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Table Of Integrals Series And Products

Interpretation of X-ray Powder Diffraction Patterns . By H. Lipson and H. Steeple. Pp. viii + 335 + 3 plates. (Mac-millan: London; St Martins Press: New York, May 1970.) £4.

X-Ray Diffraction

Small Angle Scattering of X-Rays

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Chiroptical Molecular Switches

Small angle X-ray scattering (SAXS) is an increasingly common technique for low-resolution structural characterization of molecules in solution. SAXS experiment determines the scattering intensity of a molecule as a function of spatial frequency, termed SAXS profile. SAXS profiles can contribute to many applications, such as comparing a conformation in solution with the corresponding X-ray structure, modeling a flexible or multi-modular protein, and assembling a macromolecular complex from its subunits. These applications require rapid computation of a SAXS profile from a molecular structure. FoXS (Fast X-Ray Scattering) is a rapid method for computing a SAXS profile of a given structure and for matching of the computed and experimental profiles. Here, we describe the interface and capabilities of the FoXS web server (http://salilab.org/foxs).

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FoXS: a web server for rapid computation and fitting of SAXS profiles

The Porod law states that i(h), the intensity of X-radiation scattered by an ideal multiphase noncrystalline system, for `large' momentum transfer values (≡ h) approaches lh−4 and that l is linearly related to the interphase surface areas. A more general expression is obtained which relates the value of the correlation-function derivative at the origin to the integral of the discontinuity of the electron density fluctuation along the discontinuity surface. Debye's assumption, by which the continuous electron density of the sample is approximated by a discrete-valued one, is critically discussed. The validity of the approximation results in the presence of a Porod plateau [h4i(h) = constant]. The momentum transfer range where the plateau is observed is related to the scale of lengths where the sharp-boundary idealization remains essentially unchanged. It is argued that the scattered intensity can show more than one Porod plateau and some examples of correlation functions with this behaviour are reported. The problem of the background subtraction is discussed and the corresponding coefficients are related to the electron densities relevant to the real and to the associated sharp-boundary sample.

On the Porod law

We give two general integral expressions for the first and second derivatives of the so-called stick probability functions which are commonly used in analyzing x-ray-scattering results. Then it is shown that by letting the length of the stick go to zero, the limit of the second derivative can be expressed in terms of an integral over the singularity lines of the surfaces which separate the different phases of the sample. In this way one has achieved the generalization of the well-known result that the limit of the second derivative is always zero when phase boundaries are smooth.

Correlation functions of amorphous multiphase systems

It is proved that the second-order derivative of the correlation function, relevant to an amorphous N-component sample, can have a first-order discontinuity at a point r0 (≠ 0) only when the interfaces have the following geometrical property: there exists a finite-area subset of one phase boundary, say Si, such that any point P1 of this is far away, r0, from a point P2, belonging to another boundary, say Sj, and moreover the segment P1P2 is orthogonal both to Si and to Sj. The explicit integral expression of the discontinuity is obtained. The relevance of this result to the analysis of scattered intensities showing a systematic deviation from the Porod-Debye law is pointed out.

Deviations from the Porod law due to parallel equidistant interfaces

Different functional parameterizations of the radiation intensity scattered by an N-component amorphous sample are considered. Each parameterization is such that (i) it depends only on the areas and on the angularities of the samples' interphase surfaces, (ii) it fulfils all the known physical constraints and (iii) it yields a rather simple algebraic expression both for the correlation function and for the scattered intensity. The parameterizations have been used for analysing the scattering data relevant to some three-component catalysts. Provided the volume fraction of the metal is not very small, the best fit yields satisfactory results only for some of the considered parameterizations. In this way, the determination of both the areas and the angularities of catalysts appears possible.

Salvino Ciccariello

Papers

On the Porod law

Correlation functions of amorphous multiphase systems

Deviations from the Porod law due to parallel equidistant interfaces

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

Singularities of phase boundaries and values of the second-order derivative of the correlation function at the origin