Sami Mustapha

Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids

A symmetry-adapted algorithm producing uniformly at random the set of symmetry independent configurations (SICs) in disordered crystalline systems or solid solutions is presented here. Starting from Pólyas formula, the role of the conjugacy classes of the symmetry group in uniform random sampling is shown. SICs can be obtained for all the possible compositions or for a chosen one, and symmetry constraints can be applied. The approach yields the multiplicity of the SICs and allows us to operate configurational statistics in the reduced space of the SICs. The present low-memory demanding implementation is briefly sketched. The probability of finding a given SIC or a subset of SICs is discussed as a function of the number of draws and their precise estimate is given. The method is illustrated by application to a binary series of carbonates and to the binary spinel solid solution Mg(Al,Fe)2O4.

The infrared vibrational spectrum of andradite-grossular solid solutions: A quantum mechanical simulation

Abstract Infrared spectroscopy is a powerful technique for the characterization of minerals, permitting insights into their structural and thermodynamic properties. The intrinsic complexity of mineral solid solutions makes the interpretation of their spectroscopic data a challenging task. In this work, the IR vibrational spectra of andradite-grossular (Ca3Fe2Si3O12-Ca3Al2Si3O12) solid solutions were simulated at the ab initio level with the CRYSTAL09 code by using a large allelectron Gaussian-type basis set and the B3LYP hybrid functional. All the 23 symmetry-independent configurations resulting from the substitution of 1 to 8 Fe atoms with Al atoms in the 16a octahedral site of the andradite primitive cell were considered. The IR active transverse optical frequencies and their intensities were computed. Graphical representation of the spectra, animation of the modes and isotopic substitution of the cations were used as additional interpretation tools. The dominant highfrequency modes, corresponding to Si-O stretching motions, show a simple linear behavior of both frequencies and intensities with respect to the binary composition; this trend is related to the linear behavior of the mean lattice parameter. Also the frequencies of the low-energy bands show, roughly speaking, a linear dependence on composition; however, the behavior of the dominant intensities is more complicated and strongly connected to the Al and Fe atomic fraction. When considering different possible structures at fixed composition, some spectral features display a dependence upon short-range Y cation ordering. Overall, we show how ab initio calculations permit to analyze complex systems such as solid solutions, establishing relations among structure and properties and providing critical and robust interpretations to the experimental findings.

Hydrogrossular, Ca3Al2(SiO4)3–x(H4O4)x: An ab initio investigation of its structural and energetic properties

Abstract Structural and energetic properties of the grossular-katoite solid solution are studied with a full ab initio quantum chemical approach. An all-electron basis set and the hybrid B3LYP functional are used. Calculations are performed within the primitive cell of cubic garnets. The hydrogarnet substitution, SiO4 ↔ H4O4, yields 136 symmetry-independent configurations ranging from triclinic to cubic symmetry. All of them have been structurally optimized, the relaxed geometries being characterized by pseudo-cubic conventional cells. At the present level of approximation, the most stable configurations constitute by far the largest contributions to the system properties. Considering only the most stable configurations, average geometrical features of the actual solid solution are closely approximated. The excess volume displays a highly non-ideal behavior that is favorably compared with carefully analyzed and selected experimental data. The excess enthalpy deviates from the regular model; it draws an asymmetric function of composition with two minima that can be associated to structures or compositions observed in nature. Geometrical variations and distribution of the tetrahedra are analyzed. Calculations provide independent support to the use of a split-atom model for experimental refinements on these compounds. The asymmetry of the enthalpy of mixing can be associated with two distinct distribution patterns of the tetrahedra. Hydrogen interactions also contribute to the asymmetry of the excess enthalpy, as it turns out by comparison between compositions close to fully hydrated katoite and those close to grossular. Hydrogen interactions in Si-free katoite are found to be weak as suggested by dramatic changes in the H environment associated with the introduction of SiO4 tetrahedra.

Sur les transformées de Riesz dans le cas du Laplacien avec drift

We prove Lp estimates for Riesz transforms with drift.

Gaussian estimates for heat kernels on Lie groups

Let G be a connected real Lie group and let [gfr ] be its Lie algebra. We shall denote by [qfr ] ⊂ [gfr ] the radical of [gfr ]. Let [gfr ] = [qfr ] [ltimes ] [sfr ] (where [sfr ] is semisimple or 0) be a Levi decomposition of [gfr ] (cf. [ 11 ]). When [sfr ] ≠ 0 we can apply the Iwasawa decomposition on [sfr ] (cf. [ 8 ]) [sfr ] = [nfr ] [oplus ] [afr ] [oplus ] [kfr ], where [nfr ] is nilpotent and [afr ] is abelian and normalizes [nfr ] so that [nfr ] [oplus ] [afr ] is a soluble algebra. Since [nfr ] [oplus ] [afr ] normalizes [qfr ] it is clear that [rfr ] = [qfr ] [oplus ] [nfr ] [oplus ] [afr ] is a soluble Lie algebra of [gfr ]. By Lies theorem (cf. [ 11 ]) we can find a basis on [rfr ] c = [rfr ] [otimes ] C for which the adjoint action of [rfr ] on [rfr ] c takes a triangular form. Let us denote by λ 1 ( x ); λ 2 ( x ), …, λ n ( x ), x ∈ [rfr ] the corresponding eigenvalues. The λ j s can be identified with elements of Hom ℝ ([rfr ], C ) and are called the roots of the adjoint action of [rfr ]. Let us denote by [Lscr ] = { L 1 , …, L k } the set of the non zero real parts of the λ j s. We shall say that the group G is a B-group if [Lscr ] ≠ &0slash; and if there exist α 1 , …, α k [ges ] 0, [sum ] k j =1 α j = 1, such that [sum ] k j =1 α j L j = 0. Otherwise we say that G is an NB-group. It can be shown that the above definition is independent of the particular choice of the Levi and Iwasawa decompositions that are used (cf. [ 13 ]). We shall denote by d l g = dg (resp. d r g ) the left (resp. right) Haar measure on G and by m ( g ) = d r g / d l g the modular function. Let [Xscr ] = { X 1 , X 2 , …, X n } be left invariant fields on G that verify the Hormander condition (cf. [ 15 ]) and let Δ = −[sum ] X 2 j be the corresponding sub-Laplacian. Δ is formally self adjoint on the Hilbert space L 2 ( G , d r g ) and the spectral gap of Δ is defined by formula here