prediction in chemoinformatics: Quo vadis?[END_REF]. The results suggest that the MCS-based chemical descriptors are an interesting alternative to the standard chemical fingerprints.

Fingerprint methods

The features generated by fingerprint methods are used to encode each molecule in a dataset as a k-dimensional binary vector (with k the number of features), where a 1 is marked in the i-th position if the i-th feature occurs in the example. We divide fingerprint methods into five categories, based on the way features are generated.

Dictionary-based fingerprints rely on features which have been identified a priori by domain experts as important fragments: MACCS keys consist of 166 features mostly encoded by SMARTS patterns, PubChem keys have 883 features corresponding to PubChem substructures. The remaining fingerprint types conceptually encode fragments based on the atom-bond structures in the dataset. Path-based fingerprints can enumerate all linear paths up to seven bonds including the description of rings up to 14 bonds (Linear), or the linear paths augmented with intersections of linear paths, with a maximum of five bonds per path to encode branched features (Dendritic), or features of four consecutively bonded non-hydrogen atoms along with the number of non-hydrogen branches, corresponding to a torsion angle (Torsion). Radial-based fingerprints iteratively encode features that represent each heavy atom in larger and larger structural neighborhoods, up to a given diameter (2, 4, and 6 in this study). The atom type (ECFP) and the functional class (FCFP) encoding rules were used to define the atom abstraction. In another approach, each heavy atom in a structure is characterized by an environment that consists of all other heavy atoms within a distance of two bonds. Each member of the list is encoded into a string of the form Type-freq(Type)-d, where freq(Type) is the number of times a given atom type is found at a distance d from the central atom. The atom-typing scheme used is the Sybyl Mol2 (MOLPRINT2D). Atom pair-based fingerprints encode features representing two atoms and their corresponding distance, for example the Carthart atom types and the topological distance separating them (Pairwise) or, as an extension, a triplet consisting of a set of three atoms and the topological distances separating them. As there are six different ways to order the atoms in a triplet, a canonicalization is performed, ensuring that every bit corresponds to a unique triplet (Triplets). Pairwise Maximum Common Subgraph Feature Generation (PMCSFG) [START_REF] Schietgat | A polynomial-time maximum common subgraph algorithm for outerplanar graphs and its application to chemoinformatics[END_REF] computes maximum common subgraphs (MCSs) under the block-and-bridge-preserving subgraph isomorphism between molecule pairs. For efficiency reasons the algorithm computes MCSs only from outerplanar graphs. For a given set of graphs, the method can compute all possible MCSs or a random subset [START_REF] Schietgat | Effective feature construction by maximum common subgraph sampling[END_REF].

Experiments

Dataset The European Chemicals Bureau (ECB) dataset was constructed using data from the European Chemicals Agency. We only kept chemicals annotated with standardized phrases implemented by the EU through the CLP regulation and referring to the hazard of the substance to aquatic organisms. The dataset was cleaned following standard practices such as the removal of inconsistent compounds or the addition of hydrogens on hetero-atoms. This resulted in a dataset with 372 chemicals annotated "very toxic" and 195 chemicals annotated "harmful". in five different machine learning methods: decision tree learning (DT), instance-based learning (IBL), naïve Bayes (NB), rule-based learning (RBL) and support vector machines (SVM). The learning task is to discriminate the harmful molecules from the toxic ones. We based the SVMs on the Tanimoto kernel [START_REF] Schietgat | Effective feature construction by maximum common subgraph sampling[END_REF], which computes a similarity between two fingerprints as the number of common features (i.e., the set intersection between the two molecules) divided by the total number of patterns that occur in either or both of the molecules (i.e., the set union). Tanimoto is the recommended kernel for fingerprints of small molecules. As implementation we used SVM light and C-parameter tuning. For the other learning methods, we used the Weka data mining tool, with standard configurations. We report the performance as the area under the ROC curve. 1. Area under the ROC for the different fingerprint methods and learning algorithms on the ECB dataset.

Results Table 1 shows the results of the comparison. The Torsion fingerprints perform the worst on average, while PubChem keys, MACCS keys, and PMCSFG perform the best. Interestingly, the three best performing fingerprints have the fewest features (133 to 1218). The average performance of the different learning methods varies from 0.60 to 0.87. Quantitatively, whatever the molecular description is, SVM appears to be the best performing learning method while rule-based learning leads to the poorest results. Moreover, there is a significant difference between the average score of the SVMs and the second best learner, NB.

Discussion

The preliminary results show that the dictionary-based fingerprints (PubChem keys and MACCS keys) obtain the best performance on average, with MCS fingerprints following closely behind. Apart from an adequate predictive performance, MCS fingerprints have multiple advantages. First, they can be automatically and efficiently learned from data, unlike the dictionary-based fingerprints which are selected by hand. Second, the feature set is smaller than the non-dictionary-based fingerprints. Moreover, the MCS features are typically much larger than the ones in the former fingerprints (9 atoms on average, up to 28 atoms for the ECB dataset) and hence carry more information. Third, an MCS feature corresponds to a recognizable molecular fragment, which makes it directly significant for a chemist. During the talk, we will illustrate this by exhibiting some examples of the recovery of known structural alerts for ecotoxicity, e.g., organophosphorus moieties, chlorobenzenes or phenol rings.

Acknowledgments

This work has been achieved thanks to joint financial supports: Région Basse-Normandie, ERC Starting Grant 240186 "MiGraNT", Research Fund KU Leuven, IWT (SBO Nemoa, SBO InSPECtor).


Jean-Matthieu Monnet
email: jean-matthieu.monnet@irstea.fr

Franck Bourrier

Evaluation of the effect of forest input data on rockfall simulations

Keywords: protection forest, rockfall simulation, airborne laser scanning

Rockfall simulation models are now able to take into account impacts on trees. Forest maps at operational scale can be derived from statistical inventories or remote sensing such as airborne laser scanning (ALS). The present study compares rockfall simulations results obtained with different forest inputs for three different stands. Results obtained with field and ALS-derived inputs showed little difference regarding energy values but sometimes large ones concerning the proportion of stopped blocks. It also turns out that the spatial heterogeneity and coniferous proportion of trees are critical issues which might be difficult to address at operational scale.

INTRODUCTION

The protective effect of trees against rockfall hazards has been known for a long time. However research on the interactions between tree stands and falling blocks only dates back to the 1980's. More recently, numerical models have been developed to simulate the trajectory of falling rocks in forested site [START_REF] Dorren | State of the art in rockfall -forest interactions[END_REF]. In the more advanced modelling approaches, the trees are individually integrated into the site model. Modelling forest stands in this framework thus requires building virtual forest having the same characteristics as in the site. As an alternative to making field forest inventories, tree distributions and characteristics of each tree can be estimated using airborne laser scanning (ALS). However, the errors in modelling the quantitative characteristics of the trees may lead to non-negligible errors in the simulated trajectories of rocks as they move through the trees. Such error may yield erroneous evaluation of the forest protection. The objective of this study is to compare the results of rockfall simulations performed with forest models that are derived from two types of investigation: field investigation or ALS remote sensing.

MATERIAL AND METHODS

Rockfall modelling

The software RockyFor3D is used for rockfall simulations. It is a simulation model that calculates trajectories of single, individually falling rocks, in three dimensions [Dorren et al., 2005]. The model combines physically-based algorithms with stochastic approaches, which makes Rockyfor3D a so-called "probabilistic process-based rockfall trajectory model". It simulates the rockfall trajectory by calculating sequences of classical parabolic free fall through the air and rebounds on the slope surface, as well as impacts against trees if required. Rolling is represented by a sequence of shortdistance rebounds and sliding of the rocks is not modelled. The required input data consist of a set of rasters which define slope surface, forest and block characteristics and topography [START_REF] Dorren | Rockyfor3D (v5.1) revealed -transparent description of the complete 3D rockfall model, ecorisQ[END_REF]. The main components of the model are the parabolic free fall, the rebound on the slope surface, the impact against a tree and the calculation of the fall direction [START_REF] Bourrier | Toward objective rockfall trajectory simulation using a stochastic impact model[END_REF]. Each time a simulated block surpasses or rebounds in a given raster cell, statistics of the block trajectory in that cell are recorded. Simulations are held on an archetypal virtual study site in order to focus on the effect of forest inputs on rockfall assessment. The topography is a slope of 35 degrees of compact soil with rock fragments. The roughness of the surface is expressed as obstacles with heights of 0.05 m, 0.1 m and 0.2 m for 70 %, 20 % and 10 % of the surface area respectively. Spatial resolution is 2 m. Spherical blocks with a volume of one cubic meter and of 2600 kg.m -3 density are released from five meters height along a contour line (Fig. 1). Their velocity increases across an unforested area of 50 m. Then they enter a forest patch where tree impacts might stop some blocks. The forest input data for Rockyfor3D are either a file with the trees positions and diameters, or files describing tree density, mean and standard deviatoin of diameter at breast height (DBH) in delineated polygons. The percentage of coniferous trees is in both cases supplied as a raster file.

Forest input data scenarios 2.2.1 Tree-level field data (Field_Tree scenario)

Tree positions, diameters and species are recorded on the field for three different forest stands presenting different forest structures and species (Table 1) and located in rockfall hazard areas. All trees with a diameter at breast height (DBH) above 7.5 cm (5 cm for Saint Paul) were inventoried and georeferenced. Additionally the heights of all trees were measured in Saint Agnan. In Valdrôme heights were measured in three transects including a total of 362 trees. For this scenario, the positions and diameters are known, and the coniferous percentage input is a raster image of 2 m resolution computed by retaining in each pixel the proportion of coniferous trees among trees whose centres are located inside this pixel.

Area-level field data (Field_Area scenario)

This scenario is a degraded version of the treelevel field inventory. Tree positions are unknown and diameter distribution is modelled from its mean and standard deviation. The stand surface is divided in square pixels of 50 m width (40 m for Saint Agnan in order to have an integer number of pixels for the forest stand). From the tree-level field data, the mean and standard deviation of diameter are calculated for these pixels, as well as the coniferous percentage and stem density. For this scenario, tree positions and diameters are randomly generated by the software in each pixel to provide the required inputs for the rockfall simulations. Trees positions are randomly generated with uniform probability distribution. Tree diameters are generated from a [START_REF] Hirata | Extracting forest patch attributes at the landscape level using new remote sensing techniques -an integrated approach of high-resolution satellite data, airborne lidar data and GIS data for forest conservation[END_REF]Monnet, 2011] was applied to the Saint Agnan and Valdrôme datasets. It is not used in Saint Paul as it is not relevant in coppice stands. The detection outputs are the positions and heights of the local maxima of the canopy height model computed from the ALS data. Those local maxima should correspond to the tops of the dominant trees. In order to estimate the corresponding diameters, a linear regression is fitted between the natural logarithm of DBH and the measured height for a random sample of 10% the field trees. This relationship is then applied to the detected height in order to produce the input file for the simulations. The coniferous percentage file is the same as for the Field_Area scenario (average over large pixels).

Area-based ALS data (ALS_Area scenario)

For the Saint Paul and Saint Agnan datasets, statistical inventory data were also collected in circular plots located inside the ALS flight. A circular plot is a forest area where trees within a fixed radius from the plot centre are inventoried. Stand-level forest attributes are then derived for Naesset, 2004;Monnet, 2011].

Those statistical descriptors, or ALS metrics, are computed with points whose height above ground is superior than 2 m. Height percentiles are abbreviated hg,i , where g refers to the point group (s: single, l: last of many, f: first of many) and i to the percentile value. The mean height for all points ha,mean is also computed. Density metrics are abbreviated di and correspond to the percentage of points with height lower than i % of the maximum height value on the plot. Two variables describing the canopy height model (CHM) are also computed: prop CHM>2m is the proportion of pixels with values above 2 m, and mean CHM is the mean value of the CHM on the plot. ALS metrics selected in the models and statistics about models accuracy obtained in leave-one-out cross-validation are summarized in Table 3. Those relationships are then applied to the point cloud contained in the three forest stands divided in square pixels of 20 m width in Saint Agnan and 16.7 m in Saint Paul. The positions and diameters of trees are then sampled using the same procedure as for the Field_Area scenario. The coniferous percentage input is the same as for the Field_Area and ALS_Tree scenarios.

Rockfall simulations

10 000 rockfall trajectories are simulated for each two-meter-wide pixel located on the release contour line. The number and the kinetic energy of passing blocks is recorded when the contour line immediately below the forest patch is attained. To avoid border effects, forest stands are laterally duplicated. The release line width is equal to the width of the forest stand and the measure line covers the whole width (including lateral buffers).

RESULTS

ALS estimation results

Tree detection

The assessment of detection accuracy is based on an automated procedure [Monnet, 2011]. In Saint Agnan, 136 (38.1%) of the 357 trees are correctely detected, with 18 false positives. In Valdrôme, 499 (48,7%) are detected, with 43 false positives. As usual with tree detection, dominant trees are well detected whereas trees in the lower layers are often missed (Fig. 2). For the detected trees, the height is slightly over-estimated by ALS in Valdrôme, with a mean difference of 0.2 m. In Saint Agnan the overestimation is larger, with a mean value of 1.4 m. Standard deviation of errors are 1.5 m in both cases. Linear regression with measured height h as a function of the ALS height hl is h=0.93*hl+0.49 for Saint Agnan (R 2 =0.85) and h=0.9*hl+1.79 for Valdrôme (R 2 =0.81). Heights of tall trees tend to be over-estimated, whereas small trees are rather under-estimated.

Area-based estimation

In both Saint Agnan and Saint Paul, the mean and standard deviation of the diameter are generally under-estimated by the ALS estimation models. On the contrary, the stem density is over-estimated (Table 4).

Rocks propagation

The distributions of passing block energy are displayed in Fig. 3. With the coppice stand of Saint Paul, it is noteworthy that the energy distribution is similar for all forest inputs (Table 5). Relative differences in mean energies and 95 th percentiles are below 4%. Mean energy is 37% higher when no forest effect is taken into account, compared to the Field_Tree scenario. However, the proportion of passing blocks is similar for the scenarios based on field data (around 82%) but almost all blocks roll through the forest in the ALS_Area scenario.

In the even-aged, coniferous forest of Valdrôme, the relative difference for mean energy is 7% for the Field_Area and 16% for ALS_Tree compared to the Field_Tree scenario. The mean energy for the No_Forest scenario is 64% higher. The same trend can be observed for the 95 th percentile. Regarding the number of passing blocks, values with the Field_Area and ALS_Tree inputs (80%) are between the Field_Tree (58%) and No_Forest (100%) cases.

With the uneven-aged, mixed forest of Saint Agnan, energy distributions for the Field_Tree, Field_Area and ALS_Tree scenarios are very similar, with relative difference lower than 8% for mean energy and the 95 th percentile. However, the number of passing blocks is almost 20% higher. The ALS_Area scenario is the only case where the energy and proportion of blocks have much lower values than the Field_Tree reference.

DISCUSSION

Coniferous proportion

Considering the reference scenario (Field_Tree), there is one limitation regarding the way the proportion of coniferous trees is taken into account. This parameter is critical as the energy dissipation by broadleaved trees is 1.7 times higher than for coniferous trees [Dorren & Berger, 2005]. In our study, the raster cell size for the simulation is two meters, so that the information about specie is not integrated in the simulation at the tree level, but averaged at this spatial resolution. However in Saint Agnan cases where both coniferous and broadleaved trees are located in the same pixel are rare. In case rockfalls are simulated at a coarser resolution, such as 5 or 10 m, using the average coniferous percentage may lead to inappropriate imputation of specie to the diameter class. If in the same cell two trees are present, one big broadleaved and a smaller coniferous, the coniferous proportion is 50%. The simulation software will consider that impacts on the biggest tree have an equal probability of being on a broadleaved than on a coniferous tree, and it is the same for the smallest one. This is different from considering that all impacts on the large tree are on a broadleaved tree and all impacts on the small one are on a coniferous tree. For a better integration of specie the coniferous proportion should be weighted by diameter. Besides in field inventories it is not more costly to count the specie-dependant diameter distribution than to measure diameters and count species independently.

This issue of the coniferous percentage is exemplified by the comparison of the Field_Tree and Field_Area scenarios in the mixed stand of Saint Agnan. Indeed, the energy distribution is quite similar but the proportion of passing blocks is overestimated by 14%. The major difference between those scenarios is that the average of the coniferous percentage over large pixels (40 m width) is used for the Field_Area scenario. Other differences are a slightly different diameter and spatial distribution, due the generation procedure. In order to test the effect of the coniferous percentage input data, the average coniferous proportion was used as input data with the original tree diameters and positions in an additional scenario. In that case, the energy results were closer to the Field_Area scenario, with a mean energy of 435 kJ and a proportion of passing block of 59%. This demonstrates that the issue of the coniferous percentage is critical for a better rockfall propagation assessment.

Position and diameter

Another error source is linked with the probability distribution used to sample the tree positions and diameters in the case of area-based inputs (Field_Area and ALS_Area scenarios).

Reconstructing the diameter probability distribution using a two-parameters model such as the Gamma distribution generally alters the shape of the distribution and fails to handle the case of a bimodal distribution. Besides, the interval of the Gamma distribution is from zero to plus infinity, so that small diameters can be generated below the DBH threshold used for the inventory and very large diameters may occur. Fig. 4 shows that for the coppice stand, the generated diameter distribution is smoothed compared to the real distribution, and that smaller and larger diameters are present.

The generation of the positions from a uniform probability distribution may also lead to different rockfall propagation estimations. The coppice data exhibit a clustered pattern that is not reproduced by the generation with uniform probability (Fig. 4). More simulations are required to test whether this aggregation pattern or the difference in the diameter distribution are responsible for the difference in the energy distribution of passing rocks.

In the Valdrôme stand, there is no aggregation but stem density is not spatially homogeneous inside the pixels considered in the Field_Area scenario. The density is higher near the upper border of the plot. This pattern is not accounted for in the generated positions (Fig. 5). Considering that the diameter distributions are similar, this difference in spatial heterogeneity might explain why the energy distribution is similar but with 40% more passing blocks in the case of the Field_Area scenario. Indeed in the Field_Tree case, there are higher chances that a sufficient number of consecutive tree impacts lead to a stop in the upper part. The integration of the spatial heterogeneity of tree positions is thus another important factor regarding the assessment of rockfall propagation.

Tree detection

With the tree detection in the ALS data, the spatial distribution of trees should be better modelled. Indeed in Valdrôme (Fig. 5) and Saint Agnan (data not shown), major canopy gaps appear clearly. However stem density is not correctly estimated as detection proportion depends on the vertical and horizontal arrangements of trees. In Valdrôme almost all trees are detected in sparse areas, but in denser areas the proportion decreases. Indeed, detection performance highly depends on stand structure and algorithms that automatically adjust the detection parameters have yet to be tested. As a result, spatial heterogeneity is only partially modelled as the proportion of undetected trees is both variable and unknown. The number of false detections is also an issue but it has to be tackled with the proportion of missed trees as a trade-off in parameter optimization.

The question of diameter estimation is also problematic as ALS heights are biased estimators of field heights. In our cases, the heights of the taller trees are over-estimated because trees are tilted downslope [START_REF] Hirata | Extracting forest patch attributes at the landscape level using new remote sensing techniques -an integrated approach of high-resolution satellite data, airborne lidar data and GIS data for forest conservation[END_REF]. Applying a diameter-height relationship calibrated with field measured heights thus leads to over-estimated diameters. It would be better to calibrate height estimation models directly with ALS-derived metrics (e.g. height, crown surface) in order to limit bias, but this requires tree-level field data. Speciesspecific relationships should also be considered but this requires that species is determined from ALS or other remote sensing data. The area-based method for the estimation of stand parameters from ALS point cloud statistics is theoretically unbiased, but the errors can be locally important. In Saint Agnan and Saint Paul, the ALS_Area method leads to an over-estimation of stem density, whereas the mean diameter is better estimated (Fig. 6). These errors then propagate into the rockfall propagation simulations. Results obtained with ALS_Area should be handled with care as at this scale errors can be locally important and pixels are spatially correlated.

CONCLUSION

This study at the stand level identifies the prerequisites for a better integration of forest inputs in rockfall simulations from field and ALS remote sensing data. It shows that relatively to the large difference between the reference scenarios with and without trees, the forest inputs obtained from ALS data or from field data without tree positions lead to good estimations of the mean and 95 th percentile of the energy distribution of passing blocks. However, the proportion of passing blocks displays larger errors. The study also points out some limitations due to the rockfall simulation software and to the methods used to generate the required input files from the field or ALS data.

Improvements could be made inside the software regarding the coniferous proportion. With the perspective to classify species at the tree level with remote sensing data it seems important to link this characteristic to each tree individually. In case of field inventory without tree positions, it seems important to weight the coniferous proportion with the diameter of trees. The diameter generation could also be refined by using a truncated distribution which would account for the minimum diameter of the inventory. A maximum diameter could also be imposed. Regarding the aggregation of trees in the case of coppice stands, methods developed to quantify it in field investigations could be integrated in the spatial generation procedure.

In the case of studies at larger scale, the field data do not account for the spatial heterogeneity of complex forests due to accessibility constraints and labour costs. The ALS area-based method provides stand parameters estimates at a resolution of around 20 m. The heterogeneity might only be partially rendered at local scale but over the whole area forest estimates are theoretically unbiased. With the ALS single tree detection, the heterogeneity is also only partially modelled due to algorithm and data limitations. The question of the propagation of local errors into rockfall simulation is open for both methods. However the dataset required to evaluate this will probably only be available from simulation. In the meantime, latest developments in the estimation of forest structure from airborne laser scanning and other remote sensing data should be integrated for a better characterization of species and dendrometrics.

Fig. 1

 1 Fig. 1 Simulation scheme

Fig. 2

 2 Fig. 2 Number of correctly detected trees (green), missed trees (red) and falsely detected trees (blue), in different height classes for the Saint Agnan (top) and Valdrôme (bottom) stands.

Fig. 3

 3 Fig. 3 Distribution of energies of passing blocks (blank bars) for the different input forest data scenario (columns), and for each forest stand (lines). Grey in the background correspond to the Field_Tree reference scenario.

Fig. 4

 4 Fig. 4 Coppice stand. Top: positions of field trees. Middle: generated positions. Point size is proportional to tree diameter. Bottom: reference diameter distribution (Field_Tree, grey) and generated distribution (Field_Area, blank bars).

Fig. 5

 5 Fig. 5 Inventoried tree positions (Field_Tree, top) generated positions (Field_Area, middle) and detected trees (ALS_Tree, bottom) in Valdrôme. Symbol size is proportional to tree diameter. Uphill side is on top of the image.

Fig 6

 6 Fig 6 Diameter distribution of the Field_Tree (grey), ALS_Area (top, black) and ALS_Tree (bottom, black) scenarios in Saint Agnan.

Table 1

 1 Description of forest field data

	Site	Saint Agnan Saint Paul	Valdrôme
	Structure	Uneven-aged Coppice	Plantation
	Dominant	Abies alba,	Acer opalus,	Pinus nigra
	species	Fagus	Corylus
		Sylvatica	avellanum,
			Sorbus aria,
			Quercus
			pubescens
	Size (m 2 )	80x120	50x50	50x200
	Basal area	32.5	31.2	45.5
	(m 2 .ha -1 )
	DBH (cm)	30.1±15.7	12.5±8.0	22.3±8.3
	(mean ± sd)
	Stem density	357	1800	1025
	(ha -1 )
	Table 2 Description of airborne laser scanning data
	Site	Saint Agnan and	Saint Paul
		Valdrôme
	Pulse frequency (kHz)	170	120
	Flight height (m)	600	600
	Footprint size (m)	0.3	0.3
	Flight year		2010	2009
	Gamma probability distribution. Stem density is
	respected in each pixel.
	2.2

.3 Tree-level ALS data (ALS_Tree scenario)

	ALS data were acquired over the three stands
	with a RIEGL LMS-Q560 scanner. Acquisitions
	settings are presented in Table 2. A tree detection
	algorithm based on local maximum filtering
	[Popescu & Wynne

Table 3

 3 Statistics of the ALS estimation models: adj-R 2 of linear regression, root mean square error and its coefficient of variation obtained in cross-validation.

	Block release line
	Bare ground
	area
	Forest patch
	Energy measure line

each plot. For each site, these data are used to calibrate regression models between local statistical descriptors of the ALS point clouds extracted inside the circular plots, and the corresponding forest stand parameters: mean DBH, DBH standard deviation and stem density [

Table 4

 4 Statistics on the errors of the ALS estimation models at the pixel level Site (Nb of pixels)

		Parameter	Bias	Standard
				deviation
	Saint Agnan	DBH (cm)	-1.7	4.6
	(24)	sd(DBH) (cm)	-1.8	2.4
		N (ha -1 )	190	120
	Saint Paul	DBH (cm)	-1.3	2.2
	(9)	sd(DBH) (cm)	-3.0	0.7
		N (ha -1 )	260	430

Table 5

 5 Statistics of the energy (kJ) of blocks passing through the forest stand. Mean, standard deviation, 95 th percentile and number of passing blocks expressed in percentage of the number of released blocks. The second value is the variation compared to the Field_Tree scenario, expressed in percentage.

	Sce-	Statistic	Saint	Saint Paul Valdrôme
	nario		Agnan
	Field	Mean	426	385	317
	Tree	Std. dev.	247	192	181
		95 th perc.	865	723	652
		Nb blocks	53%	84%	58%
	Field Area	Mean Std. dev. 95 th perc. % blocks	436 +2% 232 -6% 852 -2% 60% +13%	369 -4% 187 -3% 700 -3% 81% -4%	340 +7% 179 -1% 663 +2% 81% +40%
	ALS Tree	Mean Std. dev. 95 th perc. % blocks	459 +8% 259 +5% 905 +5% 63% +19%		368 +16% 180 -1% 692 +6% 80% +38%
	ALS Area	Mean Std. dev. 95 th perc. % blocks	377 -12% 229 -7% 797 -8% 46% -13%	388 +1% 174 -9% 698 -3% 97% +15%
	No Fo-rest	Mean Std. dev. 95 th perc. % blocks	641 +50% 205 -17% 1005 +16% 100% +88%	529 +37% 167 -13% 825 +14% 100% +19%	521 +64% 164 -9% 812 +25% 100% +72%

Monnet, J.M. ;Bourrier, F. (2014) Evaluation of the effect of forest input data on rockfall simulations.INTERPRAEVENT 2014, International Symposium on Natural Disaster Mitigation to Establish Society With Resilience, Japan, Nara, 25-28 November 2014

ACKNOWLEDGMENT: This work was funded by the European Commission (project Alpine Space 2-3-2-FR NEWFOR) and by the French Ministry of Defense (program RAPID, project ModTer). We thank Éric Mermin , Pascal Tardif and David Toe for field data collection.

 Vol. 70,No. 5,


Marta Aido1

Michael Kerschnitzki2

Rebecca Hoerth2

Manfred Burghammer4

Ce Dric Montero4

Sara Checa1

Peter Fratzl2

Georg N Duda1

Bettina M Willie1

Wolfgang Wagermaier3

Wolfgang Wagermaier
email: wolfgang.wagermaier@mpikg.mpg.de

Connective tissue research

Keywords: Bone mineral, calcein, mice, synchrotron sSAXS

Bone's mineral properties, such as particle thickness and degree of alignment have been associated with bone quality. Bone formation, remodeling, aging of the tissue and mineral homeostasis influence mineral particle properties leading to specific patterns across bone. Scanning small angle X-ray scattering (sSAXS) with synchrotron radiation is a powerful tool, which allows us to study bone's nanoscale mineral properties in a position-resolved way. We used sSAXS, fluorescence light microscopy and backscattered electron (BSE) imaging to study bone's mineral properties at the tibial midshaft of in vivo-loaded mice. By combining these techniques, we could detect local changes in mineral properties. Regions labeled with calcein fluorochrome have lower mean mineral thickness and degree of mineral alignment. We also observed thinner and less aligned mineral particles near blood vessels. We conclude that mineral properties (i) are altered by fluorochrome labeling and (ii) depend on the proximity to blood vessels.

Introduction

Bone has a hierarchical structure defined at the nanometer-scale as a composite material constituted by a collagen-rich organic matrix in which mineral crystals with a thickness of only a few nanometers are embedded [START_REF] Weiner | The material bone: structure mechanical function relations[END_REF][START_REF] Fratzl | Structure and mechanical quality of the collagen-mineral nano-composite in bone[END_REF]. The mineral phase is known to act as an ion reservoir and to confer stiffness and hardness to the bone (3). Bone's material quality is closely associated with the mineral crystal properties, such as size and arrangement [START_REF] Fratzl | Structure and mechanical quality of the collagen-mineral nano-composite in bone[END_REF]. Some studies indicate that mineral crystal size increases with aging [START_REF] Gourion-Arsiquaud | Spatial variation in osteonal bone properties relative to tissue and animal age[END_REF]. Newly formed tissue can be identified based on fluorochrome labeling, since these agents selectively bind to sites of new mineralized tissue formation, and are incorporated into the mineralization front. Although this technique has been routinely used for decades, it remains unclear why this selective binding occurs, although some studies suggest that this is due to the smaller crystal size formed in the initial phase of mineralization, in comparison with older mineralizing sites [START_REF] Erben | Bone-labeling techniques[END_REF].

We previously studied the bone's adaptive response to in vivo tibial loading in female mice [START_REF] Willie | Diminished response to in vivo mechanical loading in trabecular and not cortical bone in adulthood of female C57Bl/6 mice coincides with a reduction in deformation to load[END_REF]. These mice were given calcein to label new tissue formed in response to loading and growth. Our aim in the present study was to investigate mineral particle properties of this new tissue by using various high-resolution methods. Synchrotron sSAXS (1mm monochromatic X-ray beam) was combined with fluorescence and BSE imaging, to analyze mean mineral thickness and degree of alignment.

Methods

In vivo loading and sample preparation

Female 10, 26 and 78 week old C57Bl/6J mice underwent in vivo cyclic compressive loading of the left tibia (1200με) as previously described [START_REF] Willie | Diminished response to in vivo mechanical loading in trabecular and not cortical bone in adulthood of female C57Bl/6 mice coincides with a reduction in deformation to load[END_REF], their right tibia was an internal control (n=2/age). Calcein was given via intraperitoneal injection, 12 and 3 d. before euthanasia. Both tibiae were fixed in 100% ethanol, embedded in polymethyl methacrylate and sectioned using a microtome (Leica SM2500S; knife: Leica VMH 400, Nussloch, Germany). One, 10μm thick longitudinal section from each tibia was analyzed using BSE and sSAXS. Animal experiments were approved by the local legal animal welfare representative (LAGeSo Berlin, G0333/09).

sSAXS data acquisition, correction and analysis

Prior to sSAXS measurements, samples were imaged with fluorescence microscopy (Leica DMRB, Munich, Germany) to identify calcein labeled regions, which enabled us to distinguish between newly formed and mature bone (Figure 1a). A region of the tibial midshaft (125-300 μm x 30-77.5 μm) was examined by sSAXS at the Nanofocus beamline (ID 13) of the European Synchrotron Radiation Facility (ESRF). A monochromatic high energy X-ray beam with a wavelength of 0.0812nm (E=15keV) and a diameter of 1mm was used to scan samples, with a step size of 1μm within the plane perpendicular to the incident beam. The sSAXS patterns were captured with an ESRF FReLoN detector (pixel size: 51.7 μm x 51.5 μm, exposure time: 0.6 to 0.8 s). Up to 28 700 sSAXS intensity patterns were collected per sample. All 2D sSAXS patterns were corrected for dark current (CCD readout noise) and for incoming beam intensity fluctuations from ESRF source. 2D sSAXS patterns were radially and azimuthally integrated to obtain I(q) and I(Χ), respectively with custom-made software (Autofit, MPI-CI, Potsdam, Germany), with I being the scattering intensity, q the scattering length and Χ the azimuthal angle. From the resulting sSAXS intensity function I(q) and based on Porod's law, the T parameter, the average mineral particle thickness in the scattering volume (Figure 1c), was derived [START_REF] Rinnerthaler | Scanning small angle X-ray scattering analysis of human bone sections[END_REF]. The function I(Χ) was used to calculate ρ parameter, the degree of mineral particle alignment within the plane perpendicular to the primary beam. This parameter varies between 0 and 1 (7) (Figure 1b).

BSE imaging

After sSAXS measurements and to visualize the degree of mineralization (mineral content) within the bone, the entire longitudinal section of each tibia was imaged with an environmental scanning electron microscope (FEI FEG-ESEM Quanta 600, Hilsboro, OR) in BSE mode (2) under low vacuum (0.8 Torr). An operating voltage of 12.5 kV and working distance of 10 mm was used. Images collected using these multiple methods were overlaid onto each other to correlate the degree of mineralization and calcein label location (Figure 1a) with sSAXS maps of mean mineral thickness (Figure 2e) and degree of mineral alignment (Figure 2f) from the same regions.

Results

Areas labeled with calcein had lower T and o parameters

Calcein-labeled regions within the bone corresponded to regions of lower mean mineral thickness and degree of mineral alignment (Figure 2a-d). When plotting changes in T and r parameters along the first 30 μm from the bone surface, we detected a peak of low T and r at approximately 30, 15 and 8 μm in tibia from 10, 26 (Figure 2a andb) and 78 week old (Figure 2c andd) mice, respectively, which corresponded to the calcein label location. Correlating T and r maps with fluorescence microscopy images also indicated that this pattern corresponded to the location of the calcein label itself. One tibia from a 10 week old mouse with a diffuse calcein label did not adhere to this same pattern.

Mineral particles around blood vessels have lower T and ρ parameters

We observed that the regions surrounding blood vessels corresponded to areas with lower mean mineral particle thickness (Figure 2e) and lower degree of alignment of mineral particles (Figure 2f). This pattern could be observed around all the scanned blood vessels and for mice of all ages: 10 (3 blood vessels), 26 (1 blood vessel) and 78 (2 blood vessels) week old mice.

Discussion

Our results showed lower mean mineral thickness and degree of mineral particle alignment where calcein fluorochrome labels were located. A study in rats labeled with different fluorochromes also showed a decrease in the degree of mineral alignment, but not in mean mineral thickness on the border between new and old bone [START_REF] Bu ̈nger | Strontium and bone nanostructure in normal and ovariectomized rats investigated by scanning small-angle X-ray scattering[END_REF]. However, an in vitro study showed a direct relationship between alizarin red concentration and apatite like crystal size and growth kinetics [START_REF] Ibsen | Modification of bone-like apatite nanopar-ticle size and growth kinetics by alizarin red S[END_REF]. Our results indicate that the calcein might not only bind to mineralizing surfaces, but also interfere with mineral growth. The calcein may act in a similar manner as noncollagenous proteins in governing mineral growth. Interestingly, BSE imaging on selected samples did not show different mineral content at the calcein label sites (Figure 1a), but this would need to be studied further using quantitative methods such as quantitative backscattered electron imaging (qBEI) [START_REF] Fratzl | Structure and mechanical quality of the collagen-mineral nano-composite in bone[END_REF]. We observed that the regions surrounding blood vessels have thinner and less aligned mineral particles. In contrast to our findings at the calcein label location, the presence of thinner and less aligned mineral particles near the blood vessels was likely mainly related to the age of this tissue rather than alterations in crystal growth.

Due to the high resolution and time restrictions during synchrotron measurements, our sample size was limited and thus further studies examining a larger number of mice are needed to confirm these results. Also, studies examining human bone biopsies have shown different fluorochromes have different binding affinities, so other fluorochromes should be studied, including tetracycline which is used clinically [START_REF] Lindsay | Double and quadruple tetracycline labeling of bone: impact of the label itself[END_REF].

Conclusion

Using a multi-method approach, we have shown that the mean mineral particle thickness and degree of alignment were lower in the areas where particles were bound with calcein labeling. These findings suggest that fluorochrome labeling might not only bind to mineralizing surfaces, but also alters mineral properties, including stunting their growth, and thus affect bone mineral quality. We also showed that thinner and less aligned particles were present near blood vessels, indicating this tissue is younger.

Figure 1 .

 1 Figure 1. (a) Merged BSE and fluorescence image of tibial midshaft; (b) ρ parameter varies between 0 (randomly oriented mineral particles) and 1 (perfectly aligned mineral particles); (c) T parameter corresponds to mean mineral thickness.

Figure 2 .

 2 Figure 2. Graph of T parameter profile 30 μm through cortex starting at periosteal border in a loaded tibiae of a (a) 26 week old and (c) 78 week old mouse, the red line in the graphs corresponds to the average T parameter, while the shadowed pink area above and below the red line represents the standard deviation; (b) (d) T parameter map of first 30 μm of the tibial section; Lower T (e) and ρ parameter (f) near blood vessels in control tibia of a 10 week old mouse. BV, blood vessel.

Acknowledgments

The authors thank Michael Reynolds, Marzena Princ, Tobias Thiele and Annette Birkhold for assistance.

Declaration of interest

This study was funded by the German Federal Ministry (BMBF; Osteopath grant/TP6), German Research Foundation (DFG; WI 3761/1-1 and FR 2190/4-1 Gottfried Wilhelm Leibniz-Preis 2010) and the ESRF. All authors have no conflicts of interest.