The relation between the particle shape and the outer part of the small-angle X-ray scattering curve

Small-angle scattering behavior of thread-like and film-like systems

Film-like and thread-like systems are, respectively, defined by the property that one of the constituting homogenous phases has a constant thickness (δ) or a constant normal cross section (of largest chord δ). The stick probability function of this phase, in the limit δ → 0, naturally leads to the definition of the correlation function (CF) of a surface or of a curve. This CF closely approximates the generating stick probability function in the range of distances larger than δ. The surface and the curve CFs, respectively, behave as 1/rand as 1/r2asrapproaches zero. This result implies that the relevant small-angle scattering intensities behave as {\cal P}_{{\cal S}}/q^2 or as {\cal P}_{{\cal C}}/q in the intermediate range of the scattering vector magnitude (q) and as {\cal P}/q^4 in the outermostqrange. Similarly to {\cal P}, pre-factors {\cal P}_{{\cal S}} and {\cal P}_{{\cal C}} simply depend on some structural parameters. Depending on the scale resolution it may happen that a given sample looks thread like at large scale, film like at small scale and particulate at a finer scale. An explicit example is reported. As a practical illustration of the above results, the surface and the curve CFs of some simple geometrical shapes have been explicitly evaluated. In particular, the CF of the right circular cylinder is evaluated. Its limits, as the height or the diameter the cylinder approaches zero, are shown to coincide with the CFs of a circle and of a linear segment, respectively.

The chord-length probability density of the regular octahedron

The chord-length probability density of the regular octahedron is separated into three contributions, relating to the pairs of facets opposite to each other or sharing an edge or a vertex. Each of these contributions is explicitly evaluated throughout the full range of distances and the final expressions only involve inverse trigonometric functions of elementary algebraic functions. Since the chord-length probability density is proportional to the second derivative of the correlation function, knowledge of the chord-length probability density makes the numerical evaluation of the associated small-angle scattering intensity very fast and accurate.