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.

The Porod law states that i(h), the intensity of Xradiation scattered by an ideal multiphase noncrystalline system, for 'large' momentum transfer values (= h) approaches C~h-4 and that ~ 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 [hai(h) =constant]. The momentum transfer rangewhere 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.

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.

It is shown that the leading asymptotic term of the small-angle intensity scattered by any amorphous sample is determined by the parallelism among subsets of the sample interfaces. Its general expression is g, [A icos(5ih }+Sisin(5&h }]/h 4, where the 5~'s denote the distances between parallel surfaces and the A i's and the Si's are appropriate geometrical averages of the corresponding Gaussian curvatures. Since each surface is parallel to itself with a relative null distance, in the former expression the well-known Porod contribution comes out from the term relevant to 5=0. The expression is specialized to the case of those three-component samples where one of the constituting phases has a constant thickness and lies in between the remaining two phases which have no common interface. Different approximations are considered and, in the most favorable cases, it appears that the average Gaussian, mean, and squared mean curvatures of the dividing 61m can be determined.

Small-angle scattering from anisotropic samples, consisting of homogeneous particles inside a homogeneous medium with a scattering contrast (∆n) 2 , is considered. Along any direction q ≡ q/|q| of reciprocal space, at large q ( ≡ |q|) the Porod plot of the scattering intensity (i.e. q 4 I(q) vs. q) shows a plateau whose height depends on q and reads 4π 2 (∆n) 2 j,l (1/|κ G,j,l (± q)|). Here, the sum runs over all the points (labeled by (j, l)) of the surface of the j-th particle of the sample where the normal is either parallel or antiparallel to q, and κ G,j,l (± q) is the corresponding Gaussian curvature value.

It is shown that, close to the origin, the correlation function [y(r)] of any N-component sample with interfaces made up of planar facets is always a third-degree polynomial in r. Hence, the only monotonically decreasing terms present in the asymptotic expansion of the relevant small-angle scattered intensity are the Porod [-2y'(0+)/h 4] and the KirstePorod [4')(3)(0 +)/h 6] contributions. The latter contribution is non-zero owing to the contributions arising from each vertex of the interphase surfaces. The general vertex contribution is evaluated in closed form and the )'(3)(0+) values relevant to the regular polyhedra are reported.

Errors in the paper by Ciccariello, Melnichenko & He [J. Appl. Cryst.(2011),44, 43–51] are corrected.

The connection between the zero-distance value of the second-order derivative of correlation functions used in light or small-angle x-ray-scattering theory and the singularities of phase boundaries is more thoroughly exploited. It is shown that, besides sharp edges, only smooth contact points contribute. The explicit expression of the latter contribution is obtained.Small-angle x-ray-scattering experiments allow us to determine j'(0+) and j'(0+), i.e, the zero-distance values of the firstand second-order derivatives of the correlation function y(r) relevant to the sample that one is interested in. j(0+} is determined through the relation' -8n.y(0+)= lim k i(k), k~Oo where k and i (k} are the momentum transfer and the standard scattered intensity, respectively. j'(0+ ) is obtained from Porod's relation, ' y(0+) = -f [k i(k)+ 8my(0+)]dk/12m. So far the latter relation is not as commonly used as the former one, for two reasons. On the one hand, unavoidable experimental uncertainties on the large k values of i (k) make the determination of j(0+) by Eq. (1) much less accurate than that of y(0+).On the other hand, the relation between y'(0+) and some geometrical features of the phase boundaries has not been as fully analyzed as in the j (0+) case. Until a few years ago, in fact, one knew that j'(0+ } was null when the surfaces separating the different phases of the sample were smooth functions (i.e. , their parametric equations were continuous doubly differentiable functions, more briefiy, C2 functions). Moreover, it has also been argued that y'(0+ ) should be different from zero and positive whenever boundaries have sharp edges. 'Only recently the explicit relation between this )&cot[apj(y)]I .In Eq. (2), the integral is performed along the edge, and p,j(y) is the value of the dihedral angle at the edge point characterized by the curvilinear abscissa y. Equation (2) allows one to evaluate the contribution of a sharp edge to j'(0+), since y(r) is related to SPF's by y(r)=1g(n; nj) PJ(r)/(ri -) .(3) We recall that n; and (g ) in Eq. (3) are known quantities. They denote, respectively, the electron density of the ith phase and the average of the squared electron density fiuctuation.We stress, however, that we do not know whether y'(0+) is different from zero only if sharp edges are present. To the best of our knowledge the kinds of boundary singularities responsible for a j'(0+) value different from zero have not been classified yet. In this paper we shall carry through such an analysis.We shall show, indeed, that in addition to sharp edges only "smooth contact" points contribute to y(0+). Moreover, the explicit expression of these kind of surface singularity and the j'(0+) value has been obtained.In this paper it was shown that the contribution of an edge on the surface separating the ith phase from the jth one to the zero-length limit of the second-order derivative of the relevant "stick probability function" (SPF) is~1 P; (0+)=-6 V dy[1 -[n. -P; (y)] lJ 26 6384 Q~1982 The American Physical Society