Small-Angle Scattering. Ian W. Hamley

Figure 1.16 Parameters for complex form factors. (a) Gaussian bilayer, (b) Projection of a helical structure showing definitions of angles and radii in the corresponding Pringle–Schemidt form factor (Table 1.2).

      The form factors for particulate systems of different dimensionality can all be expressed in terms of hypergeometric functions [27].

      1.7.2 Limiting Behaviours

The scaling behaviour of the intensity (plotted on a double logarithmic scale) at low q for monodisperse and uniform spheres, long cylinders and flat particles (disc or layer structures) is shown in Figure 1.17, along with examples of calculated form factors (examples of experimental data corresponding to this type of structure are presented in Section 4.13). For spherical particles, the slope is almost zero (it cannot be exactly zero according to the Guinier equation, Eq. (1.24)). For a long cylinder I ∼ q⁻¹ at low q whereas for a flat particle such as a disc or a bilayer structure I ∼ q⁻² at low q.