Non-consistent approximations of self-adjoint eigenproblems: application to the supercell method

In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution. We then show that the supercell method for perturbed periodic Schrödinger operators falls into the scope of our study. We prove that this method is spectral pollution free, and we derive optimal convergence rates for the planewave discretization method, taking numerical integration errors into account. Some numerical illustrations are provided.     

Spectroscopic and ab initio characterization of the [ReH9]2- ion

The dynamics and bonding of the hydrido complex Ba[ReH9], containing the D3h face-capped trigonal prismatic [ReH9]2- ion, have been investigated by vibrational spectroscopy and density functional theory (DFT). The combination of infrared, Raman, and inelastic neutron-scattering (INS) spectroscopies has enabled observation of all the modes of the [ReH9]2- ion for the first time. We demonstrate that calculations of the isolated [ReH9]2- ion are unable to reproduce the INS spectrum and that the complete unit cell must be considered with periodic DFT to have reliable results. This is shown to be a consequence of the long-range Coulomb potential present. Analysis of the electronic structure shows that the bonding between the rhenium and the hydrogen is largely covalent. There is a small degree of covalency between the prism hydrides and the barium. The counterion is crucial to the stability of the materials; hence, variation of it potentially offers a method to fine-tune the properties of the material.     

Une méthode combinée d'éléments finis à deux grilles/bases réduites pour l'approximation des solutions d'une E.D.P. paramétrique

In the frame of mathematical optimization procedures or parameter fitting the same problem, modeled with partial differential equations depending on a parameter has to be solved many times for different sets of parameters. The reduced basis method may be successful in this frame and recent progress have permitted to make the computations reliable thanks to a posteriori estimators and to extend the method to non linear problems thanks to the “magic points” interpolation. However, in an industrial context, it may not be possible to use the code (for example of finite element type that allows for evaluating the elements of the reduced basis) to perform all the “off-line” computations necessary for an efficient performance of the reduced basis method. We propose here an alternating approach based on a coarse grid finite element the convergence of which is accelerated through the reduced basis. To cite this article: R. Chakir, Y. Maday, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Tetragonal rare earth (R) Iron borides,R1+εFe4B4 (ε≅0.1), with incommensurate rare earth and iron substructures

Crystallographic and magnetic properties of a new structural series of ternary borides with composition R_1+εFe₄B₄ (R = Ce, Pr, Nd, Sm, Gd, Tb, 0.11(Pr) ≤ε≤ 0.15(Tb) are reported. The compounds are built of incommensurate substructures of rare earth atoms (linear strings ?? c?), iron atoms (chains of edge sharing tetrahedra ?? c?),and boron atom pairs. A single crystal X-ray diffraction study of one representative (Sm_1.13Fe₄B₄) based on a commensurate structure model (composition : Sm₁₇(Fe₄B₄)₁₅, a = 7.07 Å, c ≡ 17c_Sm ≡ 15c_Fe= 58.69 Å, space group P4₂/n) revealed a periodic twist modulation of the Fe tetrahedra chains around c?. Magnetic susceptibility measurements on single crystals of another representative (Nd_1.11Fe₄B₄) revealed ferromagnetic ordering at T_c = 13 K. Above this temperature the magnetic properties are dominated by ferromagnetic inclusions (Fe₂B, Nd₂Fe₁₄B).     

Observations of the intestinal mucosa using environmental scanning electron microscopy (ESEM); comparison with conventional scanning electron microscopy (CSEM)

In order to evaluate the potential use of environmental scanning electron microscopy (ESEM) in biology, structural changes of the jejunal villi of rats were studied after periods of fasting and refeeding, using a conventional scanning electron microscope (CSEM) and ESEM. While observation using the CSEM, involves chemical fixation, drying and coating, observation of fresh, unprepared materials can be directly realized with the ESEM. Environmental microscopy provides a relatively new technology for imaging hydrated materials without specimen preparation and conductive coating. Direct observation of biological samples in their native state is therefore possible with an ESEM.

After fasting, the jejunal mucosa is dramatically reduced in size, splits and holes appearing at the tip of the villi. These changes were observed whatever the type of technique used. Artifacts due to the sample preparation for CSEM observation (drying, coating) can therefore be excluded. However, CSEM and ESEM must be used jointly. While, CSEM must be preferred for surface analysis involving high magnifications, ESEM observation, on the other hand, can prove valuable for determining the living aspect of the samples.     

Charge transfer and valence electron concentration in Chevrel phases

Single crystal X-ray diffraction measurements on M_xMo₃S₄ compounds (M = Cu, Ag, Sn, In) yield evidence for the existence of a charge transfer from the interstitial element M to the Mo₆S₈ building blocks. The importance of this transfer was estimated from the contraction of the Mo₆ octahedron as a function of the valency and the concentration of the cations. It is suggested that this effect stabilises the structure and has a strong influence on the superconducting transition temperature. Bonding is discussed using a simple valence bond model which is based on the electron requirement of the Mo₆ octahedron to form 12 covalent bonds.