Sujoy Saha

Gwenaëlle Rousse

Ignacio Blazquez Alcover

Mathieu Courty

Daniel Alves

Dalla Corte

Jean-Marie Tarascon
email: jean-marie.tarascon@college-de-france.fr

Ignacio Blazquez Alcover

Polymorphism in Li 4 Zn(PO 4 ) 2 and Stabilization of its Structural Disorder to Improve Ionic Conductivity

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

Li-ion batteries have become worldwide accepted to power today's portable electronics because they offer high energy density [START_REF] Goodenough | Challenges for Rechargeable Li Batteries[END_REF][START_REF] Zeier | A Solid Future for Battery Development[END_REF][START_REF] Larcher | Towards Greener and More Sustainable Batteries for Electrical Energy Storage[END_REF] . For the same reason, they have become the technology of choice for EV's, but safety concerns associated with the use of flammable liquid electrolytes remain a serious challenge. [START_REF] Goodenough | The Li-Ion Rechargeable Battery: A Perspective[END_REF][START_REF] Gallagher | Boeing's Dreamliner Batteries "Inherently Unsafe"-And Yours May Be Too[END_REF][START_REF] Meier | Feds Review Third Tesla Fire as Shares Fall Again[END_REF] A possible option consists in replacing the liquid electrolytes by their solid counterparts. Finding suitable solid electrolytes having high ionic conductivity will not only enable to design safer 'allsolid-state Li-ion batteries', but will also open new possibilities to play with future-generation battery chemistries like Li-S, Li-Br or Li-air [START_REF] Wang | A High Performance Lithium-Ion Sulfur Battery Based on a Li 2 S Cathode Using a Dual-Phase Electrolyte[END_REF][START_REF] Takemoto | Development of Rechargeable Lithium-Bromine Batteries with Lithium Ion Conducting Solid Electrolyte[END_REF][START_REF] Huiqiao | Rechargeable Ni-Li Battery Integrated Aqueous/nonaqueous System[END_REF] . Although solid state electrolytes are known since the observation of ionic conduction upon heating in Ag 2 S and PbF 2 by Faraday in the 1830s, [START_REF] Faraday | Experimental Researches in Electricity, Third Series[END_REF] it is only in 1960-80's that they were studied extensively. This fruitful period has seen the discovery of many new Li-ion conducting compound classes, e.g., LISICON, NASICON, LIPON, thio-LISICON etc.; but the conductivities of these solid electrolytes were far behind practical utilization and this research fall into oblivion [START_REF] Hong | Crystal Structure and Ionic Conductivity of Li 14 Zn(GeO 4 ) 4 and Other New Li + Superionic Conductors[END_REF][START_REF] Goodenough | Fast Na + -Ion Transport in Skeleton Structures[END_REF][START_REF] Tachez | Ionic Conductivity of and Phase Transition in Lithium Thiophosphate Li 3 PS 4[END_REF][START_REF] Bates | Electrical Properties of Amorphous Lithium Electrolyte Thin Films[END_REF] . However, recent safety issues of LIB's and foreseen aspiration of industries has led researchers to re-visit the field intensely in the past decade and several new interesting compounds like garnets, perovskites etc. were found [START_REF] Murugan | Fast Lithium Ion Conduction in Garnet-Type Li 7 La 3 Zr 2 O 12[END_REF][START_REF] Harada | Lithium Ion Conductivity of Polycrystalline Perovskite La 0.67-x Li 3x TiO 3 with Ordered and Disordered Arrangements of the A-Site Ions[END_REF] , but still with conductivities (~10 -4 -10 -6 Scm -1 ) trailing far behind than that of liquid electrolytes [START_REF] Bachman | Inorganic Solid-State Electrolytes for Lithium Batteries: Mechanisms and Properties Governing Ion Conduction[END_REF] . It is only recently, with the discovery of thio-phosphate Li 11-x M 2-x P 1+x S 12 (M = Si, Ge, Sn) that ionic conductivities comparable to the liquids (~10 -2 -10 -3 Scm -1 ) have been achieved. However, instability issues, synthesis and handling difficulties together with the lack of compatibility with Lielectrode has clogged its practical relevance. [START_REF] Kamaya | A Lithium Superionic Conductor[END_REF][START_REF] Hori | Phase Diagram of the Li 4 GeS 4 -Li 3 PS 4 Quasi-Binary System Containing the Superionic Conductor Li 10 GeP 2 S 12[END_REF][START_REF] Wenzel | Direct Observation of the Interfacial Instability of the Fast Ionic Conductor Li 10 GeP 2 S 12 at the Lithium Metal Anode[END_REF] Owing to their greater stability, ease of handling and larger electrochemical window, oxides are still foreseen best candidates to replace liquid electrolytes, provided an improved ionic conductivity. Hence we concentrate our search towards new Li ionic conducting oxides apart from the exhaustively explored list of above mentioned families.

Phosphates have fascinated chemists for many years, thanks to their stability, rich crystal-chemistry, and partly because of subtle interactions between phosphate framework and mobile Li + ions, which accounts for their high ionic conductivities [START_REF] Deng | Structural and Mechanistic Insights into Fast Lithium-Ion Conduction in Li 4 SiO 4 -Li 3 PO 4 Solid Electrolytes[END_REF][START_REF] Li | Mastering the Interface for Advanced All-Solid-State Lithium Rechargeable Batteries[END_REF][START_REF] Kumar | Space-Charge-Mediated Superionic Transport in Lithium Ion Conducting Glass-Ceramics[END_REF] . Along this line, we spotted Li 4 Zn(PO 4 ) 2 as a potential candidate. Although this compound was reported as early as in 1976, it was never explored in details for its transport properties despite its rich chemistry. [START_REF] Sandomirskii | Crystal structure of the mixed lithium zinc orthophosphate Li 4 Zn[PO 4 ] 2[END_REF] This compound received further attention in 1986 by West et al., who explored its thermal stability by differential thermal analysis of Li 3 PO 4 -Zn 3 (PO 4 ) 2 solid solutions. [START_REF] Torres-Trevinō | Compound Formation, Crystal Chemistry, and Phase Equilibria in the System Li 3 PO 4 -Zn 3 (PO 4 )[END_REF] Interestingly, they reported three polymorphs, low temperature the α-Li 4 Zn(PO 4 ) 2 and the high temperature β-and γ-Li 4 Zn(PO 4 ) 2 , and moreover they speculated that the β-phase derives from the α-phase by cation disordering. One year later, the same group, prepared the pure compound by solid state synthesis and explored its ionic conductivity. [START_REF] Torres-Trevinō | Kinetic, and Conductivity Studies of an Order-Disorder Transition in Li 4 Zn(PO 4 ) 2[END_REF] They found that Li 4 Zn(PO 4 ) 2 is a poor Li + conductor, but noticed an increase of conductivity for the β-phase. Fifteen years later, Jensen et al., have prepared the compound by hydrothermal method and described an accurate structural model for the α-phase. [START_REF] Jensen | Hydrothermal Synthesis of Lithium Zinc Phosphates: Structural Investigation of Twinned α-Li 4 Zn(PO 4 ) 2 and a High Temperature Polymorph β-Li 4 Zn(PO 4 ) 2[END_REF] However nobody has described the crystal structures of the high temperature phases till date. β-Li 4 Zn(PO 4 ) 2 has been speculated to be isostructural to γ-Li 3 PO 4 , [START_REF] Torres-Trevinō | Compound Formation, Crystal Chemistry, and Phase Equilibria in the System Li 3 PO 4 -Zn 3 (PO 4 )[END_REF][START_REF] Jensen | Hydrothermal Synthesis of Lithium Zinc Phosphates: Structural Investigation of Twinned α-Li 4 Zn(PO 4 ) 2 and a High Temperature Polymorph β-Li 4 Zn(PO 4 ) 2[END_REF] which is well-known since long as a good Li + conductor. [START_REF] Deng | Structural and Mechanistic Insights into Fast Lithium-Ion Conduction in Li 4 SiO 4 -Li 3 PO 4 Solid Electrolytes[END_REF] More specifically, this compound can be viewed as Li 3 PO 4 with two Li + substituted by one Zn 2+ , and since there is no twofold positions in Li 3 PO 4 to accommodate the extra Zn 2+ , the Zn 2+ and vacancies are probably disordered in fourfold Li sites. This could explain the higher conductivity reported for the β phase, bearing in mind that in solid electrolytes, disordered polymorphs possess greater conductivity than their ordered counterparts. [START_REF] Liu | Anomalous High Ionic Conductivity of Nanoporous β-Li 3 PS 4[END_REF][START_REF] Anantharamulu | A Wide-Ranging Review on Nasicon Type Materials[END_REF][START_REF] Thangadurai | Garnet-Type Solid-State Fast Li Ion Conductors for Li Batteries: Critical Review[END_REF] Hence, the high temperature disordered polymorphs are of great interest, with in particular a huge attention which is paid to the stabilization of such phases at room temperature. Various methods have been developed to stabilize the high temperature phase at room temperature, such as, quenching [START_REF] Takada | Lithium Ion Conductive Oxysulfide, Li 3 PO 4 -Li 3 PS 4[END_REF] , nanostructuring [START_REF] Liu | Anomalous High Ionic Conductivity of Nanoporous β-Li 3 PS 4[END_REF][START_REF] Kumar | A Novel Low-Temperature Solid-State Route for Nanostructured Cubic Garnet Li 7 La 3 Zr 2 O 12 and Its Application to Li-Ion Battery[END_REF] , or varying synthesis condition like using certain precursors [START_REF] Li | Mastering the Interface for Advanced All-Solid-State Lithium Rechargeable Batteries[END_REF] . Nevertheless, the most effective method is the aliovalent substitution [START_REF] Deng | Structural and Mechanistic Insights into Fast Lithium-Ion Conduction in Li 4 SiO 4 -Li 3 PO 4 Solid Electrolytes[END_REF][START_REF] Thangadurai | Fast Solid-State Li Ion Conducting Garnet-Type Structure Metal Oxides for Energy Storage[END_REF][START_REF] Wu | Gallium-Doped Li 7 La 3 Zr 2 O 12 Garnet-Type Electrolytes with High Lithium-Ion Conductivity[END_REF][START_REF] Afyon | A Shortcut to Garnet-Type Fast Li-Ion Conductors for All-Solid State Batteries[END_REF] , which creates Li + deficiency and disorder in the Li-sublattice which eventually can lead to the stabilization of the disordered phase.

In this work, a similar strategy has been carried out and we found as described herein that the partial substitution of Zn 2+ by trivalent Ga 3+ in Li 4 Zn 1-x Ga x (PO 4 ) 2 (x = 0.5) can successfully stabilize the βphase at room temperature. At lower substitution levels (x = 0.3), Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 crystallizes into an intermediate β' phase. The structure of the β and β'-phases were solved for the first time and we found that they are disordered polymorphs of the low temperature α-phase. Additionally, the β-phase shows a higher ionic conductivity along with reduced activation energy than the α-polymorph.

EXPERIMENTAL SECTION

Synthesis. Samples of Li 4 Zn(PO 4 ) 2 and Li 4 Zn 1-x Ga x (PO 4 ) 2 were prepared by classical solid state syntheses. Stoichiometric amounts of Li 2 CO 3 (Sigma Aldrich, 99%), ZnO (Sigma Aldrich, 99%), NH 4 H 2 PO 4 (Alfa Aesar, 98%) and Ga 2 O 3 (Alfa Aesar, 99.99%) were used as precursors. They were mixed homogeneously by ballmilling for 20 mins with stainless-steel balls in a stainless-steel vial using SPEX-8000M mixer-miller and pressed into a pellet subsequently. The different pellets were heated in alumina boats for 36 hours at 800 °C after a ramp of 1 °C/min, followed by quenching in air. For the high-energy ball-milling synthesis of the β-Li 4 Zn(PO 4 ) 2 , the as-prepared α-Li 4 Zn(PO 4 ) 2 sample was continuously milled for 2 hrs using ballsto-powder weight ratio of 40. Characterization. Powder X-ray diffraction (XRD) patterns were recorded in Bragg-Brentano geometry with a Bruker D8 Advance diffractometer equipped with a Cu K  radiation (λ 1 = 1.54056 Å, λ 2 = 1.54439 Å) and a LynxEye detector. Evolution of XRD patterns of the samples with temperature were monitored in situ in the same diffractometer equipped with an Anton Paar HTK 1200N furnace to control the temperature. A ramping rate of 2 °C/min was applied. Differential scanning calorimetry (DSC) measurements were done in air using a STA 449C Netzsch apparatus at a rate of 10 °C/min.

High-resolution neutron diffraction was performed on the HRPT instrument at SINQ-PSI (Villigen, Switzerland) with a wavelength λ = 1.494 Å. All patterns were refined using the Rietveld method [START_REF] Boultif | Indexing of Powder Diffraction Patterns for Low-Symmetry Lattices by the Successive Dichotomy Method[END_REF] as implemented in the FullProf program [START_REF] Rodríguez-Carvajal | All Programs Can Be Obtained from[END_REF] . Bond Valence Energy Landscape maps (BVEL) were generated according to the method developed by Adams, [START_REF] Adams | From Bond Valence Maps to Energy Landscapes for Mobile Ions in Ion-Conducting Solids[END_REF] using the program BondSTR as implemented in the FullProf Suite [START_REF] Rodríguez-Carvajal | All Programs Can Be Obtained from[END_REF] . For the BVEL maps, calculations of Li conduction paths were done after the removal of pre-existing Li in the unit cell and also after removing Zn and Ga ions located at Li sites. Anionic neighbors up to 8 Å were considered.

Electrochemical Impedance Spectroscopy (EIS) was used to determine the Li-ion conductivity of the samples. The AC impedance measurements were made using a Bio-Logic MTZ-35 impedance analyzer equipped with a HTF-1100 furnace. To perform the measurements, pellets of ~8 mm diameter and ~1.5 mm thickness were obtained by cold-pressing using a uniaxial pressure of ~5 bars in a hydraulic press and were densified by sintering at 700 °C for 6 hours while heated and cooled at a rate of 1 °C/min. In one exception of low-temperature sintering, the pellet of ball-milled β-Li 4 Zn(PO 4 ) 2 was sintered at 200 °C for 4 days. All the sintered pellets were found to have a similar density of 75-80% with respect to the crystallographic density of Li 4 Zn(PO 4 ) 2 (2.88 g/cm 3 ), hence our reported conductivity values were not corrected by the pellet porosities.

The AC impedance measurements were performed using platinum (Pt) blocking electrodes, by applying a sinusoidal signal of 500 mV amplitude over a frequency range of 10 MHz to 1 Hz. A thin foil of carbon was placed between the Pt electrode and the pellet in order to improve the electrode contact. The measurements were carried out in the temperature range of 100-600 °C with an accuracy of ±1 °C and were soaked for 15 mins at each measurement temperature prior triggering each sweep of frequency dependent impedance measurement at the temperature.

RESULTS AND DISCUSSIONS

Structural characterization. X-ray and neutron powder diffraction were used to examine the crystal structures of Li 4 Zn(PO 4 ) 2 in its low temperature α-and high temperature β-forms. The XRD pattern of the as-prepared pristine sample, α-Li 4 Zn(PO 4 ) 2 (room temperature form), can be perfectly fitted with a monoclinic unit cell having a P 2 1 /c space group and lattice parameters a = 8.11990(16) Å, b = 10.24192(13) Å, c = 8.10549(15) Å,  = 104.8589(9)°, V = 651.538(2) Å 3 , with Z = 4, using the structural model reported earlier [START_REF] Jensen | Hydrothermal Synthesis of Lithium Zinc Phosphates: Structural Investigation of Twinned α-Li 4 Zn(PO 4 ) 2 and a High Temperature Polymorph β-Li 4 Zn(PO 4 ) 2[END_REF] . The combined Rietveld refinement of the neutron and XRD patterns is shown in Figures 1a and1b and the resulting structural parameters are reported in Table 1. The structure of α-Li 4 Zn(PO 4 ) 2 is plotted in (10) Occupancy

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
The α-Li 4 Zn(PO 4 ) 2 phase is known to undergo phase transition at high temperature to β-Li 4 Zn(PO 4 ) 2 , [START_REF] Jensen | Hydrothermal Synthesis of Lithium Zinc Phosphates: Structural Investigation of Twinned α-Li 4 Zn(PO 4 ) 2 and a High Temperature Polymorph β-Li 4 Zn(PO 4 ) 2[END_REF][START_REF] Torres-Trevinō | Kinetic, and Conductivity Studies of an Order-Disorder Transition in Li 4 Zn(PO 4 ) 2[END_REF] but the crystal structure of β-Li 4 Zn(PO 4 ) 2 was never reported, to our best knowledge. Hence we have carried out systematic investigations of the phase transition and finally have been able to deduce a structural model for the β-Li 4 Zn(PO 4 ) 2 phase. Differential scanning calorimetry (DSC), shown in Figure 3a, of the as-prepared Li 4 Zn(PO 4 ) 2 sample clearly depicts the α → β transitions at ca. 443 °C on heating, with an activation enthalpy of 11.7 kJ/mol (endothermic). On cooling, the DSC trace shows the reversible process with a thermal hysteresis of ∼15 °C indicating that the transition is of first order. This phase transition was monitored by temperature dependent XRD experiments and the diffractograms are gathered in Figure 3b. Upon heating, XRD patterns of α-Li 4 Zn(PO 4 ) 2 show a small shift towards lower angles due to thermal expansion, until the temperature reaches 430°C. Above this temperature, the XRD pattern changes and looks similar to the one previously reported for the β form. On cooling, this process is reversible and the α-phase is recovered below 420°C, see Figure S1. Note that our efforts to quench the high temperature β phase at room temperature were unsuccessful; therefore we directly used the pattern recorded at 600°C to solve the structure for the β phase. The β phase presents a powder pattern similar to the α-phase, except for some peaks that have disappeared, especially in the low angle region. This relative similarity in diffraction pattern hints for a structural relationship between the -and the -structures. The XRD pattern of β-Li 4 Zn(PO 4 ) 2 at 600°C could be indexed using the Dicvol program [START_REF] Boultif | Indexing of Powder Diffraction Patterns for Low-Symmetry Lattices by the Successive Dichotomy Method[END_REF] with an orthorhombic unit cell with lattice parameters a = 10.30571(4) Å, b = 6.50156(3) Å and c = 5.00496(2) Å, leading to a unit cell volume V = 335.348(2) Å 3 that is half the volume of the room temperature monoclinic unit cell and therefore can accommodate two formulas per unit cell (Z = 2). The orthorhombic unit cell vectors are linked to the monoclinic α-Li 4 Zn(PO 4 ) 2 ones by the following relation:

( 𝐚 ortho 𝐛 ortho 𝐜 ortho ) = ( 0 1 0 ½ 0 -½ ½ 0 ½ ) ( 𝐚 mono 𝐛 mono 𝐜 mono )
Systematic extinctions were found to be consistent with P n m a space group. The orthorhombic unit cell is plotted on top of the α structure in Figure 4a, and where, looking at two adjacent orthorhombic unit cells, it clearly appears that the high temperature β phase should involve a disorder between Zn and Li2 atoms to comply invariance through lattice translation. Therefore the XRD and neutron diffraction patterns (NPD) of the β-phase recorded at 600°C were then simultaneously refined by the Rietveld method, introducing mixing between Zn and Li cations. 104.9

105.0 The deduced structural model is shown in Figure 4b, the corresponding Rietveld refinements in Figures 1c and1d and structural parameters are gathered in Table 2. Lastly, all temperature-dependent XRD patterns were Rietveld refined and the evolution of lattice parameters of the Li 4 Zn(PO 4 ) 2 with temperature is summarized in Figure 3c using the monoclinic description for the β phase for an easier comparison. It can be noticed that the unit cell volume monotonously increases with T and exhibits a slight discontinuity at the transition, as expected from a first order phase transition. Moreover it should be noted that this volume increase results from anisotropic changes in the b, c and β unit cell parameters. Having found that the β-Li 4 Zn(PO 4 ) 2 is a disordered counterpart of the α-Li 4 Zn(PO 4 ) 2 polymorph, it became interesting to stabilize this phase at room temperature. Since ball-milling is known to increase disorder by repeated particles fracturing and welding, as was demonstrated recently for sulphates [START_REF] Lander | Synthesis and Electrochemical Performance of the Orthorhombic Li 2 Fe(SO 4 ) 2 Polymorph for Li-Ion Batteries[END_REF] , we tried to stabilize the β phase by ball-milling. More specifically, the as-prepared α-Li 4 Zn(PO 4 ) 2 was ball milled for 2 hours. The XRD pattern of the obtained powder shows broad peaks that can be nicely fitted with the structural model of the β-Li 4 Zn(PO 4 ) 2 polymorph deduced earlier (Figure S2). From the width of the Bragg peaks, we could further deduce that the ball-milled powder is made of particles with crystallites size of 10(±2) nm. Chemical substitution was then explored as an alternative to ball milling to introduce disorder. More precisely, we attempted the partial substitution of Zn 2+ with M = Al 3+ , Ga 3+ and In 3+ in Li 4 Zn(PO 4 ) 2 with the hope to create disorder in Li-sublattice by introducing Li + deficiency for charge compensation and hence to promote the β-phase stabilization. All samples of nominal compositions of Li 4-x Zn 1-x M x (PO 4 ) 2 were prepared by quenching the pellet from the synthesis temperature (800°C). We found that Al 3+ and In 3+ are unable to form solid solutions with Li 4 Zn(PO 4 ) 2 , leading to impurity phases consisting of AlPO 4 and not yet identified phase, respectively. This contrasts with the incorporation of Ga 3+ that was successful for x ≤ 0.5 mol pfu., beyond which secondary phases were found to form. Figure 5b shows the room temperature XRD patterns of as-prepared Li 4-x Zn 1-x Ga x (PO 4 ) 2 samples with 0 ≤ x ≤ 0.5. The Bragg peaks gradually shift towards higher 2 angles, indicating that the unit cell volume decreases with insertion of Ga 3+ , as expected from the smaller ionic radii of Ga 3+ versus Zn 2+ (Figure 5c). Moreover, low angle XRD peaks gradually broaden and their intensity decreases with increasing the Ga content, prior to finally disappear completely for 0.3 mol Ga pfu. Note that for x = 0.3 the collected XRD pattern appears similar, at first sight, to the one of the high temperature β-Li 4 Zn(PO 4 ) 2 phase, hence suggesting that Gasubstitution is efficient to preserve some Li/Zn disorder even at room temperature. Taking a closer look 011) and ( 013) peaks, at 2 values of 16.2° and 29.8°, respectively, rules out the P n m a space group, and suggests its P 2 1 2 1 2 1 subgroup. A structural model was therefore constructed and Rietveld refined against both neutron and XRD patterns, simultaneously. The best structural model (Figure 4c), as deduced by combined neutron and X-ray diffraction refinements (Figure 6), was obtained by mixing Li and Zn on one site, and Li and Ga on another site. This structure, whose structural parameters are gathered in Table 3 and which will be denoted as β'-hereafter, only differs from the β-structure by the fact that the 8d site in P n m a splits into two crystallographic sites of multiplicity 4, indicating that introducing Ga segregates Zn and Ga on two distinct positions in P 2 1 2 1 2 1 , both of them being mixed with Li. Further substitution of Zn with Ga leads to disappearance of the ( 011) and ( 013) peaks in the XRD patterns, as evident from Figure 5b. This confirms the stabilization of the β-polymorph at room temperature with Ga concentration ≥0.35 mol pfu. The Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 compound produces a XRD pattern fully consistent with the P n m a space group involving a full Li/Zn/Ga disorder, as the high temperature pure Zn compound. In order to deduce a structural model for Ga-substituted β-phase, we performed a combined Rietveld refinement of the neutron and XRD patterns of Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 recorded at room temperature, as shown in Figure S3. The patterns could be fitted perfectly in P n m a space group with lattice parameters of a = 10.08917(3) Å, b = 6.51079(2) Å, c = 4.935843(16) Å, and V = 324.16(5) Å [START_REF] Larcher | Towards Greener and More Sustainable Batteries for Electrical Energy Storage[END_REF] . The structural parameters are summarized in Table 4.

(°)  (b) (c) (a)
At this stage, it is worth reminding that all Ga-substituted samples mentioned previously were prepared by quenching from their synthesis temperature (800°C). Samples prepared by cooling slowly to room temperature (1 °C/min) and denoted as sc-Li 4-x Zn 1-x Ga x (PO 4 ) 2 hereafter show slight but notable discrepancy in the XRD patterns compared to those prepared by quenching, for Ga contents of x = 0.3 and above. Two compounds, sc-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 and sc-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ), were examined with special care. At room temperature, both samples present a XRD pattern characteristic of the β'-phase (P 2 1 2 1 2 1 (011) and (013) peaks at 2 = 16.3° and 29.9°, highlighted in green) along with many additional small peaks indicated with stars in Figure 7a, 7b. To establish the origin of these additional peaks and to understand the relations with the quenched samples, we performed a temperature-dependent XRD study of the compositions mentioned above. Upon heating, the XRD pattern of the sc-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 sample transforms into the β structure (P n m a space group) at 380°C, as indicated by the disappearance of all additional peaks and the (011) & (013) peaks of the β'-phase (Figure 7a). For the sc-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 sample, the evolution of the XRD pattern is slightly more complex since we observe two successive transitions. The first one occurs at 380°C and corresponds to the disappearance of the β'related peaks and most of the small peaks, indicating a transition to a β-like structure (Figure 7b). However, few tiny peaks (e.g. at 2 = 13°) still persist above 380°C and disappear only at 750°C. At 800°C, the pattern is fully alike the β one with all peaks being indexed in the P n m a space group. The tiny additional peaks shown for both slow-cooled compounds should therefore be considered as two separate sets of peaks, as they disappear at different temperatures for 0.5 Ga. The set that disappears at 380°C can be indexed with a propagation vector k 1 = (0.37, 0.47, 0) related to the P 2 1 2 1 2 1 cell with lattice parameters a = 4.94 Å, b = 6.50 Å, and c = 10.11 Å. The second set of peaks, which disappears at higher temperature for Ga content of 0.5, can be indexed with k 2 = (0, 0, 0.52). The corresponding Rietveld refinement of sc-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 is shown in Figure S5. Therefore these additional peaks can be tentatively considered as superstructure peaks, arising from complex Li/Zn/Ga/vacancies ordering and/or Ga/Zn/Li chemical composition fluctuations. Further studies are needed to fully solve the underpinned complexity. Similarly, the quenched samples were examined by the temperaturedependent XRD experiments (Figures 7c,d). The β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 sample was found to transform into the β-phase at 380 °C, as witnessed by the disappearance of the ( 011) and ( 013) peaks. This is fully confirmed by the combined refinement of the neutron and the XRD patterns recorded at 600°C. The results of the refinement with P n m a space group is presented in Figure S6, with structural parameters gathered in Table S1. Regarding the quenched β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 , upon heating it transforms to the β' structure at 300 °C (therefore β'-should be the thermodynamically stable at room temperature) before transforming back to β at 340°C (Figure 7d). These experiments clearly indicate that quenching enables the stabilization of the high temperature phases at room temperature. Figure 8 summarizes the temperature-composition phase diagram we could establish from our experiments, for both the slowcooled and quenched samples. Finally, to confirm the phase transitions with temperature in the quenched Ga-substituted samples, we measured the DSC curve of all quenched samples. The data have been summarized in Figure 5a and are in perfect agreement with high temperature XRD diffraction experiments. Transition temperatures deduced from DSC are reported as red stars in Figure 8. We see that the phase-transition temperatures to the high temperature β phase slightly decrease with increased Ga content, while the enthalpy of the phase transition reduces drastically and finally disappears for x Ga = 0.45. This confirms that the substitution creates Li + deficiency and an inherently disordered Li-sublattice, which in turn Intensity (a.u.)

2 (°) (a) (b) (c) (d)
reduces the free energy associated with the ordering of Li-sublattice, eventually leading to stabilization of the disordered phase. Hence stabilization of the β-phase at room temperature can become possible by quenching. Ionic Conductivity. We measured the ionic conductivity of all compounds by ac impedance spectroscopy over a broad frequency and temperature range. The impedance spectra were recorded on sintered pellets between ionically blocking platinum electrodes (see the experimental section for samples preparation). Figure 9a shows a typical Nyquist impedance plot of the sintered pellets of the pristine Li 4 Zn(PO 4 ) 2 sample at 300 °C. Each recorded spectra shows a depressed semicircle at high frequencies followed by a Warburg tail at lower frequency region. The low frequency spike represents the impedance response due to the blocking of mobile Li ions at the electrode interface with Pt electrode and it confirms that the ac conductivity is ionic in nature. This was further confirmed by direct current (DC) polarization experiments, which indicates that the electronic contribution is negligible, see SI. The total ionic conductivity (σ tot ) was obtained by using the value of intercept of the Warburg tail with the semicircle. We have explored the impedance spectra further by examining the depressed semicircle at high frequency. The depression of the semicircle cannot be accounted by the inhomogeneities at the sample-electrode interface, hence the need of deconvolution of the semicircle to separate the different contributions. Since the impedance response of polycrystalline materials is expected to be composed of grain (bulk) and grain boundaries (surface) contributions, we decouple the semicircle to two contributions by fitting the whole impedance spectra to an equivalent circuit model (R g //Q g )(R gb //Q gb )W o (inset of Figure 9a); where R and Q represent the resistance and constant phase element associated with the grains (g) or grain boundaries (gb) of the sample, respectively, and W o is the Warburg resistance. The fitting produces an capacitances of ~10 -12 F and ~10 -11 F associated with the constant phase element, for the high and low frequency deconvoluted semicircles, respectively and hence can be assigned to grains '

(P 2 1 2 1 2 1 ) k 1 , k 2  (P 2 1 /c) T (°C)
x Ga (mol pfu.) (P 2 1 2 1 2 1 )

 (P 2 1 /c) T (°C)
x Ga (mol pfu.)

 (P n m a) (a) (b)
(σ g ) and grain boundaries (σ gb ) contributions, respectively [START_REF] Irvine | Electroceramics: Characterization by Impedance Spectroscopy[END_REF] . A typical fitting is shown in Figure 9a. This assignment was further confirmed by calculating the characteristic resonance frequencies (f r ) of the semicircles which generally appear at the apexes of the deconvoluted semicircles; as indicated in Figure 9a. However, with increasing temperature, both grains and grain boundary responses gradually shift towards higher frequencies and finally move beyond the measurable frequency window of the instrument leaving only the Warburg tail observable. Hence deconvolution of grain conductivity (σ g ) and grain boundary conductivity (σ gb ) was not possible above ~300-400 °C for the samples and only σ tot was obtained. The variations of the ionic conductivity for the pristine Li 4 Zn(PO 4 ) 2 over the temperature range of 200 -375 °C are shown in Figure S7A and in Figure 10. There is a linear variation for the ln(T) with temperature, indicative of a thermally activated Arrhenius type behavior σ(T)=A/T•exp(-E a /k B T), where k B is the Boltzmann constant, A is a pre-exponential factor and T is temperature). The room temperature  b,  gb, and σ tot values of ~5.5x10 -18 , ~3.5x10 -19 , and ~1.8x10 -19 S•cm -1 , respectively, were deduced by linear extrapolation of the Arrhenius plot to the room temperature, with a similar E a of ~1.3 eV for all the conduction processes. Altogether, this implies that this compound is not a good conducting material at low temperature, at least in its α-form.

We next measured the conductivity of the pristine Li 4 Zn(PO 4 ) 2 sample over a broader temperature range of 200 -600 °C and reported the measured σ tot values in Figure 9b. A sudden change in the slope of the Arrhenius plot occurs at ~425 °C; temperature at which the α→β structural transition was shown to occur. Note that the high temperature β-Li 4 Zn(PO 4 ) 2 polymorph shows a much higher conductivity with an E a of 1.1 eV and a room temperature σ tot of 1.8x10 -15 S•cm -1 . For sake of completion we also measured the transport properties of our new room temperature β-Li 4 Zn(PO 4 ) 2 phase obtained by ball-milling of the α-Li 4 Zn(PO 4 ) 2 phase (Figure 9c). Interestingly, the ball-milled β-sample shows an extrapolated RT σ tot value of 2.4x10 -15 S•cm -1 , nearly similar to that obtained from the high-temperature β-phase of Li 4 Zn(PO 4 ) 2 with however a slightly lower activation energy (Ea = 1.0 eV). We exploited this data further by decoupling the grains and grain boundaries contributions and found activations energies of 1.1 and 1.0 eV, respectively, with corresponding room temperature conductivities of 1.7x10 -14 S•cm -1 (σ g ) and 3x10 -15 S•cm -1 (σ gb ). Overall, this further confirms the better transport properties of the βpolymorph even when prepared at room temperature. Encouraged by these findings, we also explored the transport properties of the quenched Li 4-x Zn 1-

x Ga x (PO 4 ) 2 samples. The results, shown in Figure 10, reveal a steady increase of the ionic conductivities together with a progressive smearing of the jump in conductivity at the α→β'/β transition with increasing the Ga 3+ substitution (x Ga ). More specifically, the change of slope in conductivity at the structural transition gradually decreases till a Ga 3+ content of 0.15 pfu. and then disappears completely with further increase of Ga content. This is consistent with the results obtained from XRD experiments which show that the β'-phase is formed beyond x Ga =0.25 and also from DSC measurements which indicate a shift/smearing of the phase transition with increasing x Ga content. Further insight onto these transport properties of members of the Li 4-x Zn 1-x Ga x (PO 4 ) 2 series can be gained by further analyzing the impedance data with the key values reported in Table 5. Firstly, there is an increase by several orders of magnitude of the σ g room temperature values with increasing the x Ga content so that σ g of ~1-2 x10 -12 S•cm -1 can be obtained for x Ga = 0.05, and 0.10 as compared to σ g = 5.55•10 -18 S•cm -1 for x Ga =0, while structure-wise the samples are still pertaining to the α-phase in the composition range. Such an effect is most likely nested in the increasing amount of disorder that we can spot by XRD from the broadening of Bragg reflections characterizing the α-polymorph (Fig 5b). Moreover, we notice a drastic fall in the E a to 0.8 and 0.9 eV, respectively for the above mentioned compositions. Interestingly, we could observe that σ gb tracks σ g and increases to ~4-8 x10 -14 S•cm -1 at RT, with σ tot reaching ~4-8 x10 -14 S•cm -1 at RT. Upon further increasing the x Ga content beyond 0.05, the room temperature σ g values slightly increase prior to saturate at values of ~6 x10 -12 S•cm -1 (β-phase, Table 5) while structure-wise the compounds evolve from the α _ phase to the β'-phase. Similarly  gb follows the same variation over this composition range while the activation energy remains nearly constant. It is only for samples having higher Ga contents (x Ga > 0.40) and crystalizing in the β-structure that we observe a noticeable decease in σ g to ~2 x10 -12 S•cm -1 and an increase in E a from 0.9 to 1.0 eV. Altogether, these findings fail to establish a direct correlation between structure and conductivity and further analysis is required to interpret this trend. + in the structure, obtained using an iso-surface value of 1.05, 0.7, and 0.9 eV above the minimum energy, respectively. For color codes refer to Figure 4.

To retrieve qualitative information concerning the connection existing between structure and conductivity, BVEL was employed. The BVEL maps representing the Li + conduction pathways in the Li 4-

x Zn 1-x Ga x (PO 4 ) 2 polymorphs are shown in Figure 11. For -Li 4 Zn(PO 4 ) 2 , infinite 2-D Li + percolation paths, involving Li2, Li3 and Li4 sites and not the sterically hindered Li1 site, appear along the b-c plane at 1.05 eV above the minimum energy. This is fully consistent with its layered crystal structure and Li + percolation occurs along the layers, whereas hopping of Li + between layers is not possible because of the Zn hindrance between the Li layers. This explains the high E a and very low σ g of the sample.

For the high temperature β-Li 4 Zn(PO 4 ) 2 and '-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 phases, BVEL maps indicate a 3-D Li + percolation at 0.7 and 0.9 eV above the minimum energy, respectively. This comes in no surprise, because, in β-and β'-phases, the mixing between Li and Zn/Ga atoms on the same crystallographic positions breaks the layered nature of the α-polymorph, for which Li atoms are confined in the b-c planes. Thus in the β-and β'-polymorphs, Li ions can hop from sites to sites three-dimensionally. This is consistent with the lower E a in the β-and β'-sample compared to the α-phase. Moreover, in β'/βstructures that present the same (PO 4 ) 3-framework, Li3 is the only Li site not being mixed with Zn or Ga which accommodates the vacancies resulting from the charge compensation, (Figure 11), and BVEL maps indicate that ionic conduction takes place preferentially through this site. Therefore, this explains the little changes in σ g values when the structure changes from β' to β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 .

Although BVEL could account for the greater conductivity for the β'/β-polymorph as opposed to the α-one, the feasibility to obtain ionic conductivities for the α-polymorph as high as the β-phase at low Ga substitution level is not well understood. First let's recall that ionic conductivity is the result of a delicate balance between vacancies, disorder, and defects among others [START_REF] Bachman | Inorganic Solid-State Electrolytes for Lithium Batteries: Mechanisms and Properties Governing Ion Conduction[END_REF] . Numerous ionic conductors show order-disorder phase transition with the high temperature phase having usually a higher ionic conductivity, owing to its higher structural symmetry which flattens the potential landscape hence leading to reduced activation barrier for Li + jump as well as to its enhanced disorder that increases the number of energy equivalent Li-sites having partial occupancies. [START_REF] Phani Dathar | Li-Ion Site Disorder Driven Superionic Conductivity in Solid Electrolytes: A First-Principles Investigation of β-Li 3 PS 4[END_REF] The Ga-substitutions, besides promoting structural disorder in the samples, introduces Li vacancies. However, the role of the vacancies remains questionable for the above observation as the ionic conductivity is nearly similar for α-Li 3.9 Zn 0.9 Ga 0.1 (PO 4 ) 2 and β'-Li 3.9 Zn 0.9 Ga 0.1 (PO 4 ) 2 . To understand the respective contributions of vacancies and disorder, we plotted the variation of the log(σ g ) as a function of the concentration of Li vacancies and of the broadening of the Bragg peaks characteristic to the α-polymorph as an estimation of disorder in the samples, see Figure S7b. Note that the ionic conductivity drastically increases with increasing the amount of vacancies (x = 0.05) with afterwards another increase from x = 0.15 which seems to be related to an increase in structural disorder. We thus believe that the disorder, as evidenced from the peak broadening which indicates the presence of antiphase boundaries and/or small disordered β'/β domains in the structure, as was also observed in rocksalt-type oxides, [START_REF] Jacquet | The Li 3 Ru y Nb 1-y O 4 (0 ≤y ≤1) System: Structural Diversity and Li Insertion and Extraction Capabilities[END_REF] could trigger β'/β-like conduction even at low Ga-substitution level.

Additionally, the conductivity of a sample depends upon the surface contact of a grain with its neighbors, hence stressing the importance of grain boundaries vs. grain. Although ionic conductivity measurements are often reported as total conductivity ( tot ) omitting grain ( g ) and grain boundary values ( gb ). Nevertheless, a simultaneous increase of (σ b ) and (σ gb ) with increasing the doping content is frequently reported was reported [START_REF] Imashuku | Improvement of Grain-Boundary Conductivity of Trivalent Cation-Doped Barium Zirconate Sintered at 1600°C by Co-Doping Scandium and Yttrium[END_REF][START_REF] Avila-Paredes | Dopant-Concentration Dependence of Grain-Boundary Conductivity in Ceria: A Space-Charge Analysis[END_REF] . Our samples, being synthesized and sintered identically, are expected to have similar grain size and surface contacts. Thus for compounds having the highest conducting grains, the number of charge carriers crossing the grain boundaries will be greater as well, hence providing a possible explanation to the systematic tracking of grain and grain boundary conductivities observed in our samples.

CONCLUSIONS

In summary, we have explored in great details the rich crystal chemistry of Li 4 Zn(PO 4 ) 2 and described how we could improve its ionic conductivity. We showed that this compound undergoes an order-disorder phase transition at ~443 °C, accompanied by a jump in its ionic conductivity. We solved the crystal structure of the high temperature orthorhombic β-polymorph, which appears to be derived from the low temperature monoclinic α-structure by preserving the PO 4 3-backbone, but presenting a disordered Li/Zn sublattice. The β-polymorph was found to have a higher conductivity than the α-phase.

We have demonstrated the feasibility to stabilize the β-polymorph at room temperature either by ball-milling the α phase, or by partial substitution of Zn 2+ by Ga 3+ in Li 4 Zn(PO 4 ) 2 . With 0.5 mol pfu Ga, we could successfully stabilize the pure β-polymorph at room temperature. The evolution of crystal structure from α-Li 4 Zn(PO 4 ) 2 to β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 through an intermediate β'-Li 3.3 Zn 0.7 Ga 0.3 (PO 4 ) 2 phase has been thoroughly characterized. The XRD pattern of Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 , recorded at RT, could be refined with the structural model deduced for β-Li 4 Zn(PO 4 ) 2 phase, with the Zn and Ga ions fully mixed in a disordered Li-sublattice. The β-Li 4-x Zn 1-x Ga x (PO 4 ) 2 samples exhibit ionic conductivities of ~2-5 x10 -13 S•cm -1 at RT, by several orders of magnitudes higher than α-Li 4 Zn(PO 4 ) 2 . The higher crystal symmetry of the disordered β-Li 4 Zn(PO 4 ) 2 enables to switch the conduction mechanism from 2-D to 3-D, thus rationalizes its higher conductivity.

Despite such a rich chemistry of Li 4 Zn(PO 4 ) 2 and its Ga substitutes, needless to mention that this level of ionic conductivity is still far from desired performance to consider for practical applications. However, this report illustrates the role of disorder to ionic conduction and demonstrates how to use chemical substitutions to trigger room temperature stabilization of high temperature disorderedpolymorphs. We hope that these insights will further boost the search of disordered compounds as new ionic conductors beyond the limited library of existing ionic conductors, keeping in mind their limitations to practical implications.

ASSOCIATED CONTENT

Figure 2 .

 2 It consists of corner sharing ZnO 4 and PO 4 tetrahedra forming layers along the b-c plane, which are stacked along [100]. These layers are separated by Li atoms which are also in tetrahedral coordination.

Figure 1 .

 1 Figure 1. Rietveld refinement of the XRD and NPD patterns of α-and β-Li 4 Zn(PO 4 ) 2 , measured at room temperature (a, b) and at 600°C (c, d). The red circles, black continuous line, blue line and green tick bars represent the observed, calculated, and difference patterns, and Bragg positions, respectively.

Figure 2 .

 2 Figure 2. Crystal structure of Li 4 Zn(PO 4 ) 2 ; (a) along [1 ̅ 01], (b) along [001]. Li is shown in yellow; ZnO 4 and PO 4 tetrahedra are colored in blue and green, respectively.

Figure 3 .

 3 Figure 3. (a) DSC curve of Li 4 Zn(PO 4 ) 2 , measured in air with a heating and cooling rate of 10 °C/min. (b) Evolution of the XRD patterns of Li 4 Zn(PO 4 ) 2 between 100 and 600 °C on heating. Values of temperatures corresponding to the patterns are indicated at the right. The patterns colored in blue and red corresponds to the -and -polymorphs, respectively, and the peaks that disappear for the -phase has been indicated by blue backgrounds. (c) Evolution of lattice parameters with respect to temperature. The lattice parameters have been converted to monoclinic unit cell setting pertaining to the α-crystal structure for comparison. The blue and red regions correspond to the -and -phases, respectively.

Figure 4 .

 4 Figure 4. Crystal structure of (a) α-Li 4 Zn(PO 4 ) 2 , (b) β-Li 4 Zn(PO 4 ) 2 and (c) β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 ; in (a), the monoclinic α unit cell is plotted in black and the orthorhombic β cell is shown in red. For sake of clarity, only half of the unit cell content (0 ≤ y ≤ ½ in the monoclinic setting) is shown. Li, Zn and Ga are shown in blue, yellow and pink, respectively and PO 4 tetrahedra are colored in green.

  -Li 4 Zn(PO 4 ) 2 P 2 1 /c b) β-Li 4 Zn(PO 4 ) 2 P n m a c) β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 P 2 1 2 1 2 1

Figure 5 .

 5 Figure 5. (a) DSC curves of Li 4-x Zn 1-x Ga x (PO 4 ) 2 samples, measured in air with a heating and cooling rate of 10 °C/min. Values of x corresponding to the patterns are indicated at the right. (b) XRD patterns of Li 4-x Zn 1-x Ga x (PO 4 ) 2 samples, measured at room temperature. Values of x corresponding to the patterns are indicated at the right.The patterns colored in blue, green and red corresponds to the -, '-, and -polymorphs, respectively, and the peaks that disappear from the -and -phases have been indicated by blue and green backgrounds, respectively. (c) Evolution of lattice parameters with respect to Ga content. The lattice parameters have been converted to monoclinic unit cell setting for comparison. The blue, green, and red regions correspond to the -, '-, and -phases, respectively.

3 . 7

 37 Zn 0.7 Ga 0.3 (PO 4 ) 2 pattern, it appears that all peaks can be indexed with the same orthorhombic cell as β-Li 4 Zn(PO 4 ) 2 with slightly smaller lattice parameters, a = 4.9391(2) Å, b = 6.4803(3) Å, c = 10.1468(5) Å, and V = 324.77(3) Å 3 . However the presence of (

Figure 7 .

 7 Figure 7. Evolution of XRD patterns of Ga-substituted samples: (a) sc-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 , (b) sc-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 , (c) quenched-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 and (d) quenched-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 with temperature. The patterns colored in green and red corresponds to the '-and -polymorphs, respectively, and the peaks that disappear for the -phase has been indicated by green backgrounds.

Figure 8 .

 8 Figure 8. Temperature-Composition phase diagram of Li 4-x Zn 1-x Ga x (PO 4 ) 2 compounds, deduced from temperature-dependent Xray diffraction and DSC experiments (red stars). The black shaded region indicates the presence of secondary phases for Ga contents above 0.5 mol pfu.

Figure 9 .

 9 Figure 9. (a) Representative AC impedance spectra ofLi 4 Zn(PO 4 ) 2 sample recorded at 300 °C, the fitting has been shown by light green line on the curve. The deconvolution to different contributions, as obtained from the fitting, has been shown in the same curve and also zoomed in the inset. Contributions from the grains and grain boundary, has been indicated by green and red semicircles, respectively, and Warburg tail by black line. Some typical frequencies has also been indicated, the highest two of them are the resonance frequencies corresponding to grains and grain boundaries, respectively. Inset shows the equivalent circuit used to fit the spectra. Arrhenius plot of σ tot of Li 4 Zn(PO 4 ) 2 (b) and ball-milled β-Li 4 Zn(PO 4 ) 2 (c); in (b) the blue and red background refers to α-and β-Li 4 Zn(PO 4 ) 2 polymorph, respectively.

2 T

 2 -Li 4 Zn(PO 4 ) (°C)

Figure 10 .

 10 Figure 10. Arrhenius plot ionic conductivity showing σ tot (a), σ g (b) and σ gb (c) of Li 4-x Zn 1-x Ga x (PO 4 ) 2 . Values of x are indicated in (a).

Figure 11 .

 11 Figure 11. BVEL of (a) -Li 4 Zn(PO 4 ) 2 , (b) high temperature β-Li 4 Zn(PO 4 ) 2 and (c) '-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 at the percolation energies. The yellow domains indicates the migration paths for Li+ in the structure, obtained using an iso-surface value of 1.05, 0.7, and 0.9 eV above the minimum energy, respectively. For color codes refer to Figure4.

Table 1 . Structural Parameters for α-Li 4 Zn(PO 4 ) 2 , deduced from the Rietveld refinement of the neutron pattern at room temperature. Isotropic temperature factors (B iso ) were constraint to be equal for a same chemical species. α-Li 4 Zn(PO 4 ) 2 ,

 1

				room temperature
				Space group P 2 1 /c
	a = 8.11990(16) Å, b = 10.24192(13) Å, c = 8.10549(15) Å,  = 104.8589(9)°
				V = 651.538(2) Å 3 , Z = 4
	atom	site	x	y	z	B iso
	Zn1	4e	0 0.5938(8)	0.1567(7)	0.0933(8)	0.72(9)
	P1	4e	0.2125(6)	0.4079(4)	0.4726(7)	0.30(4)
	P2	4e	0.2706(7)	0.5916(4)	0.0353(7)	0.30(4)
	O1	4e	0.1628(7)	0.1505(6)	0.1313(6)	0.535(17)
	O2	4e	0.0620(7)	0.3992(4)	0.3115(7)	0.535(17)
	O3	4e	0.3614(6)	0.3328(6)	0.4378(6)	0.535(17)
	O4	4e	0.2518(6)	0.5549(4)	0.5046(6)	0.535(17)
	O5	4e	0.1230(6)	0.6536(6)	0.0888(7)	0.535(17)
	O6	4e	0.5688(6)	0.1017(4)	0.3127(7)	0.535(17)
	O7	4e	0.6926(6)	0.3308(6)	0.1152(6)	0.535(17)
	O8	4e	0.2249(6)	0.0465(4)	0.4769(6)	0.535(17)
	Li1	4e	0.3484(20)	0.1466(17)	0.341(2)	0.39(10)
	Li2	4e	0.1164(19)	0.3395(18)	0.1121(19)	0.39(10)
	Li3	4e	0.0203(19)	0.0738(9)	0.269(2)	0.39(10)
	Li4	4e	0.150(2)	0.8355(16)	0.157(2)	0.39

Table 2 . Structural Parameters for β-Li 4 Zn(PO 4 ) 2 , deduced from the combined Rietveld refinement of the XRD and neutron patterns recorded at 600 °C. Isotropic temperature factors (B iso ) were constraint to be equal for a same chemical species.

 2

			-Li 4 Zn(PO 4 ) 2 , high temperature form
				Space group P n m a
			a = 10.2974(4) Å, b = 6.5021(3) Å, c = 5.0011(2) Å,
			V = 334.85(2) Å 3 , Z = 2
	atom	site	x	y	z	B iso	Occupancy
	P1	4c	0.40840(15)	3/4	0.8105(3)	1.87(7)	1
	O1	8d	0.3415(2)	0.5553(3)	0.7009(5)	2.77(7)	1
	O2	4c	0.5522(2)	3/4	0.7330(6)	2.55(9)	1
	O4	4c	0.6039(3)	1/4	0.8808(5)	2.63(9)	1
	Li1/Zn1	8d	0.1546(2)	0.4995(3)	0.8074(4)	3.10(8)	0.75/0.25
	Li3	4c	0.9207(17)	1/4	0.780(4)	4.45(8)	0.5

Table 3 . Structural Parameters for β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 deduced from the combined Rietveld refinement of the XRD and neutron patterns recorded at RT.

 3

'-Li 3.

7 Zn 0.7 Ga 0.3 (PO 4 ) 2

					, room temperature
				Space group P 2 1 2 1 2 1
			a = 4.9391(2) Å, b = 6.4803(3) Å, c = 10.1468(5) Å,
				V = 324.77(3) Å 3 , Z = 2
	atom	site	x	y	z	B iso	Occupancy
	P	4a	0.1879(4)	0.0077(6)	0.3419(2)	0.92(5)	1
	O1	4a	0.3045(11)	0.1887(7)	0.4158(4)	1.23(3)	1
	O2	4a	0.2705(6)	0.0131(8)	0.1959(2)	1.23(3)	1
	O3	4a	0.7091(11)	0.2999(8)	0.0953(4)	1.23(3)	1
	O4	4a	0.1229(5)	0.4896(8)	0.1434(3)	1.23(3)	1
	Li1/Ga1	4a	0.1826(15)	0.246(16)	0.5784(5)	0.97(8)	0.85/0.15
	Li2/Zn2	4a	0.3092(8)	0.2531(9)	0.1032(3)	0.97(8)	0.65/0.35
	Li3	4a	0.231(6)	0.532(6)	0.822(3)	2.1(7)	0.35

Table 4 . Structural Parameters for β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 deduced from the combined Rietveld refinement of the XRD and neutron patterns recorded at RT.

 4

-Li

3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 ,

  Rietveld refinement of (a) XRD, and (b) NPD patterns of β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 , measured at room temperature. The red circles, black continuous line, blue line and green tick bars represent the observed, calculated, and difference patterns, and Bragg positions, respectively.

	Figure 6.
					room temperature
				Space group P n m a
		a = 10.08925(3) Å, b = 6.51080(2) Å, c = 4.935878(15) Å,
			V = 324.162( 0.054) Å 3 , Z = 2
	atom	site	x	y	z	B iso	Occupancy
	P1	4c	0.40726(18)	0.75	0.8106(4)	1.16(7)	1
	O1	8d	0.3384(3)	0.5511(3)	0.6966(5)	1.89(5)	1
	O2	4c	0.5531(3)	0.75	0.7319(7)	1.89(5)	1
	O4	4c	0.6072(3)	0.25	0.8766(6)	1.89(5)	0.1
	Li1/Zn1/Ga1	8d	0.1589(2)	0.5013(3)	0.8123(4)	1.37(8)	0.750/0.125/0.125
	Li3	4c	0.940(3)	0.25	0.900(7)	1.37(8)	0.5

Table 5 . Ionic conductivities of the Li 4-x Zn 1-x Ga x (PO 4 ) 2 samples at RT obtained by extrapolation of the Arrhenius plot and at 200 °C and their activation energies (E a ). The blue, green and red background indicates α-, β'-, and β-phases, respectively.

 5

		x Ga	RT	σ g (S•cm -1 ) 200 °C	E a (eV)	RT	σ gb (S•cm -1 ) 200 °C	E a (eV)	RT	σ tot (S•cm -1 ) 200 °C	E a (eV)
		0	5.55•10 -18	2.14•10 -9	1.32 3.49•10 -19	9.54•10 -11	1.30 3.61•10 -19	9.14•10	-11	1.3
		0.05 1.44•10 -12	5.7•10	-7	0.87 4.23•10 -14	7.42•10 -9	0.82 4.10•10 -14	7.33•10	-9	0.82
	α-	0.1 1.86•10 -12	1.34•10	-6	0.92 8.3•10 -14	2.38•10 -8	0.85 8.10•10 -14	2.34•10	-8	0.85
		0.15 6.46•10 -12	3.63•10	-6	0.89 2.65•10 -13	3.13•10 -8	0.79 1.75•10 -13	3.10•10	-8	0.82
		0.2 5.16•10 -12	1.93•10	-6	0.95 3.46•10 -13	6.93•10 -8	0.82 3.39•10 -13	6.69•10	-8	0.83
	β'	0.25 6.0•10 -12 0.3 5.65•10 -12	5.93•10 4.27•10	-6 -6	0.93 4.88•10 -13 0.91 3.93•10 -13	1.13•10 -7 1.40•10 -7	0.84 4.54•10 -13 0.86 3.57•10 -13	1.11•10 1.35•10	-7 -7	0.84 0.87
	β	0.4 5.55•10 -12 0.5 1.63•10 -12	9.18•10 5.94•10	-6 -6	0.95 2.07•10 -13 1.02 1.41•10 -13	8.67•10 -8 1.30•10 -7	0.88 2.37•10 -13 0.93 1.30•10 -13	8.59•10 1.27•10	-8 -7	0.87 0.93

ACKNOWLEDGMENTS

The authors thank Alexis Grimaud for fruitful discussions and for a careful reading of the manuscript, and Virginie Viallet for fruitful discussions. Vladimir Pumjakushin is acknowledged for his help on neutron diffraction experiments on HRPT at SINQ. S.S. thanks the Réseau sur le Stockage Electrochimique de l'Energie (RS2E) for funding of Ph.D.

Supporting Information

The Supporting Information is available free of charge on the ACS Publications website: DC Polarization measurement method; Evolution of XRD patterns of Li 4 Zn(PO 4 ) 2 with temperature during cooling; Rietveld refinement of the XRD pattern of the ball-milled -Li 4 Zn(PO 4 ) 2 sample, measured at RT; Rietveld refinement of the XRD, and NPD patterns of β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 at RT; Crystal structure of β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 , Rietveld refinement of the XRD pattern of sc-Li 3.7 Zn 075 Ga 0.3 (PO 4 ) 2 at RT using the P 2 1 2 1 2 1 unit cell and k 1 and k 2 propagation vectors; Rietveld refinement of the XRD, and NPD patterns of β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 at 600 °C; Structural Parameters for '-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 at 600 °C; Ionic conductivity of -Li 4 Zn(PO 4 ) 2 ; BVEL of β-Li 3.5 Zn 0.5 Ga 0.5 (PO 4 ) 2 at the percolation energies; DC polarization plot and Arrhenius plot of DC conductivity of the β'-Li 3.7 Zn 0.7 Ga 0.3 (PO 4 ) 2 (PDF)

Notes

The authors declare no competing financial interest.

Table of Contents


Jean-Pierre Luminet
email: jean-pierre.luminet@lam.fr

Gravitational Music

Keywords: Black holes, Gravitational waves, Holography, Quantum gravity, Contemporary Music

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Science and Contemporary Music

Modern science upsets our vision of the world and influences the theory and the musical practice on an often-metaphoric mode. A recent, striking illustration was the workshop L'influence des théories scientifiques sur le renouvellement des formes musicales dans la musique contemporaine, held in November 2014 at the Centre de Musique Contemporaine in Paris [3]. As noticed by the organizer of the conference Mártha Grabócz, in the years 1970-1980 major advances in the field of physics, such as the catastrophe theory by R. Thom, the chaos theory by I. Prigogine, the fractal geometry by B. Mandelbrot, the big bang models of cosmology exposed in a best-seller book by S. Weinberg or the fascinating black hole studies popularized by S. Hawking, R. Penrose, J. A. Wheeler or myself, have led many composers to study scientific works and try to transpose them into musical creations [START_REF] Grabócz | Entre naturalisme sonore et synthèse en temps réel: Images et formes expressives dans la musique contemproaine[END_REF].

As I had the privilege to give the opening lecture of the workshop [START_REF] Luminet | Astrophysique et musique contemporaine[END_REF], I began to emphasize the fact that the recent deep-sky observations by large ground-based and space telescopes, as well as by neutrino detectors, gravitational interferometers, particle accelerators, and the astrophysical modeling aimed to interpret such experimental data, revealed that space was no more the harmonious, peaceful and unchanging place dreamt by generations of scholars from Pythagoras to Kepler and Newton, but the arena of discordant, arrhythmic and utterly violent phenomena. Taking this for granted, a number of creations in contemporary music have tried to reflect this new, disturbing picture of cosmic frenzy. I gave several examples of contemporary works that used mathematical and/or astrophysical structures to create new forms of musical language. I had myself the chance to have specific collaborations with leading composers, such as Gérard Grisey for Le Noir de l'Étoile (1990), written for six percussionists, magnetic tape and on-site broadcast of astronomical signals emitted by pulsars [START_REF] Luminet | Musique avec pulsar obligé[END_REF]. Pulsars are rapidly rotating neutron stars, i.e. compact remnants of supernovae explosions that emit electromagnetic radiation in the form of periodic radio pulses. It is technically possible to monitor the pulses and figure out a way to translate them live, into sound, as the genesis of a piece of music. This was done in a striking way in Grisey's masterpiece.

The next speaker at the CDMC workshop was Hèctor Parra. The first time I had met him was in 2011, when he visited me in my office of Paris-Meudon Observatory, and I was delighted to hear that the reading of my book Le destin de l'univers, trous noirs et énergie sombre [START_REF] Luminet | Le Destin de l'Univers : trous noirs et énergie sombre[END_REF] had played a role in the composition of his work of 2011 Caressant l'horizon [START_REF] Besada | Musique, science et crise, quel rapport?[END_REF]. Before this, Hèctor had composed Hypermusic Prologue (2009), a chamber opera written in collaboration with the Harvard physicist Lisa Randall, a specialist of the so-called string theory [START_REF] Parra | Hypermusic Prologue. Òpera projectiva en set plans[END_REF]. Indeed these two pieces emerged from the keen interest of Hèctor (whose father was an engineer, as he told me later) for theoretical physics, and more particularly on the new vision of the world offered by Einstein's theory of general relativity, with such spectacular consequences as space-time curvature, time elasticity, gravity waves, black holes, wormholes or cosmological big bang models. As Hèctor explained [START_REF] Luminet | Caressant l'horizon : entretien entre Jean-Pierre Luminet et le compositeur Hèctor Parra[END_REF], in Caressant l'horizon he tried to transpose the physical feeling one could have by being crossed by strong gravity waves generated by the coalescence of a binary black hole.

From black holes to quantum gravity

Science often creates its own language full of technical words, but also uses terms that belong to the layman vocabulary, although with a very different meaning that may be confusing for the non-informed reader. For instance, the nice and poetic title of Hèctor's piece Caressant l'horizon refers to usual terms such as "caresser" (to caress) and "horizon". However, in the vocabulary of black hole physics, "horizon" refers to the surface of a black hole (more precisely called the event horizon), viewed as a purely geometric border between the external universe and the inner zone of no return. And the seemingly attractive word "caress" refers to the fact that any object doomed to graze the event horizon of a black hole will suffer a violent and destructive gravitational pull, not at all a pleasant loving touch! Twenty-five years ago I myself studied how a star grazing the event horizon of a giant black hole undergoes huge tidal forces and is transformed into a stellar pancake before being fully destroyed in a titanic explosion [START_REF] Carter | Pancake Detonation of Stars by Black Holes in Galactic Nuclei[END_REF]. Our telescopes have since captured such scenes.

Nevertheless, beyond such possible confusions of terms, it is important to feed the numerous gateways between science and art, even when we deal with technical concepts hardly comprehensible to the layman. Hèctor is fascinated by the relationship between the notions of a strongly distorted, curved or "atomized" space-time in the neighborhood of black hole and big bang singularities, and the nature of life itself, taking place in the apparently peaceful oasis of our planet Earth.

Einstein's theory of general relativity provides a consistent picture of the propagation of gravitation. One century ago, he discovered special solutions of the gravitational field equations representing waves of space-time curvature, travelling at the velocity of light. The major difficulty for detecting gravitational waves is their extreme weakness. The best astrophysical generators of gravitational waves are closely coupled pairs of black holes, that radiate enough gravitational energy for its effects to be directly detectable, at least in the last phase of coalescence of the binary system and the merger into a single black hole. The gravitational luminosity from the collision of two stellar mass black holes is expected to be more than the electromagnetic luminosity of the full universe filled with its hundreds of billion galaxies! However, the amplitude of the signal received on Earth is so weak that it is only in September 2015 that two detectors in the United States, achieved the required sensitivity to catch several signals of black hole coalescences, whose waveforms strikingly corresponded to the theoretical calculations previously performed by supercomputers [START_REF][END_REF]. Thus gravitational astronomy was born, opening a new window on the universe and one of the most promising developments of astrophysics for the decades to come.

Another spectacular prediction of general relativity (although not yet checked by astronomical observations) is the possible existence of "wormholes", long a staple of science fiction, as they arise as solutions to Einstein's equations describing the inner structure of rotating black holes. Such "bridges" or "throats" could join two different regions of space-time, that in some cases one is at liberty to interpret also as "parallel universes". In the upper universe, the throat has the appearance of a black hole consuming matter, light and energy; below, it appears as a white fountain expelling matter, light and energy. Thus a wormhole can be thought of as a tunnel with two ends at separate points in space-time, and the tunnel would function as a shortcut offering fantastic possibilities for interstellar travel. Now, when the curvature radius of space-time reaches huge values, as it is the case near the center of a black hole or very close to the big bang, then matter, energy, and the gravitational field must all be quantized, and this requires a comprehensive treatment of new theories called « quantum gravity ».

Quantum mechanics governs the evolution of the microscopic world. It describes local interactions between elementary particles, which are considered to be point-like, within the geometry of a fixed background, namely, the flat space-time of Newtonian mechanics or of special relativity. Throughout the twentieth century, developments in quantum mechanics provided a unified description of electromagnetic and short-range nuclear interactions, leading to the powerful "Standard Model" of particle physics. Its technological successes are innumerable, including lasers, transistors, supercomputers, etc. At the astronomical scale, however, quantum effects play no role at all, and physics is reduced to its classical version. Gravity reigns supreme, and this is described by general relativity, a geometric theory of space-time that models gravity, not as a force acting in a space with a fixed background, but instead as the curvature of space-time, whose value at each point changes when matter is displaced. It too has shown remarkable power in describing astrophysical and cosmological phenomena.

Faced with these two domains of quite different scales and theoretical frameworks-which, through an abuse of the language, are commonly called the "infinitely small" and the "infinitely large"-the physicists may apply either quantum theory or relativity depending on whether they are dealing with small or large-scale phenomena. If this strategy is perfectly appropriate for a practical approach, it is not at all satisfactory for a theoretical one. One wants physics as a whole to be coherent. From this perspective, there must be a theory of a higher order of which quantum mechanics and general relativity are but approximations, valid only in their respective domains. This theory is called quantum gravity.

True comprehension of the Big Bang and of black holes precisely requires a theory of quantum gravity. Given the technical difficulty of the problem, it was necessary to wait until the beginning of the 1990s for physicists to learn how to transform classical theories into quantum theories. For this they needed to develop complex mathematical tools, such as knot theory, gauge group theory, topology, and so on. For a half-century, two main routes to quantifying gravity have been followed.

The first privileges the geometric approach of general relativity, and seeks thus to quantify space and time themselves. The fundamental idea is that there exist "atoms of space", namely entities at a level in which quantum effects radically change the texture of space-time itself, making it granular. The second route privileges the approach of quantum field theory, and thus considers gravitation as a force field between the objects, which it seeks to quantify in terms of an exchange of particles within a passive, exterior space-time. The fundamental ingredient is that the particles are extended objects, not points, with the radical consequence that space has more than three dimensions. These two methods have followed twisted and difficult paths. The formalism of loops for the first approach, and that of strings for the second represent their current state.

In these two cases as well, quantum gravity theories -which are still under constructionput in question the entire foundation of our understanding of the universe. That is to say, they both imply, in very different ways, the fascinating concept of a "multiverse". This term refers to a hypothetical group containing a very large, perhaps infinite, number of "possible" universes, of which ours would just be one individual example. The multiverse-the collection of all the universes in existence-would represent all that exists: space in the fullest sense of the term, time, matter, energy, and all the possible physical laws, together with the fundamental constants of nature which determine them.

Eventually, in some approaches (e.g. string theory), it is hypothesized that all the information about the matter and energy within the three-dimensional volume of a black hole is encoded as a hologram upon its two-dimensional surface, the event horizon. Such an idea, initially applied to black holes only in the hope to solve the so-called information problem, has been studied by hundreds of physicists, with hopes that it could be generalized for any physical system occupying a finite volume of space-time. According to this so-called "holographic principle", all the physics within a N-dimensional volume, including gravitational phenomena, could be entirely described by a simpler quantum theory without gravitation that operates only on the (N-1)-dimensional border of the volume [START_REF] Luminet | The Holographic Universe[END_REF].

The Inscape Project

It is fascinating to see how the most difficult and speculative concepts of modern physics, such as quantum gravity, holography or multiverse models briefly mentioned above, can be transposed into artworks. Of course it is not about strict equivalences, it has more to do with metaphors and analogies.

The work of the artist often translates the fragility of human existence, its extreme limitation in time. When we begin thinking of our existence in a cosmic perspective, we end inevitably in a kind of derisory status of human condition, so limited in space and time. Nevertheless, it is in this so short lifetime that our brain seems to be able to decipher the hidden laws of our immense universe, which normally should overtake us completely. Thus our "cosmic feeling", if it can generate some anxiety or more simply humility, may also provide some pride coming precisely from the fact that, in spite of our short-lived and infinitely modest dimension, we may succeed in achieving some big things.

Physics is precisely such a big thing, but for me another very big thing is the musical composition, that I place over any other kind of artistic creation, because it is the one which seems to me the most elaborate, the most emotional also although the most abstract, and it is probably my training in mathematics and theoretical physics that feeds my fascination for the musical language, that I had the opportunity to acquire very early, in my childhood.

Anyway, the various feelings generated by our own condition of limited human beings extend far beyond our practice as a musician or a physicist. Could "being human" just mean to be moved by the always-deeper knowledge of the surrounding nature, what we are, where we are, what we see? It seems to me that great music can give and amplify a sensation of emotional, physical and intellectual self-fulfillment, at the same time as it can give us the sensation to be overtaken. It is partly on this psychological basis that with Hèctor Parra we undertook to build together a narrative structure as a basis for an orchestral composition entitled Inscape, planned for 2018.

We had a first working session in January 2016, at my home place in Marseille. The musicologist José Luis Besada, a post-doc researcher at IRCAM whose project was to study in vivo the genesis of the work, emphasizing on the cognitive sharing and transfers between the composer and his scientific collaborators, accompanied us.

Hèctor had the idea to transpose into sounds a realistic journey through the mysteries of space-time curved by gravity, taking account of the various related physical phenomena. Another key idea was to create an acoustic illusion in which the audience would have the feeling to be in a hologram. Indeed, using the sophisticated tools of electroacoustic music, we wanted to look for a sound equivalent of a hologram, which is of course primarily visual. As holography assumes that the information contained in a volume can be entirely encoded on its border, we began to think about various ways to create acoustic holograms. This could be achieved for instance by a specific arrangement of the orchestra and space position of the instruments. We could also have musical sequences, which resume a former sequence already played by the orchestra in the full volume of the concert hall, but replayed only on the edges although displaying the same quantity of information.

How to realize that? Hèctor proposed that the acoustic superstructure played on the edge could echo the breath of the public. All the sound information generated by the spectators, their breaths, whispers, even coughing, would be encoded in some way on the edge of the concert hall -Universe. I suggested that it could be also interesting to transpose the idea that the universe can be perceived at three very different scales, from the very small (micro scale) to the very large (macro scale), passing by the intermediate human scale (meso scale), and to make listen how we can acoustically travel through these scales.

After a night of reflection, the next morning Hèctor and I exchanged our ideas, and we noticed that we had, independently one of the other, ended in a common structure that followed a sequence of physical events associated to a gravitational journey through a giant black hole. Here are some elements of the (very tentative) scenario:

Start from the meso scale by hearing some biological activity (human noise, whispers, etc.) with sounds confined within a small volume. Then the music grows and amplifies at the macro-scale to simulate the genesis of a massive star. At this moment an instability occurs and gravitational collapse begins. Acceleration, transposition of relativistic effects such as time dilation and space contraction, emission of gravitational waves and formation of the event horizon of a black hole. Once the black hole formed, astrophysical processes take place around: formation of a turbulent gaseous accretion disk, orthogonal jets of gases at relativistic speeds, slow growth of the black hole by accretion of matter and energy. It becomes supermassive. Destruction of full stars by tidal effects and star-star collisions: flambéed stellar pancakes, accidental supernovae. Approach and collision of two galaxies, each harboring a supermassive black hole. Ballets of their central black holes. Loss of progressive energy by emission of gravitational waves. Acceleration, orbital spiraling. Huge burst of gravitational energy when the two black holes merge into a single one. Plunging into the giant black hole. All the information of the matter and energy, which formed the black hole, is deposited on the event horizon, in the form of bits. Entering the wormhole: passage through an annular singularity, taking out of the wormhole by a white fountain. Birth of a "baby-universe" which a dodecahedral topology of space. Expansion of the new universe, at first slowed down, then accelerated by dark energy. Decreasing of dark energy, general collapse. Atomization of the space, inversion of gravitational collapse and Big Bounce.

Work in Progress

A second working session took place in Paris one year later, in January 2017, at IRCAM, with Thomas Goepfer as the computer music director of the project. We worked in a room equipped with a system of spatialization of sound able to create the illusion of an acoustic hologram. With Thomas and Hèctor, we looked for new electronic treatments in order to transpose acoustically gravity waves and the idea of holography. Eventually Hèctor asked me to draft a text to be recorded in various ways, speeds and tones that would be distorted by electronic treatment. I provided (in French) the following text (in a very approximate English translation).

"The star that was light became dark, silent, fathomless. Black hole, funnel of cold hells. Once crossed its horizon, an endless fall towards a bottomless center. Mixed, inverted, time and space tighten and collide. The first state of the world vaporizes into interlaced elementary grains. What become matter, energy, waves that fall? Are there a bottom, a final stop of fall, and a crushing singularity? Yet an absolute end cannot exist. An inexhaustible blooming is the only possible fruit. At the bottom of the black hole a tunnel now opens, a shortcut leading to somewhere else in the universe, or even in other universes. When any border vanishes in all directions, there is no more remedy to dizziness. New universes mature, as full and delicious as newborn fruits. Our Big Bang is just the moment when occurred a bounce. From then on, the metamorphosis of the many-worlds is faster than we can think. When sprouting out of their quantum matrix, baby-universes are endowed with inconceivable shapes. The ignorant considers them flat, Nature thickens their curvatures. And more and more widely space extends, expanding, always expanding, still and forever beyond. The sky, brimming with dark energy, becomes of a frightening transparency."

From now on, Hèctor Parra is working on the musical composition and I am eager to be in the concert hall for the world creation of Inscape in Barcelona on 19 May 2018, followed in June by performances in Lille and Paris.