Salvino Ciccariello

Supercritical carbon dioxide behavior in porous silica aerogel

Analysis of the tails of the small-angle neutron scattering (SANS) intensities relevant to samples formed by porous silica and carbon dioxide at pressures ranging from 0 to 20 MPa and at temperatures of 308 and 353 K confirms that the CO2 fluid must be treated as a two-phase system. The first of these phases is formed by the fluid closer to the silica wall than a suitable distance δ and the second by the fluid external to this shell. The sample scattering-length densities and shell thicknesses are determined by the Porod invariants and the oscillations observed in the Porod plots of the SANS intensities. The resulting matter densities of the shell regions (thickness 15–35 A) are approximately equal, while those of the outer regions increase with pressure and become equal to the bulk CO2 at the higher pressures only in the low-temperature case.

Small-angle scattering from three-phase samples: application to coal undergoing an extraction process

The correlation functions of samples made up of three homogeneous phases, with a fixed geometrical configuration and different scattering densities, are linearly related among themselves. The same is true of the corresponding scattering intensities. The property holds approximately true if the phase boundaries of some of the previous samples are slightly modified. The analytical expressions of the coefficients involved in the relevant linear combinations are derived and applied to obtain the volume fractions and the scattering densities of the three phases of a coal from its small-angle scattering intensities collected during an extraction process.

Oscillatory deviations from Porod's law in the intensities scattered by some glasses

The oscillatory deviations with respect to the Porod asymptote, observed in the small-angle X-ray intensities scattered by some selenium ruby glasses, are explained in terms of a finite discontinuity of the second-order derivative of the correlation function. The best fit of the intensity allows the value of the discontinuity to be determined as well as the point at which it occurs. The physical implications of these results are discussed.

The Lerch function and the thermodynamical functions of the ideal quantum gases

The unified description of the main thermodynamical functions of the Bose and Fermi ideal gases, obtained by Lee [J. Math. Phys. 36, 1217 (1995)] in terms of the polylogarithmic functions, can also be obtained by analytic continuation in the chemical potential owing to the analytic properties of the Lerch function that is simply related to the polylogarithmic ones. By this procedure we also show that the Fourier coefficients of the thermal Green function of the ideal Bose gas convert into those of the Fermi one.

Small-angle scattering intensity behaviours of cylindrical, spherical and planar lamellae

Analytical asymptotic expressions for the small-angle scattering intensities of cylindrical, spherical and planar lamellar grains are determined. Denoting the lamellar spacing by D and the number of lamellae by N, it is found that in the corresponding Porod plots, the positions of the main peaks, whatever the shape, are nearly given by 2k\pi/D, where k is a positive integer. At a fixed number of lamellar grains, the heights of the main peaks in the three cases increase with N as N3, N4 and N2, respectively. The satellite peaks are much more structured for cylindrical and spherical lamellae than for planar ones. The momentum-transfer (h) range in which the asymptotic expressions turn out to be accurate is h\ge 2\pi/ND.

In Situ Small-Angle X-ray Scattering Observations of Pt/NaY Catalysts During Processing: Sintering of Pt

Small-angle X-ray scattering observations on Pt/NaY catalysts, made in situ during calcination and reduction stages of processing, demonstrate the usefulness of this technique in following morphological changes. Observations show that the same platinum species (Pt 0 under the preparation conditions used) is present in the early stages of calcination, carried out at relatively high heating rates, as after reduction, and that the ultimate dispersity of the metal is already reached within 0.5 h of the start of calcination. Increasing aggregation of metal particles occurs at calcination temperatures higher than 573 K, leading to average particle sizes too large to fit the supercages of the zeolite framework. With the assumption that the metal is a Maxwellian distribution of spheres, values of the distribution parameters giving the best fit to the scattering for each catalyst sample are found ; from these parameters, average particle radii are calculated.

The correlation functions of plane polygons

The correlation function of a bounded plane figure of arbitrary shape and its derivatives are studied using their integral expressions. From these follows that the correlation function and its first derivative are continuous functions, while the second and the higher order derivatives can show finite and/or algebraic singularities [with exponents equal to −(n+1/2) and n∊Z¯+] in the presence of some well defined geometrical features of the figure boundary. The limit values of the first, second, and third derivatives at the origin are obtained. They are similar to those of the three dimensional case. In the case of a plane polygon of arbitrary shape the correlation function has a closed analytical form essentially equal to the sum of the values taken by two analytically known functions at appropriate sets of points. Two simple cases are explicitly worked out for illustration.

Complete subsets of a diffraction pattern

A positive atomic density ρ(r) = ∑j = 1Nnjδ(r-rj) in a D-dimensional space can be exactly reconstructed from an appropriate finite subset (complete set) of its Fourier series coefficients {Uh}hD or even (limited to the support {rj}j = 1,...,N) from a finite subset of moduli |Uh|. It is necessary first to determine a complete set of Fourier coefficients Uh (possibly inside the unavoidable high-resolution cut-off ||h||<L) and then, by these coefficients, to determine the unknown density. We report some procedures which are able to single out complete sets. They are based on a property of Goedkoops vector lattice {|Ah}hD, defined so that Ah|Ak = Uk-h. The property states that if the vector with index h = (h1,...,hD) is a linear combination of the vectors relevant to a set of indices with a particular shape, then all the vectors relevant to the hyperquadrant h≡{h | hα≥hα, α = 1,...,D} are linear combinations of the vectors relevant to 0}h. Moreover, the determination of ρ from a complete set passes through the solution of a system of polynomial equations in D variables, whose roots determine the position vectors. We show how to convert this problem into the simpler problem of sequentially solving a set of polynomial equations in one variable.

Polydisperse analysis of small‐angle intensities scattered by natural coals

The small‐angle x‐ray intensities, scattered by four of the most typical natural coals, are analyzed assuming that coals behave as polydisperse distributions of cubic particles having a layered structure with an interlayer spacing partly dependent on particle sizes. The particle distributions are determined by a rather fast numerical algorithm, and in the considered momentum‐transfer range, they appear independent of the interlayer spacing. The model reproduces quite well only the intensities which are neither particularly structured nor fractal at very small momentum transfers. Micropore, mesopore, and macropore relative sizes are estimated starting from the obtained particle distributions.

Generalization of a theorem of Caratheodory

Caratheodory stated the conditions to be obeyed by n complex numbers c1, ..., cn in order that they can be written uniquely in the form cp = ∑mj=1ρjjp with p = 1, ..., n, js being different unimodular complex numbers and ρjs strictly positive numbers. We give the conditions to be obeyed for the former property to hold true if ρjs are simply required to be real and different from zero. The number of the possible choices of the signs of ρjs are at most equal to the number of the distinct eigenvalues of the Hermitian Toeplitz matrix whose (i, j) th entry is cj−i, where c−p by definition is the complex conjugate of cp and c0 = 0. This generalization is relevant to neutron scattering. Its proof is made possible by a lemma stating the necessary and sufficient conditions to be obeyed by the coefficients of a polynomial equation for all the roots to lie on the unit circle. This lemma is an interesting side result of our analysis.