Index theory for quasi-crystals I. Computation of the gap-label group

In this paper, we give a complete solution to the gap labelling conjecture for quasi-crystals. The method adopted relies on the index theory for laminations, and the main tools are the Connes-Skandalis longitudinal K-theory index morphism together with the measured index formula.     

Combined operando studies of new electrode materials for Li-ion batteries

The performances of Li-ion batteries depend on many factors amongst which the important ones are the electrode materials and their structural and electronic evolution upon cycling. For a better understanding of lithium reactivity mechanism of many materials the combination of X-Ray Powder Diffraction (XRPD) and Transmission Mössbauer Spectroscopy (TMS) providing both structural and electronic information during the electrochemical cycling has been carried out. Thanks to the design of a specific electrochemical cell, derived from a conventional Swagelock cell, such measurements have been realised in operando mode. Two examples illustrate the greatness of combining XRPD and TMS for the study of LiFe_0.75Mn_0.25PO₄ as positive electrode and TiSnSb as negative electrode. Different kinds of insertion or conversion reactions have been identified leading to a better optimization and design of performing electrodes.     

Differential characters in K-theory

We define the new notion of R/Z-differential K-characters and study some properties. In particular, we show that the spectral eta invariant is an R/Z-secondary invariant in this theory.     

Index Theory and Non-Commutative Geometry I. Higher Families Index Theory

We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a k-oriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a Chern–Connes character from the k-theory of the compactly supported smooth functions on the holonomy groupoid of the foliation to the Haefliger cohomology of the foliation. Combining these with the Connes–Skandalis topological index map and the classical Chern character gives a commutative diagram from which the index theorem follows immediately.     

A higher Lefschetz formula for flat bundles

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.

     

How M?ssbauer spectroscopy can improve Li-ion batteries

In the domain of Li-ion batteries, M?ssbauer spectroscopy is mainly used for the characterization of electrode materials and the analysis of electrochemical reactions. Depending on the properties under investigation, different approaches are often considered, which are based on ex situ, in situ and operando measurements. The specific electrochemical cells and sample preparations used for such measurements are described in this paper. Applications to selected examples of cathode and anode materials are presented in order to show how M?ssbauer spectroscopy, when associated with other techniques, provides essential information to understand the mechanisms and improves the performances of Li-ion batteries.     

Calcul du label des gaps pour les quasi-cristauxComputation of the gap-labelling for quasi-crystals

We give in the present Note a proof of the Bellissard gap-labelling conjecture for quasi-crystals. Our main tools are the measured index theorem for laminations on the one hand, and the naturality of the longitudinal Chern character on the other hand. To cite this article: M.-T. Benameur, H. Oyono-Oyono, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 667–670.     

Role of iron in Na <Subscript>1.5</Subscript>Fe <Subscript>0.5</Subscript>Ti <Subscript>1.5</Subscript>(PO <Subscript>4</Subscript>) <Subscript>3</Subscript>/C as electrode material for Na-ion batteries studied by operando Mössbauer spectroscopy

The role of iron in Na _1.5Fe _0.5Ti _1.5(PO ₄)₃/C electrode material for Na batteries has been studied by ⁵⁷Fe Mössbauer spectroscopy in operando mode. The potential profile obtained in the galvanostatic regime shows three plateaus at different voltages due to different reaction mechanisms. Two of them, at 2.2 and 0.3 V vs Na ⁺/Na ⁰, have been associated to redox processes involving iron and titanium in Na _1.5Fe _0.5Ti _1.5(PO ₄)₃. The role of titanium was previously elucidated for NaTi ₂(PO ₄)₃ and the effect of the substitution of Fe for Ti was investigated with ⁵⁷Fe Mössbauer spectroscopy. We show that iron is an electrochemically active center at 2.2 V with the reversible Fe ³⁺/Fe ²⁺ transformation and then remains at the oxidation state Fe ²⁺ along the sodiation until the end of discharge at 0 V.     

Quantum-Chemical Study of the FeNCN Conversion-Reaction Mechanism in Lithium- and Sodium-Ion Batteries

We report a computational study on 3d transition-metal (Cr, Mn, Fe, and Co) carbodiimides in Li- and Na-ion batteries. The obtained cell voltages semi-quantitatively fit the experiments, highlighting the practicality of PBE+U as an approach for modeling the conversion-reaction mechanism of the FeNCN archetype with lithium and sodium. Also, the calculated voltage profiles agree satisfactorily with experiment both for full (Li-ion battery) and partial (Na-ion battery) discharge, even though experimental atomistic knowledge is missing up to now. Moreover, we rationalize the structural preference of intermediate ternaries and their characteristic lowering in the voltage profile using chemical-bonding and Mulliken-charge analysis. The formation of such ternary intermediates for the lithiation of FeNCN and the contribution of at least one ternary intermediate is also confirmed experimentally. This theoretical approach, aided by experimental findings, supports the atomistic exploration of electrode materials governed by conversion reactions.