Rafael B Nuernberg

Juan F Basbus

Kevin C Lux

Miguel P Sainz

Gabriel J Cuello

Ana C M Rodrigues

Andrea A Piarristeguy

Michel Ribes

Annie Pradel

Correlation between structural features and ionic transport in lithium ionconducting glass-ceramics from the Li1+xCrxGeTi1-x(PO4)3 system

Keywords: Li1+xCrxGeTi1-x(PO4)3 system, NASICON-like structure, crystallographic parameters, Li diffusion pathway, ionic conductivity

NASICON-structured glass-ceramics based on the Li1+xCrxGeTi1-x (PO4)3 (0≤x≤1) system are synthesized by the melt-quenching method followed by glass crystallization. Since Ti +4 and Cr +3 have approximately the same crystal radius in octahedral coordination, this aliovalent substitution should avoid considerable changes in the cell volume, thus allowing to evaluate the true contribution of the substitution of a tetravalent cation by a trivalent one on other structural factors. The crystal structure of these glass-ceramics is investigated by Xray and high-resolution neutron diffraction while the electrical properties are accessed by impedance spectroscopy. Fourier differences and bond valence energy landscape analyses are used to determine the additional Li position and occupancies caused by the aliovalent substitution of Ti +4 by Cr +3 . The most important change in the structural features caused by the increase in the lithium content in the chemical formula is the occupation of the 36f position followed by the partial depopulation of the 6b sites. Furthermore, the grain-related ionic conductivity and activation energy for lithium conduction are notably dependent of the aliovalent substitution. The correlations between these two main findings are discussed herein.

Introduction

The unique combination of high energy and power density makes lithium-ion batteries the most attractive technology for several applications, such as portable electronics, power tools, electric vehicles, and large-scale energy production for intermittent power generation. The development of lithium-ion batteries has boosted studies of solid-state batteries because the non-flammability of their solid electrolytes offers a fundamental solution to safety concerns.

In this context, ion conducting solids are materials of increasing interest due to their possible use as solid electrolytes in high-energy solid-state batteries. Since these materials are generally conductors of a single ion species, only lithium ions have significant mobility while other ions form a rigid three-dimensional network, resulting in greater electrochemical and thermal stability. Among various types of solid-state lithium-ion conductors, the ceramics with NASICON-type structure have attracted the interest of researchers due to their chemical stability in air and moisture, as well as suitable ionic conductivity [START_REF] Nitta | Li-Ion Battery Materials: Present and Future[END_REF][START_REF] Knauth | Inorganic Solid Li Ion Conductors: An Overview[END_REF][START_REF] Manthiram | Lithium Battery Chemistries Enabled by Solid-State Electrolytes[END_REF][START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF] .

The base chemical formula of these compounds can be written as LiM2(PO4)3, where M is a tetravalent cation (M +4 = Ge, Ti, Zr, Sn or Hf). The symmetry of these materials is rhombohedral for Ge and/or Ti as tetravalent cations. However, when ionic radii of tetravalent cations become bigger, such as Sn, Hf and Zr, a triclinic distortion is produced to better accommodate Li ions. In both cases, the structure can be described as a covalent skeleton containing MO6 octahedra linked by corners to PO4 tetrahedra and interstitial positions in which the mobile cations are distributed [START_REF] Pérez-Estébanez | A Systematic Study of Nasicon-Type Li1 + XMxTi 2 -X(PO4)3 (M: Cr, Al, Fe) by Neutron Diffraction and Impedance Spectroscopy[END_REF][START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][START_REF] Arbi | NMR and Electric Impedance Study of Lithium Mobility in Fast Ion Conductors LiTi2 -XZrx(PO4)3 (0 ≤ x ≤ 2)[END_REF][START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . As reported in literature, Li + ions may occupy different interstitial positions within this network. In the case of rhombohedral symmetry, the R3 ̅ c space group (SG), the Wyckoff positions (WP) of Li correspond to the M1 site 6b (0, 0, 0) [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][9][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF][START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF] , the M2 sites 18e (0.645, 0, 1/4) [START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF][START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF] and the M3 sites 36f (0.7, 0.34, 0.7) [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][9][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF] . These sites alternate along the channels forming three-dimensional conduction pathways where Li + ions might move jumping from one site to another (i.e., Li hopping mechanism). The energy landscape of the conduction channels depends on the skeleton framework, which in turn, is governed by the size and electronegativity of the M atoms in the MO6 octahedra [START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF]9,[START_REF] Kang | Integrated Study of First Principles Calculations and Experimental Measurements for Li-Ionic Conductivity in Al-Doped Solid-State LiGe2(PO4)3 Electrolyte[END_REF] .

Among the above-mentioned tetravalent cations, Ti +4 and Ge +4 tend to lead to the basic NASICON-like compound with the highest lithium conductivity and smallest activation energy [START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF][START_REF] Arbi | NMR and Electric Impedance Study of Lithium Mobility in Fast Ion Conductors LiTi2 -XZrx(PO4)3 (0 ≤ x ≤ 2)[END_REF] . Additionally, the aliovalent substitution of the tetravalent M cation by a trivalent one (A +3 = Al, Ga, In, Sc, Y, La, Cr or Fe) gives rise to the general formula Li1+xAxM2-x(PO4)3, leading to an increase of ionic conductivity due to additional Li + charge carriers [START_REF] Knauth | Inorganic Solid Li Ion Conductors: An Overview[END_REF][START_REF] Pérez-Estébanez | A Systematic Study of Nasicon-Type Li1 + XMxTi 2 -X(PO4)3 (M: Cr, Al, Fe) by Neutron Diffraction and Impedance Spectroscopy[END_REF][START_REF] Nuernberg | A New NASICON Lithium Ion-Conducting Glass-Ceramic of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . Among the aliovalent-substituted series the Li1+xAlxTi2-x(PO4)3 (LATP) [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Narváez-Semanate | Microstructure and Ionic Conductivity of Li1+xAlxTi2-x(PO4)3 NASICON Glass-Ceramics[END_REF] and Li1+xAlxGe2-x(PO4)3 (LAGP) [START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF][START_REF] Cruz | Controlled Crystallization and Ionic Conductivity of a Nanostructured LiAlGePO4 Glass-Ceramic[END_REF][START_REF] Kuo | Lithium Ion Diffusion Mechanism and Associated Defect Behaviors in Crystalline Li1+ XAlxGe2-x(PO4)3 Solid-State Electrolytes[END_REF] have been the most studied systems. However, some studies have suggested that the enhancement of ionic conductivity in this system is also due to changes in the energy barrier for lithium ion conduction and not only to the increase of Li + concentration in the NASICON-type phase [START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF][START_REF] Kang | Integrated Study of First Principles Calculations and Experimental Measurements for Li-Ionic Conductivity in Al-Doped Solid-State LiGe2(PO4)3 Electrolyte[END_REF] . Moreover, in some cases it is difficult to attribute the real contribution of the aliovalent substitution since this strategy usually results, besides the increase in the concentration of charge carriers, also in important structural changes such as the unit cell volume [START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF] . Recent Neutron Powder Diffraction (NPD) results in LATP compounds have started showing interesting information about the correlation between electrical properties and structural factors, such as Li occupancy and the energy landscape of the structure 9,10 . Based on these ideas, the purpose of the present work is to perform a structural investigation of several compositions of a new NASICON system Li1+xCrx(GeyTi1-y)2-x (PO4)3, which has recently been proposed by us [START_REF] Nuernberg | A New NASICON Lithium Ion-Conducting Glass-Ceramic of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . This system is compatible with the glass-ceramic route synthesis as long as sufficient amounts (around 50% mol.) of oxide glass formers such as germanium and phosphorus oxides are present [START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . Thus, the general Li1+xCrxGeTi1-x(PO4)3 formula was chosen where only Ti is replaced by Cr, instead of replacing Ti and Ge. Moreover, since Ti +4 and Cr +3 have approximately the same crystal radius in octahedral coordination, this strategy should avoid considerable changes in the cell volume, thus allowing to evaluate the true contribution of the substitution of a tetravalent cation by a trivalent one on other structural factors. Finally, the electrical properties of these electrolytes are accessed by electrochemical impedance spectroscopy (EIS) and the ionic transport is correlated with the structural investigation.

Experimental Methods

Preparation of Glass and Glass-Ceramics

Seven glass-ceramic compositions were formulated according to the Li1+xCrxGeTi1-x(PO4)3

(LCGTP) general formula with x = 0, 0.2, 0.4, 0.5, 0.6, 0.8 and 1. to relieve thermal stresses (Tg > 600 o C) [START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . After cooling, the chromium containing glasses exhibited a dark greenish color, which increased in intensity with the increment of Cr content.

All the LCGTP glasses were heat-treated as bulk samples at 900 o C for 12 hours to obtain fully crystallized glass-ceramics. The heat treatment temperature was defined based on our previous work, in which we investigated the influence of heat treatment temperature on ionic conductivity [START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . After crystallization, LCGTP samples became opaque with a greenish color less intense than that of the precursor glass.

Structural characterization

The phase formation and purity of powder samples from all the heat-treated glass-ceramics were evaluated by X-Ray Diffraction (XRD). The patterns were recorded at room temperature in air using a Panalytical X'Pert MPD Pro equipment with a Bragg Brentano geometry, Cu Kα radiation, Ni filter, and X'Celerator detector operated at 45 kV and 30 mA.

The angular 2θ range of 10-120 o was continuously scanned during 6 hours with a minimum 2θ step size of 0.001 o . Moreover, the crystallographic parameters of the glass-ceramics were investigated by neutron powder diffraction (NPD) [START_REF] Piarristeguy | Structural Investigation of LCGTP Glass-Ceramics[END_REF] . NPD patterns were collected in air at room temperature at the high-resolution D2B diffractometer (wavelength of 1.594 Å) available at the Institut Laue-Langevin (ILL) [START_REF] Hewat | A New High Resolution Neutron Powder Diffractormeter at Ill Grenboble[END_REF] . The powder samples were loaded in vanadium cans used as sample holders in the beam. In addition, the neutron beam wavelength and the instrumental profile were calibrated using standard powders. NPD data were previously pre-treated with LAMP software [START_REF] Richard | Analysis and Visualisation of Neutron-Scattering Data[END_REF] and then analyzed using the Rietveld method and Fullprof software [START_REF] Pérez-Estébanez | A Systematic Study of Nasicon-Type Li1 + XMxTi 2 -X(PO4)3 (M: Cr, Al, Fe) by Neutron Diffraction and Impedance Spectroscopy[END_REF][START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Rodríguez-Carvajal | Recent Advances in Magnetic Structure Determination by Neutron Powder Diffraction[END_REF] . The rhombohedral NASICON-type R3 ̅ c (nº 167) was used as a common structural model for the samples and crystallographic parameters, such as lattice parameters, atomic positions, occupancies, and atomic displacements were refined. Besides, the crystallite size and isotropic microstrain were also refined. The Thompson-Cox-Hastings pseudo-Voigt profile convoluted with an axial divergence asymmetry function was used as a line shape for the diffraction peaks [START_REF] Finger | A Correction for Powder Diffraction Peak Asymmetry Due to Axial Divergence[END_REF] . The NPD backgrounds were determined by a linear interpolation of selected points. Initially, NPD data refinements were carried out for the basic structural framework consisting of Cr/Ge/Ti, P, and O atoms. Then, Fourier maps of the difference between the real structure and a structural model without Li were used to determine the Wyckoff positions (WPs) of Li atoms which were missing in the starting model. Afterward, Li atoms were included in the final model and the occupancies were determined using further refinements. For a matter of simplicity, the absolute Li content, considering the occupancies of both sites, is constrained to the nominal lithium content (1+x).

Bond valence energy landscapes (BVEL) were generated adopting the Bond_Str program (FullProf package) to infer lithium diffusion pathways and confirm WPs of Li [START_REF] Adams | From Bond Valence Maps to Energy Landscapes for Mobile Ions in Ion-Conducting Solids[END_REF][START_REF] Katcho | An Investigation of the Structural Properties of Li and Na Fast Ion Conductors Using High-Throughput Bond-Valence Calculations and Machine Learning[END_REF] . The BVEL plots of LCGTP compounds were calculated from crystallographic parameters refined for the NPD patterns using a differential bond valence mismatch ΔV = 1 eV and an ionic neighbor up to 10 Å. Both difference Fourier maps and BVEL maps were visualized using the VESTA software [START_REF] Momma | VESTA 3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data[END_REF] .

Electrical Characterization

The electrical conductivities of the glass-ceramics were assessed by electrochemical impedance spectroscopy (EIS), using a Solartron 1260 Impedance/Gain Phase Analyzer in the frequency range between 1 MHz and 100 mHz, with an applied root mean square voltage of 500 mV. EIS measurements are performed in the temperature range of -73 o C -27 o C to resolve the grain conductivity, which is difficult to access at higher temperatures due to the high conductive nature of these glass-ceramics. The sub-ambient temperatures were accessed through a closed-loop cooling system (Janis Research CCS-400H/204) coupled to a LakeShore 331 temperature controller. EIS measurements were performed on flat polished samples with parallel faces and gold electrodes sputtered on both sides to ensure electrical contact. Samples were about 0.1 cm thick and had a contact area of 0.1 to 0.2 cm 2 . The spectra were fitted using ZView 3.5f impedance spectroscopy software by using a suitable equivalent circuit. These shifts indicate smaller interplanar distances for the LCGTP glass-ceramics compared to the LiTi2(PO4)3 compound. The shrinkage of the LCGTP cell parameters must be related to the germanium content as Ge +4 has a smaller crystal radius (0.670 Å) than Ti +4 (0.745 Å) [START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF][START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF][START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] . On the other hand, no considerable change is observed in the position of the peaks when comparing XRD patterns of LCGTP glass-ceramics among them, showing evidence that the replacement of Ti +4 by Cr +3 causes minimum changes in the structure, as one might have anticipated. This occurs because Ti +4 and Cr +3 have roughly the same crystal radii in octahedral coordination (0.755 Å for Cr +3 and 0.745 Å for Ti +4 ) [START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF][START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF][START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] .

Results and Discussion

The structure refinements of Li1+xCrxGeTi1-x(PO4)3 glass-ceramics (0≤x≤0.5) are performed based on neutron powder diffraction (NDP) analyses data, as the localization of Li atoms using the X-ray diffraction technique is not feasible due to their low X-ray scattering factor.

Initially, only the atoms comprising the structure framework were considered: (i) Cr/Ge/Ti in 12c (0, 0, 0. [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF]9,[START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF] . After the refinement of the structural model with the atoms that make up the skeleton, difference Fourier maps were generated considering the experimental diffraction pattern and the calculated ones based on the structure framework. The negative scattering length of Li atoms considerably increases the contrast with Ge/Cr, P and O elements, resulting in negative densities in Fourier maps [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF]9,[START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF] . Figure 1 shows the structural model used in the refinement and the negative isosurface of the remaining scattering length densities for LiGeTi(PO4)3 composition (LGTP). Based on these analyses, the most common Li site reported in these compounds become easily identified at the position labeled M1 with fractional coordinates (0,0,0) and Wyckoff symbol 6b 6,9-12 . The other two Li positions usually reported in literature are the M2 site at 18e (0.64525, 0, 0.25) [START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF][START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF] and the M3 site at 36f (0.07, 0.34, 0.07) 6,9-11 , but they are generally found only for doped compounds. The M2 site is located at the middle of two M sites, while the M3 sites lie on either side of the M2 18e positions [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF] . Both M1 and M2 sites are cut by the lattice plane (0 1 -4), where 3D lithium diffusion pathways have been found to extend throughout the structure as zig-zag channels passing by M1 6b and M2 18e but also by M3 36f positions 9 . Figure 2 depicts difference Fourier maps of remaining neutron scattering negative density at the 2D section (0 1 -4) corresponding to the LGTP and LCGTP04 compounds. The most intense negative density peaks at this lattice plane are located around the positions 6b (0, 0, 0) for the LGTP sample (Figure 2a) and besides 6b also 36f (0.06, 0.39, 0.09) for the LCGTP04 composition (Figure 2b). Therefore, the M3 36f sites seems to be favored over their adjacent M2 18e sites as already reported in literature for other aliovalent substitutions [START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF]9,[START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF] . The difference Fourier maps for the samples LCGTP02 and LCGTP05 have similar features to those for the LCGTP04 sample. On the contrary, the LGTP sample shows practically no negative density in the 36f position. Indeed, the further refinement of the neutron pattern of this sample (Figure S2a), constraining the nominal lithium content for both 6b and 36f positions, gives near zero occupancy in the 36f position. In contrast, when the same procedure is done for the sample LCGTP04 (Figure S2b), the occupancy of the 36f position increases at the expense of the 6b occupancy, resulting in a higher absolute Li content at the 36f position than at the 6b position (see Table 2). Figure 3 presents the dependence of cell parameter variations and Li occupancies on x in the LCGTP compositional system. For 0 ≤ x ≤ 0.5, the cell parameters a = b oscillate from 8.37 to 8.41 Å and the cell parameter c from 20.68 to 20.83 Å, resulting in volumetric changes inferior to 0.5 % (Figure 3a). Comparing these structural variations with the ones of about 5 % (x = 0.5) for the aliovalent substitution of Ge +4 by Cr +3 in the Li1+xCrxGe2-x(PO4)3 system [START_REF] Xu | Preparation and Characterization of Lithium Ion-Conducting Glass-Ceramics in the Li 1 + XCr XGe 2 -x(PO 4) 3 System[END_REF] , where the crystal radii in octahedral coordination are quite distinct (0.755 Å for Cr +3 and 0.670 Å for Ge +4 ), one can indeed state that the replacement of Ti +4 by Cr +3 is not associated to substantial volumetric changes of the unity cell. On the other hand, the Li occupancy increases in the 36f position (Li2 site) at the expense of the occupation of the 6b position (Li1 site) (Figure 3b). For the LGTP compound (x = 0), the error in the Li2 occupancy is higher than the occupancy itself. Therefore, Li1 occupancy can be assumed as full and Li2 as negligible, which is in good agreement with the generated Fourier map of this compound (Figure 2a). Since the substitution of Ti by Cr is accompanied by an increment of lithium content and the 6b position is already fulfilled for the LGTP compound, the additional lithium should indeed be accommodated in the 36f position. Notwithstanding, the increase in the lithium content not only causes an occupation of the 36f position but also triggers a partial depopulation of the 6b sites. This behavior has been already spotted in the LAGP compositional system [START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF] . Nevertheless, in the present case, the changes in the occupancy are basically the only crucial structural change if we consider that the slight changes of volume (< 0.5 %) could be neglected. Besides, an alternative approach is used for predicting lithium positions, as well as the preferable lithium pathways. The method assumes that the diffusion in solids might occur on a pathway where deviations between bond valence and nominal valence remain as low as possible. Therefore, BVEL were generated using the experimentally determined parameters from NPD refinement 9,28 . Figure 4 displays the complete structural model of the LGTP glassceramic based on R3 ̅ c SG, as well as BVEL isosurfaces connected in a 3D percolating network. The BVEL isosurface of 1 eV above the minimum energy predicts 3D diffusion pathways across the structure, which pass through the Li atomic positions 6b and 36f revealed in the Fourier maps analyses. This result corroborates Li occupancy in the 6b and 36f position, although the 18e position that is located between two 36f sites is also percolated by the 3D diffusion pathway 9,10 . This is consistent with results reported for LATP compounds where a very similar percolation network passing by the 6b position and the 36f position were also found 9 . In general, the BVEL maps are quite similar for all LCGTP glass-ceramics. Figure 5 depicts the absolute and the relative negative BVEL maps at the lattice plane (0 1 -4) calculated based on the experimentally determined structural parameters for LiGeTi(PO4)3 and Li1.4Cr0.4GeTi0.6(PO4)3 glass-ceramic compositions. As one might observe, absolute negative BVEL maps of the glass ceramic samples LGTP (Figure 5a) and LCGTP04 (Figure 5b) differ considerably in the minimum energy (-4.6 eV for LGTP and -7.5 eV for LCGTP04), but the minimum is always located at the 6b site. The other two glass ceramics, LCGTP02 and LCGTP05 present minimum energies of -6 and -9.2 eV, respectively, characterizing a continuous decrease of the minimum energy with the chromium content. Nevertheless, if the BVEL maps are normalized by each respective minimum energy, the differences in the BVEL maps of the glass ceramic samples LGTP (Figure 5c) and LCGTP04 (Figure 5d) practically disappear. The same is true for any other comparison with the BVEL maps of the other two samples (LCGTP02 and LCGTP05) (not shown). Therefore, no significant difference in the energy landscapes is remarked between BVEL maps of LCGTP glass ceramics. These results indicate that the substitution of Ti +4 by Cr +3 does not significantly alter the energy landscape for Li ions. The 6b and 36f sites indicated in the maps create a zig-zag trajectory through the structure.

Electrical Characterization

EIS analyses of the investigated glass-ceramics reveal a typical ion-conducting behavior, characterized by a straight and steep increase of the imaginary part of impedance (spike) at low frequencies, jointly to the impedance response of the sample from medium-to highfrequency ranges (Figure S3). This low-frequency response is commonly denoted as a polarization effect and is due to the blockage of mobile ions at the interface electrolyte/metallic electrodes [START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF][START_REF] Nuernberg | A New NASICON Lithium Ion-Conducting Glass-Ceramic of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . This feature in the impedance response indicates that the ions are the predominant charge carriers in this glass-ceramic. Moreover, reported measurements applying Hebb-Wagner method to estimate the electronic conductivity on Cr and Al NASICON-substituted compounds reveal that the electronic contribution is between five to six orders of magnitude lower than their respective ionic conductivity [START_REF] Zhang | High Lithium Ion Conductivity Solid Electrolyte of Chromium and Aluminum Co-Doped NASICON-Type LiTi2(PO4)3[END_REF] . The impedance response of the LGTP and LCGTP samples show a depressed and distorted semicircle that suggests more than one contribution at high and medium frequencies. The grain and grain boundary contributions to the ionic conductivity were determined based on its impedance response using a suitable equivalent circuit for fitting their respective resistance and capacitances. The best fit was obtained with three R|CPE contributions in series, besides the electrode contribution (CPE(p)), one for the grain (R(g)|CPE(g)) and two others are ascribed to different contributions of the grain boundaries (R(gb1)|CPE(gb1) and R(gb2)|CPE(gb2)). The origin of the two different contributions related to grain boundaries is unknown but a residual glassy phase is possibly present in the glass-ceramics [START_REF] Nuernberg | A New NASICON Lithium Ion-Conducting Glass-Ceramic of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF] . Nevertheless, only the grain-related contribution of ionic conductivity is considered in this study because this contribution is related with structural parameters while the grain boundary-related contribution is mostly linked with microstructural aspects [START_REF] Nuernberg | A New NASICON Lithium Ion-Conducting Glass-Ceramic of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF][START_REF] Mariappan | Correlation between Micro-Structural Properties and Ionic Conductivity of Li1.5Al0.5Ge1.5(PO4)3 Ceramics[END_REF] . The dependence of grain-related contribution of ionic conductivity on the inverse of temperature has been plotted following the Arrhenius relation expressed in Equation 1, where kB is the Boltzmann constant, T is the absolute temperature, A is the pre-exponential factor and Eag is the grain contribution-related activation energy [START_REF] Nuernberg | Numerical Comparison of Usual Arrhenius-Type Equations for Modeling Ionic Transport in Solids[END_REF] .

𝜎𝑇 = 𝐴 exp (

-𝐸a g 𝑘 B 𝑇 )

Figure 6 shows Arrhenius-type plots for the grain-related ionic conductivity of LCGTP glassceramics and their associated activation energy as a function of the x content. The impact of Ti replacement by Cr in the grain-related ionic conductivity is evident at the first glimpse (Figure 6a). Samples containing chromium present ionic conductivity between two or three orders of magnitude higher (depending on the temperature) than the LGTP glass-ceramic. At 300K for instance, the undoped LGTP glass-ceramic presents a grain-related ionic conductivity of 2.7 x 10 -6 Ω -1 cm -1 , while LCGTP02, LCGTP04 and LCGTP05 glassceramics show grain-related ionic conductivities of 4.2 x 10 -4 Ω -1 cm -1 , 5.7 x 10 -4 Ω -1 cm -1 and 8.5 x 10 -4 Ω -1 cm -1 , respectively. The dependence of their ionic conductivity on temperature is also markedly different, which results in a higher value of activation energy for the LGTP sample (Figure 6b). The activation energy drops considerably with the x content for low values of x, but slightly increases when comparing sample LCGTP04 to LCGTP05.

Correlation between structural parameters and ionic transport

The correlation between structural parameters and ionic transport for compounds with NASICON-like structure has been a topic of debate in the literature [START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF][START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF] . One of the first correlations ever proposed links the activation energy with the unity cell volume of NASICON structure. For simple LiM2(PO4)3 NASICON compounds (M +4 = Ge, Ti, Zr, Sn or Hf), it has been shown that there is an optimal volume, normally achieved when Ti +4 is the octahedral cation, leading to a minimization of the activation energy for ionic conduction [START_REF] Takada | Progress and Prospective of Solid-State Lithium Batteries[END_REF][START_REF] Arbi | NMR and Electric Impedance Study of Lithium Mobility in Fast Ion Conductors LiTi2 -XZrx(PO4)3 (0 ≤ x ≤ 2)[END_REF] . However, when the aliovalent substitution is used, other structural parameters, such as lithium occupancy in additional sites, also play a crucial role in decreasing the activation energy. In this context, the LAGP and LATP are the most studied systems [START_REF] Pérez-Estébanez | A Systematic Study of Nasicon-Type Li1 + XMxTi 2 -X(PO4)3 (M: Cr, Al, Fe) by Neutron Diffraction and Impedance Spectroscopy[END_REF][START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][9][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF][START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF][START_REF] Kang | Integrated Study of First Principles Calculations and Experimental Measurements for Li-Ionic Conductivity in Al-Doped Solid-State LiGe2(PO4)3 Electrolyte[END_REF][START_REF] Kuo | Lithium Ion Diffusion Mechanism and Associated Defect Behaviors in Crystalline Li1+ XAlxGe2-x(PO4)3 Solid-State Electrolytes[END_REF] . For these cases, there is a consensus on the fact that the lower activation energy in the doped compounds is caused by the occupation of additional sites (36f, 18e, 6a etc) besides the commonly occupied M1 6b site [START_REF] Pérez-Estébanez | A Systematic Study of Nasicon-Type Li1 + XMxTi 2 -X(PO4)3 (M: Cr, Al, Fe) by Neutron Diffraction and Impedance Spectroscopy[END_REF][START_REF] Arbi | Structural Factors That Enhance Lithium Mobility in Fast-Ion Li 1+xTi2-XAlx(PO4)3 (0 ≤ x ≤ 0.4) Conductors Investigated by Neutron Diffraction in the Temperature Range 100-500 K[END_REF][START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Case | Structure and Ion Transport of Lithium-Rich Li1+xAlxTi2-x(PO4)3 with 0.3<x<0.5: A Combined Computational and Experimental Study[END_REF][START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF][START_REF] Kang | Integrated Study of First Principles Calculations and Experimental Measurements for Li-Ionic Conductivity in Al-Doped Solid-State LiGe2(PO4)3 Electrolyte[END_REF][START_REF] Kuo | Lithium Ion Diffusion Mechanism and Associated Defect Behaviors in Crystalline Li1+ XAlxGe2-x(PO4)3 Solid-State Electrolytes[END_REF] . In the case of chromium doped NASICON-like lithium conductors, studies linking ionic transport and structural parameters are much scarcer. Only a few studies have reported some correlation between the cell volume parameter and activation energy for ion conduction [START_REF] Nuernberg | A Systematic Study of Glass Stability, Crystal Structure and Electrical Properties of Lithium Ion-Conducting Glass-Ceramics of the Li1+xCrx(GeyTi1-y)2-x(PO4)3 System[END_REF][START_REF] Xu | Preparation and Characterization of Lithium Ion-Conducting Glass-Ceramics in the Li 1 + XCr XGe 2 -x(PO 4) 3 System[END_REF] . As discussed in Section 3.1, the most important structural change observed with Cr-doping in the LCGTP system is the occupancy of 36f sites followed by the depopulation of the 6b sites. This particular behavior has also been observed in LAGP [START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF] and LATP 6 systems. The authors of these studies concluded that the decrease in activation energy caused by the aliovalent substitution of Ge +4 or Ti +4 by Al +3 is triggered by the accommodation of extra Li + ions in new energetically favored sites.

Computer simulations adopting the first-principle density functional theory or classical molecular dynamics to identify the Li + diffusion pathways have proposed cooperative motions of 6b and 36f Li + as the underlying mechanism for the activation barrier change caused by aliovalent substitution [START_REF] Kang | Integrated Study of First Principles Calculations and Experimental Measurements for Li-Ionic Conductivity in Al-Doped Solid-State LiGe2(PO4)3 Electrolyte[END_REF][START_REF] Kuo | Lithium Ion Diffusion Mechanism and Associated Defect Behaviors in Crystalline Li1+ XAlxGe2-x(PO4)3 Solid-State Electrolytes[END_REF][START_REF] Vyalikh | NMR Studies of Li Mobility in NASICON-Type Glass-Ceramic Ionic Conductors with Optimized Microstructure[END_REF] . This is likely to be the main reason for the drop in the activation energy of the LCGTP02 and LCGTP04 glass-ceramics regarding the LGTP glass-ceramic. The further increase in activation energy, observed for the LCGTP05 sample, is still not well understood. However, a possible explanation could be the existence of optimal ratio between 6b and 36f occupancies, which would favor the cooperative motion [START_REF] Redhammer | A Single Crystal X-Ray and Powder Neutron Diffraction Study on NASICON-Type Li1+xAlxTi2-x(PO4)3 (0 ≤ x ≤ 0.5) Crystals: Implications on Ionic Conductivity[END_REF][START_REF] Kuo | Lithium Ion Diffusion Mechanism and Associated Defect Behaviors in Crystalline Li1+ XAlxGe2-x(PO4)3 Solid-State Electrolytes[END_REF] .

Nevertheless, a minimum in the activation energy and a maximum in conductivity for the dependence on the aliovalent substitution have been also found around x = 0.5 in LAGP [START_REF] Francisco | Energetics of Ion Transport in NASICON-Type Electrolytes[END_REF][START_REF] Kuo | Lithium Ion Diffusion Mechanism and Associated Defect Behaviors in Crystalline Li1+ XAlxGe2-x(PO4)3 Solid-State Electrolytes[END_REF] and LATP 6,10 compounds.

Conclusions

The aliovalent substitution of Ti +4 by Cr +3 in the Li1+xCrxGeTi1-x(PO4)3 (0 ≤ x ≤ 1)

compositional system has been investigated on samples obtained by the glass-ceramic synthesis route. After melting and quenching, the glasses were heat-treated at 900 o C for 12 hours to obtain fully crystallized glass-ceramics. The crystal structure of the resulting glassceramics has been investigated by X-ray and high-resolution neutron powder diffraction, whereas their electrical properties were determined using impedance spectroscopy. XRD analysis confirms a typical diffraction pattern of NASICON-like rhombohedral structure in the whole range of compositions, although single-phase NASICON are achieved only for x ≤ 0.5. In this range of composition, the cell volume parameter does not vary considerably, confirming our hypothesis that, because of their equivalent crystal radius, the replacement of Ti +4 by Cr +3 should avoid important volumetric changes, allowing to evaluate the true contribution of the aliovalent substitution on other structural factors.

Lithium positions were determined using Fourier difference maps to high-resolution neutron diffraction data. The refinement of the structural model indicates a full occupancy of the 6b site and a negligible one for the 36f site for the unsubstituted compound (LGTP). However, since the substitution of Ti +4 by Cr +3 is accompanied by an increment of lithium content and the 6b position is already fulfilled in the LGTP, the additional lithium is accommodated in the 36f position. Besides, the increase in the lithium content not only causes an occupation of the 36f position but also triggers a partial depopulation of the 6b sites. Furthermore, the grain-related activation energy for ionic conductivity of Cr-substituted glass-ceramics drops considerably compared to the unsubstituted compound. Therefore, the partial occupancy of the 36f sites along with the partial depopulation of 6b sites should be a determining factor for decreasing the energy barrier for the ion conduction mechanism in these compounds, facilitating the ionic transport.

3 . 1

 31 Structural characterization of LCGTP glass-ceramics XRD analyses were previously performed aiming to select single-phase glass-ceramics for subsequently neutron diffraction study. A typical diffraction pattern of NASICON-like rhombohedral structure, LiTi2(PO4)3-type phase (COD card 96-722-2156), was detected in all the seven LCGTP glass-ceramics (see Figures S1). However, compositions with x > 0.5 also show diffraction peaks of secondary phases, most likely corresponding to the LiCrP2O7type phase (COD card 96-221-2724). Besides, the differences between the structural parameters of the LiTi2(PO4)3 compound and the present glass-ceramics become evident at higher diffraction angles, where the peak positions of LCGTP glass ceramic are shifted to higher angles compared to those of the LiTi2(PO4)3 compound (COD card 96-722-2156).

Figure 1 -

 1 Figure 1 -The structural model used based on R3 ̅c SG for Li1+xCrxGeTi1-x(PO4)3 glass-ceramics with 0 ≤ x ≤ 0.5. Dark blue spheres in octahedral coordination represent Ti/Ge/Cr (12c), purple spheres in tetrahedral coordination represent P (18e) and red spheres represent O (36f). Isosurfaces of neutron scattering negative density of LiGeTi(PO4)3 composition are shown in a light blue color (6b). The orientation of the lattice plane (0 1 -4) displaying the 2D section of Fourier maps used to identify Li positions is also shown.

Figure 2 -

 2 Figure 2 -Fourier maps calculated based on scattering density differences between the experimental data and the structural model with the atoms that make up the skeleton at the lattice plane (0 1 -4) for LiGeTi(PO4)3 (x = 0) (a) and Li1.4Cr0.4GeTi0.6(PO4)3 (x = 0.4) (b) compositions. The 6b and 36f sites are indicated in the maps making up a zig-zag trajectory through the structure.

Figure 3 -

 3 Figure 3 -Dependence of cell parameters variations (a) and the Li occupancies (b) on x in the LCGTP compositional system. Lines in the plot are only guides for the eyes. Error bars in Figure 3a are smaller than symbols.

Figure 4 -

 4 Figure 4 -Structural model of LiGeTi(PO4)3 glass-ceramic based on R3 ̅c SG (a) and its respective BVEL isosurfaces percolating 6b and 36f sites (b). Dark blue spheres in octahedral coordination represent Ti/Ge (12c), purple spheres in tetrahedral coordination represent P (18e), red spheres represent O (36f) and cyan blue and white spheres represent Li in the 6b and 36f positions, respectively. BVEL isosurfaces connected in a 3D percolating network are shown in yellow. The orientation of the lattice plane (0 1 -4) displaying the 2D section of BVEL maps used to identify Li-diffusion pathways is also shown.

Figure 5 -

 5 Figure 5 -BVEL maps calculated based on experimentally determined structural parameters at the lattice plane (0 1 -4) for LiGeTi(PO4)3 (LGTP) (a and c) and Li1.4Cr0.4GeTi0.6(PO4)3 (LCGTP04) (b and d) compositions.

Figures a and b

  Figures a and b show absolute negative BVEL maps while Figures c and d show relative negative BVEL maps.

Figure 6 -

 6 Figure 6 -Arrhenius-like plots of the grain-related ionic conductivity of Li1+xCrxGeTi1-x(PO4)3 (0 ≤ x ≤ 0.5) glass-ceramics (a) and the dependence of their respective grain-related activation energy (Eag) on x (b).

Table 1

 1

	presents the

Table 1 -

 1 Nominal glass compositions based on systematic increases of x in the Li1+xCrxGeTi1-x(PO4)3 system.

	The samples are labeled according to their Cr content.
	Samples	x	Li 1+x	Cr x	Ge 1	Ti 1-x	P 3	O 12
	LGTP	0	1.0	0	1.0	1.0 3 12
	LCGTP02 0.2	1.2	0.2	1.0	0.8 3 12
	LCGTP04 0.4	1.4	0.4	1.0	0.6 3 12
	LCGTP05 0.5	1.5	0.5	1.0	0.5 3 12
	LCGTP06 0.6	1.6	0.6	1.0	0.4 3 12
	LCGTP08 0.8	1.8	0.8	1.0	0.2 3 12
	LCGP	1	2.0	1.0	1.0	0	3 12

Table 2 -

 2 Crystallographic parameters obtained by the Rietveld Method. The goodness of fit parameters as profile factor Rp and weighted profile factor Rwp are indicated. Uiso corresponds to the equivalent isotropic displacement parameters.

	Sample Type WP	Occ.	x	Atomic position y	z	Uiso (×10 2 ) (Å 2 )	Lattice parameter Volume cell V (Å 3 ) a = b (Å) c (Å)	Rp (%)	Rwp (%)
		Li1 6b 0.98(5)	0		0		0	4.3(6)
		Li2 36f 0.003(5) 0.0648(6) 0.3440(6) 0.0752(5)	8(5)
	LGTP	Ti Ge	12c	0.5 0.5	0		0	0.1382(6) 2.4(2)	8.4064(1) 20.6827(2) 1265.79(2) 4.92 6.35
		P		1	0.2871(4)	0		0.25	1.1(1)
		O1 18e	1	0.1822(3) 0.9920(4) 0.1900(1) 1.5(1)
		O2 36f	1	0.1863(3) 0.1620(3) 0.0814(1) 1.1(1)
	LCGTP02	Li1 6b 0.74(5)	0		0		0	3(2)	8.3824(2

) 20.7306(6) 1261.48(5) 4.63 6.12 Li2 36f 0.08(5) 0.0648(4) 0.3440(5) 0.0752(2) 15 (13)

		Ti		0.4
		Ge	12c	0.5	0	0	0.1407(5) 1.5(2)
		Cr		0.1
		P 18e	1	0.2864(6)	0	0.25	2.2(1)
		O1 36f	1	0.1784(5) 0.9860(5) 0.1895(2) 1.9(1)
		O2 36f	1	0.1872(4) 0.1611(5) 0.0824(1) 2.0(1)
		Li1 6b 0.60(5)	0	0	0	12(4)
		Li2 36f 0.13(5) 0.0647(4) 0.3440(4) 0.0750(1) 13(4)
		Ti		0.3
	LCGTP04	Ge Cr	12c	0.5 0.2	0	0	0.1414(4) 2.6(2)	8.3792(2) 20.7263(6) 1260.28(5) 3.77 4.90
		P 18e	1	0.2868(6)	0	0.25	1.3(1)
		O1 36f	1	0.1794(5) 0.9878(4) 0.1895(2) 1.8(1)
		O2 36f	1	0.1869(4) 0.1604(4) 0.0825(1) 1.7(1)
		Li1 6b 0.48(5)	0	0	0	16(4)
		Li2 36f 0.17(5) 0.0649(4) 0.3442(3) 0.0749(1) 23(4)
		Ti		0.25
	LCGTP05	Ge Cr	12c	0.50 0.25	0	0	0.1422(3) 1.4(2)	8.3733(2) 20.8290(5) 1260.28(5) 3.83 4.79
		P 18e	1	0.2849(5)	0	0.25	1.8(1)
		O1 36f	1	0.1789(4) 0.9844(5) 0.1998(2) 2.7(1)
		O2 36f	1	0.1876(3) 0.1602(4) 0.0839(1) 2.4(1)

Acknowledgements

The authors are very grateful to the São Paulo Research Foundation (FAPESP) for its financial support under CEPID Process no. 2013/07793-6. R. B. Nuernberg acknowledges Brazil's National Council for Scientific and Technological Development (CNPq) for the doctoral scholarship granted under projects no.140456/2014-7. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001. CAPES is also acknowledged for the doctoral scholarship granted under project no. 88881.132930/2016-01. The Institut Laue-Langevin also supported this work under proposal number 5-22-755.

6

Supplementary Material


Xavier Rolland

Pierre Maurel

Camille Maumet

TOWARDS EFFICIENT FMRI DATA RE-USE: CAN WE RUN BETWEEN-GROUP ANALYSES WITH DATASETS PROCESSED DIFFERENTLY WITH SPM?

Keywords: Brain Imaging, Analytical Variability, Reproducibility, Validity, Null Hypothesis, Pipeline

The increased amount of shared data creates an opportunity to reuse existing data to reach larger sample sizes and hence increase statistical power in neuroimaging studies. However, doing so may require to perform analyses using subject data processed differently. Here, we performed between-group analyses under the null hypothesis (making any detection a false positive), with data from the Human Connectome Project (HCP) (n=1080) processed with different pipelines. We compared the estimated false positive rates obtained to the theoretical false positive rate, to assess whether the variability in processing pipelines (called analytical variability) impacts the validity of the analyses. We found that some differences in parameter values caused invalidity, suggesting that analytical variability has to be taken into account before combining subject data processed with different pipelines.

INTRODUCTION

Task-based functional Magnetic Resonance Imaging (fMRI) studies the activation of brain regions while a task is performed. Blood-Oxygen Level Dependent (BOLD) fMRI uses MRI to measure a BOLD signal (whose variations in time are related to brain activity), at each position of the brain. Multiple steps of processing are performed on the data: preprocessing, subject-level (first-level) and group-level (second-level) analysis. Series of steps performed for a complete analysis, or parts of it, are called pipelines (e.g. subject-level pipelines cover preprocessing and first-level analysis).

Many concerns have been raised over the last few years regarding the lack of reproducibility of fMRI studies [START_REF] Carp | On the plurality of (methodological) worlds: estimating the analytic flexibility of fmri experiments[END_REF][START_REF] Bowring | Exploring the impact of analysis software on task fmri results[END_REF][START_REF] Strother | Evaluating fmri preprocessing pipelines[END_REF]. One important cause of this issue is the overall low statistical power of fMRI studies induced by low sample sizes, which increases the likelihood that any positive result is false [START_REF] Button | Power failure: why small sample size undermines the reliability of neuroscience[END_REF]. Larger sample sizes may be achieved by taking advantage of shared neuroimaging datasets, and combining subject data from multiple different sources for new studies. However, these datasets may include already processed subject data, and combining them may require to perform analyses using data processed differently. Multiple choices are possible at each processing step (different orders of operations, different parameter values, different software packages).

Multiple studies have explored how this variability in processing and analysis protocol (called analytical variability) may impact the reproducibility of neuroimaging results. For example, in fMRI, [START_REF] Botvinik-Nezer | Variability in the analysis of a single neuroimaging dataset by many teams[END_REF] showed substantial differences in results obtained across teams when completing a similar analysis with a different pipeline. Other studies have explored the variability resulting from specific processing and analysis parameters [START_REF] Glatard | Reproducibility of neuroimaging analyses across operating systems[END_REF][START_REF] Bowring | Exploring the impact of analysis software on task fmri results[END_REF][START_REF] Carp | On the plurality of (methodological) worlds: estimating the analytic flexibility of fmri experiments[END_REF]. Frameworks for the optimization of pipelines by estimation of performance metrics associated with reproducibility have also been developed [START_REF] Churchill | An automated, adaptive framework for optimizing preprocessing pipelines in taskbased functional mri[END_REF].

Here, we focused on how analytical variability in subjectlevel pipelines can impact the compatibility of subject data in between-group studies. If researchers want to use processed subject data coming from different sources, they must ensure that subject-level processing differences will not increase the probability of obtaining false positive results.

In order to assess the validity of between-group studies combining subject data processed differently, we processed raw data from the Human Connectome Project [START_REF] Van Essen | The wu-minn human connectome project: an overview[END_REF] with various pipelines which differed on a set of predefined parameters. After subject-level processing, we carried out a series of between-group analyses under the null hypothesis, with different pairs of subject-level pipelines. False positive rates were then used to assess the validity of these between-group analyses (with inflated false positive rates indicating the invalidity of the combination of pipelines).

MATERIAL

This study was performed using data from the HCP [START_REF] Van Essen | The wu-minn human connectome project: an overview[END_REF]. We used unprocessed fMRI data associated with the motor task and structural data for all available subjects (n = 1080). Multiple preprocessing and first-level analyses were performed (see section 3.2).

METHODS

In order to test the validity of between-group analyses using subject data processed differently, we performed analyses under null hypothesis assuming no difference in means across pipelines. We used the detection rate as an estimate of the Fig. 1. Steps performed for the analysis: subject-level analysis on subject data with different pipelines, between-group analyses with subject data processed differently for multiple pairs of pipelines, and repetitions of each analysis 1000 times to estimate the false positive rates.

false positive rate and compared it to the expected false positive rate under the null hypothesis (Fig. 1). Analyses were performed with SPM12 r7771 (RRID: SCR 007037, [START_REF] Penny | Statistical parametric mapping: the analysis of functional brain images[END_REF]) with Octave 5.1.0 (RRID: SCR 014398), under Debian 10.6.

Subject-Level Pipelines

All subject data were processed through multiple pipelines, which carried out preprocessing and first-level analysis. The preprocessing steps were: spatial realignment of the functional data, coregistration of realigned data towards the structural data, segmentation of the structural data, nonlinear registration of the structural and realigned functional data towards a common space and smoothing of the normalized functional data. For each pipeline, we selected the first-level contrast corresponding to the right hand in the motor task.

Our pipelines varied on the following parameters: • Smoothing kernel: Full-Width at Half-Maximum (FWHM) was 5mm or 8mm. • Number of motion regressors in the General Linear Model (GLM) for the first-level analysis: 0, 6 (3 rotations + 3 translations) or 24 (3 rotations + 3 translations, 6 derivatives and the 12 corresponding squares of regressors). • Presence or absence of the temporal derivatives of the Haemodynamic Response Function (HRF) in the GLM. Apart from these parameters, each pipeline used the default settings. In total, those combinations provided a set of 12 different subject-level pipelines ( 2 FWHM × 3 numbers of motion regressors × 2 HRF). The steps performed and parameter values were chosen so as to represent typical pipelines found in the literature [START_REF] Carp | On the plurality of (methodological) worlds: estimating the analytic flexibility of fmri experiments[END_REF].

Between-group Analyses and False Positive Rates Estimation

In the remainder of the text, we will refer to as the "default pipeline", the subject-level pipeline with the following parameter values: 5mm FWHM for smoothing kernel, 24 motion regressors and no temporal derivatives for the HRF. We performed between-group analysis to compare two groups of 50 subjects, with each group processed respectively by the default pipeline and by one of the 11 alternative pipelines. The 100 subjects from the pair of groups were randomly sampled without replacement, uniformly among the 1080 subjects. For this reason, our analyses are under the null hypothesis of no difference between groups. In addition to comparisons with the default pipeline against the 11 different pipelines (referred to as "different pipeline analyses" in the following), to assess the compatibility between pipelines, we performed analysis with the default pipeline for both groups (referred to as "same pipeline analysis" in the following), as a sanity check.

We looked at the between-group difference in means for the right hand contrast. We performed two one-tailed t-test (for the comparisons default > alternative and alternative > default) with unequal variance using a voxelwise p < 0.05 FWE-corrected threshold. For conciseness, the results are presented below as one two-tailed t-test with a p < 0.1 FWE-corrected threshold.

For each pair of pipelines, the between-group analyses were repeated 1000 times with different groups of subjects to estimate the empirical false positive rate, which was the proportion of analyses over the 1000 repetitions with at least one significant voxel between both groups. This false positive rate is expected to be equal to 0.1 under the null hypothesis. The set of 1000 pairs of 50-subject groups used to estimate the empirical false positive rate was identical for all pairs of pipelines.

For same pipeline analysis, and for comparisons with the alternative pipeline varying from default on one or two parameter values, we also created variants of P-P plots to observe the behavior of the distribution of second-level statistical values. For each comparison, we created a subset of 1,000,000 statistical values chosen randomly over our 1,000 second-level analysis statistical maps. We obtained and sorted the p-values associated with these statistical values for the expected distribution under the null hypothesis (Student distribution with n=98 degrees of freedom). We made variants of P-P plots plotting the difference between obtained and expected -log(p-values), against the expected -log(p-values).

RESULTS

False positive rates

False positive rates for comparisons between the default pipeline and each of the 11 alternative pipelines are presented in Table 1 in both groups) led to conservative results with a false positive rate of 0.040, which is consistent with the literature [START_REF] Eklund | Cluster failure: Why fmri inferences for spatial extent have inflated false-positive rates[END_REF].

For analyses combining pipelines with a single varying factor, the temporal derivatives of the HRF was the least impacting of all three factors (temporal derivative, smoothing or motion regressors) with a false positive rate of 0.035 (third row, second column). Smoothing was the most impacting factor with a false positive rate of 0.113 (third row, third column). For motion regressors, the two possible alternative values led to false positive rates of 0.044 (motion reg = 6, second row, first column) and 0.109 (motion reg=0, first row, first column).

For analyses combining pipelines with multiple varying factors, all comparisons with different smoothing kernel (FWHM 8mm in the alternative pipeline, third and fourth column) led to invalid results, with false positive rates above 0.1. Other parameters being equal, comparisons with same smoothing always led to smaller false positive rates (column 2 versus 4 and column 1 versus 3). Similarly, comparisons with different numbers of motion regressors (especially with no motion regressors in the alternative pipeline) gave higher false positive rates than comparisons with the same number of motion regressors (line 1 versus line 3). More generally, multiple varying factors always led to higher false positive rates, suggesting that the combination of differences between pipelines leads to a combination of effects.

P-P plots

P-P plots are shown on Figure 2. Positive differences outside the confidence interval indicate invalidity (-log(p-values) higher than expected) whereas negative differences indicate conservativeness.

For same pipeline analysis (using the default pipeline for both groups), the results observed were conservative. The effects of parameter differences on P-P plots are similar to those observed on false positive rates: no clear effect of the modeling of the HRF, effect of differences in smoothing and numbers of motion regressors, and combination of effects with combination of differences. The P-P plots show that the results observed with a specific thresholding in Table 1, regarding the effect of parameter differences, are representative of general tendencies for statistical values overall.

DISCUSSIONS

We observed that differences between subject-level pipelines on specific parameters (smoothing, number of motion regressors) had an effect which caused invalid results when combined in between-group analyses. Therefore, pairs of pipelines with different values for these parameters cannot be combined for analyses with differences between groups: it would be impossible to know if the between-group differences have an effect, as it would be confounded with the effect of pipeline differences.

Our results suggest that some combinations of pipelines should be avoided, while other could be used. Differences in modeling of the HRF had no observable effect in our experiments. This is not the case for smoothing and motion regressors, which gave invalid results.

We performed analyses with different subject-level processing pipelines, where the pipelines are confounded with the groups. This situation may happen in practice if we want to do group comparisons where each group is associated with a specific dataset, which is itself associated with a specific subject-level pipeline. Other situations may also happen in which there are multiple pipelines used on the subjects within each group. Such situations may be investigated in future work.

Since there are many more parameters which may vary between subject-level pipelines in practice (different models for the HRF, performing or not specific substeps such as slicetiming correction, software packages, etc), effects of pipeline differences would likely be more important in real conditions.

Here, we focused our investigations on measuring deviations from the theoretical false positive rate under the null. While this is important in order to assess the validity of the statistical approaches, the lack of sensibility is also an important methodological issue that will require further investigations in future work.

Conclusion

Our study shows that processing applied to data must be taken into consideration when combining them, due to the potential invalidity of the results. The framework that we defined may be used with other variations of parameters or other paradigms in future work, and also using different software packages, to assess the generalizability of our results. Also, we may try to create methods to model and correct the effect of analytical variability, which would allow us to combine data without having to consider the differences in terms of processing.

COMPLIANCE WITH ETHICAL STANDARDS

This study was performed using data from the Human Connectome Project (HCP). Written informed consent was obtained from participants and the original study was approved by the Washington University Institutional Review Board. We agreed to the HCP Open Access Data Use Terms.

  and expected -log10(p-values) Default 8mm for smoothing, presence of derivatives 8mm for smoothing, no motion regressors

Fig. 2 .

 2 Fig.2. Variants of P-P plots for distributions obtained with various analyses between the default pipeline and pipelines with one (top) or two (bottom) varying parameter values (indicated in legend), against the expected distribution, with 0.95 confidence interval (in grey). The curve for same pipeline analysis (alternative pipeline equal to default) is plotted in red.

  . Same pipeline analysis (using the default pipeline

		Smoothing, 5 mm Smoothing, 8 mm
		No der.	Der.	No der.	Der.
	0 motion reg.	0.109	0.097	0.237	0.245
	6 motion reg.	0.044	0.046	0.139	0.145
	24 motion reg.		0.035	0.113	0.131
	Table 1. Empirical false positive rates for analyses com-
	paring the 11 alternative pipelines to the default pipeline
	(smoothing=5mm, no der. and 24 motion reg.) at FWE-
	corrected p < 0.1 two-tailed. Invalid results (> 0.1) are in
	bold. The false positive rate obtained for same pipeline analy-
	sis (default pipeline in both groups, corresponding to the grey
	cell) was equal to 0.040. reg.= regressors, der.=derivatives.

7. ACKNOWLEDGEMENTS Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. Xavier Rolland was supported by Region Bretagne (ARED Varanasi) and by EU H2020 project OpenAIRE-Connect (Grant agreement ID: 731011).