Intersect distributions and small-angle X-ray scattering theory

A new method has been developed for calculating the intersect distribution function G(M) which is used in the theory of the small-angle X-ray scattering from suspensions of identical non-interacting randomly oriented particles with uniform electron density. Intersects, or chords, are straight lines which have both ends on the particle boundary, and G(M)dM is the probability that an intersect has a length between M and M+ dM. Since G(M) can be shown to contain all information about these suspensions which is obtainable from small-angle X-ray scattering measurements, the intersect distribution function can be used to study the relation between the scattered intensity and the particle shape and dimensions. The new calculation technique, which employs some results from integral geometry, is much simpler than methods previously used to find G(M) or the characteristic function γ₀(r), which contains essentially equivalent information. For particles with a smooth convex boundary, the first two terms are obtained in the expansion of G(M) in powers of M and are evaluated for a plane lamina and a three-dimensional particle. The approximate expressions for G(M) are used to determine some properties of the scattered intensity in the outer part of the scattering curve.