Sur un nouvel hétéropolytungstate lacunaire à squelette X2W21 dérivé du polytungstate [H6X2W22O76] (X = Sb,Bi)

A novel vacant heteropolytungstate of skeleton X₂W₂₁ derived from the polytungstate [H₆X₂W₂₂O₇₆]^8– (X = Sb,Bi). Five heteropolytungstates Na₈[H₆Sb₂W₂₁O₇₃]·16.5 H₂O, Na₈[H₆Sb₂W₂₁O₇₃]·19 H₂O, Na₈[H₆Bi₂W₂₁O₇₃]·16.5 H₂O, Li₂Na₆[H₆Sb₂W₂₂O₇₆]·14.5 H₂O, and Na₅[H₇Sb₂NiW₂₁O₇₃(H₂O)₃]·20 H₂O were prepared and studied by X-ray diffraction. The values of isotropic temperature factors of the external tungsten atom and of its three terminal oxygen atoms were compared to those of same-type atoms, firstly of the same polytungstate, and secondly inside this five-compound family. This analysis led to the conclusion that the vacant assembly X₂W₂₁O₇₃ (X = Sb,Bi) does exist and that the vacancy may be filled up with a first row transition metal such as nickel. To cite this article: Y. Jeannin, C. R. Chimie 7 (2004).     

Infinite groups with large balls of torsion elements and small entropy

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. Received: 3 August 2005     

Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface

In one space dimension we address the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficient is scaled as ε². The domain is made of two purely periodic media separated by an interface. Depending on the connection between the two cell spectral equations, three different situations arise when ε goes to zero. First, there is a global homogenized problem as in the case without an interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition on the interface. Third, there is an exponential localization near the interface of the first eigenfunction. Received: January 10, 2001; in final form: July 9, 2001?Published online: June 11, 2002     

Loudness reduction induced by a contralateral tone

The induced reduction in the loudness (ILR) of a weaker tone caused by a preceding stronger tone was measured with both tones in the same ear (ipsilateral ILR) and also in opposite ears (contralateral ILR). The two tones were always equal in duration and were presented repeatedly over several minutes. When the tone duration was 200 ms, for 24 listeners the loudness reduction averaged 11 dB under ipsilateral ILR and 6 dB under contralateral ILR. When the duration was 5 ms, ILR was 8 dB whether ipsilateral or contralateral. For each duration, ipsilateral and contralateral ILR were strongly correlated (r around 0.80).     

An improved multimodal method for sound propagation in nonuniform lined ducts

An efficient method is proposed for modeling time harmonic acoustic propagation in a nonuniform lined duct without flow. The lining impedance is axially segmented uniform, but varies circumferentially. The sound pressure is expanded in term of rigid duct modes and an additional function that carries the information about the impedance boundary. The rigid duct modes and the additional function are known a priori so that calculations of the true liner modes, which are difficult, are avoided. By matching the pressure and axial velocity at the interface between different uniform segments, scattering matrices are obtained for each individual segment; these are then combined to construct a global scattering matrix for multiple segments. The present method is an improvement of the multimodal propagation method, developed in a previous paper [Bi et al., J. Sound Vib. 289, 1091-1111 (2006)]. The radial rate of convergence is improved from O(n(-2)), where n is the radial mode indices, to O(n(-4)). It is numerically shown that using the present method, acoustic propagation in the nonuniform lined intake of an aeroengine can be calculated by a personal computer for dimensionless frequency K up to 80, approaching the third blade passing frequency of turbofan noise.     

Rainbow refractrometry on particles with radial refractive index gradients

The rainbow refractrometry, under its different configurations (classical and global), is an attractive technique to extract information from droplets in evaporation such as diameter and temperature. Recently a new processing strategy has been developed which increases dramatically the size and refractive index measurements accuracy for homogeneous droplets. Nevertheless, for mono component as well as for multicomponent droplets, the presence of temperature and/or of concentration gradients induce the presence of a gradient of refractive index which affects the interpretation of the recorded signals. In this publication, the effect of radial gradient on rainbow measurements with a high accuracy never reached previously is quantified.     

Diselenophosphate‐Induced Conversion of an Achiral [Cu20H11{S2P(OiPr)2}9] into a Chiral [Cu20H11{Se2P(OiPr)2}9] Polyhydrido Nanocluster

A polyhydrido copper nanocluster, [Cu₂₀H₁₁{Se₂P(OiPr)₂}₉] ( 2_H ), which exhibits an intrinsically chiral inorganic core of C₃ symmetry, was synthesized from achiral [Cu₂₀H₁₁{S₂P(OiPr)₂}₉] ( 1_H ) of C_3h symmetry by a ligand‐exchange method. The structure has a distorted cuboctahedral Cu₁₃ core, two triangular faces of which are capped along the C₃ axis, one by a Cu₆ cupola and the other by a single Cu atom. The Cu₂₀ framework is further stabilized by 9 diselenophosphate and 11 hydride ligands. The number of hydride, phosphorus, and selenium resonances and their splitting patterns in multinuclear NMR spectra of 2_H indicate that the chiral Cu₂₀H₁₁ core retains its C₃ symmetry in solution. The 11 hydride ligands were located by neutron diffraction experiments and shown to be capping μ₃‐H and interstitial μ₅‐H ligands (in square‐pyramidal and trigonal‐bipyramidal cavities), as supported by DFT calculations on [Cu₂₀H₁₁(Se₂PH₂)₉] ( 2_H′ ) as a simplified model.     

[Cu32(H)20{S2P(OiPr)2}12]: The Largest Number of Hydrides Recorded in a Molecular Nanocluster by Neutron Diffraction

An air‐ and moisture‐stable nanoscale polyhydrido copper cluster [Cu₃₂(H)₂₀{S₂P(OiPr)₂}₁₂] ( 1_H ) was synthesized and structurally characterized. The molecular structure of 1_H exhibits a hexacapped pseudo‐rhombohedral core of 14 Cu atoms sandwiched between two nestlike triangular cupola fragments of (2×9) Cu atoms in an elongated triangular gyrobicupola polyhedron. The discrete Cu₃₂ cluster is stabilized by 12 dithiophosphate ligands and a record number of 20 hydride ligands, which were found by high‐resolution neutron diffraction to exhibit tri‐, tetra‐, and pentacoordinated hydrides in capping and interstitial modes. This result was further supported by a density functional theory investigation on the simplified model [Cu₃₂(H)₂₀(S₂PH₂)₁₂].