Section 2.6.2.7.1. Particle shape

All X-ray and neutron small-angle scattering curves can be approximated by a parabolic fit in a narrow Q range near Q = 0 (Porod, 1951): I(Q) I(0) (1 − a²Q²/3 + ). In the case of single-particle scattering, a Gaussian approximation to the scattering curve is even more precise (Guinier & Fournet, 1955) in the zero-angle limit: where is the radius of gyration of the particle's excess scattering density.

The concept of and the validity of the Guinier approximation is discussed in more detail in the SAXS section of this volume (Section 2.6.1). It might be mentioned here that the frequently used rule for the validity of the Guinier approximation is no more than an indication and should always be tested by a scattering calculation with the model obtained from the experiment: Spheres yield a deviation of 5% of the Gaussian approximation at , rods at ; ellipsoids of revolution with an elongation factor of 2 can reach as far as .

More detailed shape information requires a wider Q range. As indicated before, Fourier transforms may help to distinguish between conflicting models. In many instances (e.g. hollow bodies, cylinders), it is much easier to find the shape of the scattering particle from the distance distribution function than from the scattering curve [see §2.6.2.7.3].