Parameterizations of scattering intensities and values of the angularities and of the interphase surfaces for three-component amorphous samples

Ciccariello, S.; Benedetti, Alvise

doi:10.1107/s0021889885010184

Cited by 28 publications

(36 citation statements)

References 13 publications

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“…It was first introduced by Porod (1965) and measured by Tchoubar & M6ring (1969). Subsequently, it has been thoroughly discussed and used for analysing the behaviour of some catalysts by Ciccariello & Benedetti (1985). Mathematically, the angularity can be defined as the t In passing we note that for hyperbolic and parabolic contact points (28) cannot be applied and this is clearly signalled by the fact that it becomes meaningless, since ~(o(r) is negative or null in the two cases.…”

Section: Vf['y~( T+)-tt( T-)]mentioning

confidence: 99%

Small-angle techniques for the asymptotic analysis of X-ray diffraction peaks

Ciccariello¹

1990

Acta Cryst Sect A

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Any wide-angle X-ray scattering (WAXS) peak, relevant to a powder sample of crystallites with negligible internal disorder, is the Fourier transform of the so-called oriented stick probability function (oSPF) of the filled part of the sample, with the stick orientated along the reflexion direction. From this observation the following consequences are obtained: the correlation function used in small-angle X-ray scattering (SAXS) is the average of the former oSPF's over all possible stick orientations; any peak profile asymptotically vanishes as Sr h-2, where Sr is the (specific) area of the interphase surface presented by the sample along the reflexion direction; oscillatory deviations, behaving as &,ll cos (hL)h -2, are present only when a subset (having area &,ll) of the interface, after having been translated by L along the reflexion direction, superposes on itself; the angularity of the interphase surface can be measured by a natural modification of the Porod integral relation. For samples really isotropic, the above quantities should not depend on the reflexion direction and thus they should be equal to those measured by SAXS experiments. These results are applied to three ideal isotropic powder samples made up, respectively, of monodisperse spherical, cubic and cylindrical crystallites as well as to the analysis of two WAXS peaks diffracted by two real samples of zirconia powders.

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Section: Vf['y~( T+)-tt( T-)]mentioning

confidence: 99%

Small-angle techniques for the asymptotic analysis of X-ray diffraction peaks

Ciccariello¹

1990

Acta Cryst Sect A

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“…Equations (49) and (50) are mathematically equivalent to (8) and can be also presented in the form of (10). Since 49) and (50) show that there are p(p-1)/2 independent () Ŝ functions which is exactly the number of the linearly independent "stick probability functions" [39,41]. We should also 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 …”

Section: Multiphase Systemsmentioning

confidence: 99%

“…In the later case [38,41,50], the most frequently used model involves exponential correlation function , but Gaussian and other functions were also used. The studies went in this direction, since it is not possible to determine by SAXS the phase surface areas in system with more than two phases without assuming a structure model.…”

Section: Deletedmentioning

confidence: 99%

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Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Tatchev¹

2008

Philosophical Magazine

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“…The interphase surface areas obey (Goodisman & Brumberger, 1971) Sij/4 V = (dPit/dr)r = o (6) while the second derivative of Pv at r = 0 is related (Ciccariello, Cocco, Benedetti & Enzo, 1981) to the angularity of the corresponding surface. It is also useful to write (Ciccariello & Benedetti, 1985) ~(r)= E ~ijQij …”

Section: ~' Pij(r)(ni-nj) 2 V (5) Y(r) = 1 --~ Q~iq~j(n I __ Nj)2 Umentioning

confidence: 99%

Small-angle X-ray scattering analysis of catalysts: comparison and evaluation of models

Brumberger¹,

Delaglio²,

Goodisman³

et al. 1986

J Appl Cryst

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Small-angle X-ray scattering (SAXS) can be used to obtain interphase surface areas of a system, such as a supported-metal catalyst, composed of internally homogeneous phases with sharp interphase boundaries. Measurements of SAXS for samples of porous silica, alumina, platinum on silica, and platinum on alumina are reported. A variety of models and forms for the correlation function, the Fourier transform of which gives the X-ray scattering, are considered, and theoretical and measured intensities are compared. A criterion of fit for comparing models with different numbers of parameters is proposed. For the two-phase (unmetallized) systems the 'Debye-random' model must be rejected. Modifications of the Debye (exponential) correlation function are also not particularly good compared to an exponential-plus-Gaussian form, not derivable from a physical model, and forms based on Voronoi cell models. Since intensities can be fit to experimental error with a five-parameter correlation function, it seems incorrect to ascribe significance to the result of fitting a function with six or more parameters. It is shown that values for the single interphase surface area can be obtained independently of a model. However, fitting intensities using a modelbased correlation function gives information about the structure of the system. The two-cell-size Voronoi and the correlated Voronoi cell models are useful in this regard. For the systems containing metal, fiveparameter correlation functions again suffice to fit intensities. However, for three-phase systems a model or physical assumption is necessary to obtain values for the three surface areas from X-ray scattering intensities. The area of the surface between support and void is quite insensitive to the assumptions employed and the metal-support surface area somewhat less so, but values for the metal-void surface area $23 are consistent only to one significant figure from model to model. If the support in the three-phase catalyst is known to be unchanged from support in the absence of metal, a 'support-subtraction' model can be used to obtain reliable values for $23. In the present systems, the assumption does not seem to be borne out.

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Parameterizations of scattering intensities and values of the angularities and of the interphase surfaces for three-component amorphous samples

Cited by 28 publications

References 13 publications

Small-angle techniques for the asymptotic analysis of X-ray diffraction peaks

Small-angle techniques for the asymptotic analysis of X-ray diffraction peaks

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Small-angle X-ray scattering analysis of catalysts: comparison and evaluation of models

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