Small-Angle Scattering. Ian W. Hamley

Radius, R. length, L. Scattering contrast, Δρ Infinitely thin circular disc Radius, R Scattering contrast, Δρ Lipid bilayer with Gaussian scattering density profile (within a disc) P(q) = [Pdisc(q)/(Δρ)²]Pcs(q) where the cross‐section form factor Pcs(q) is Cross‐section parameters illustrated in Figure 1.16 Helices and helical tapes (Pringle–Schmidt form factor [54]) Here and ɛ₀ = 1 and ɛn = 2 for n ≥ 1 H helix length, P helix pitch, other parameters defined in Figure 1.16

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.

Figure 1.17 Form factors calculated for homogeneous particles along with limiting slopes. Form factors are calculated for spheres of radius R = 30 Å, cylinders of radius R = 30 Å and length L = 1000 Å and discs with thickness T = 30 Å and radius R = 1000 Å. The calculated profiles have been offset vertically for convenience. The minima in principle have zero intensity, but are truncated due to numerical calculation accuracy and for convenience plotting on a logarithmic intensity scale. The profiles were calculated using SASfit [21].

      These scaling behaviours for extended rod‐like and flat particles can be derived as discussed in the following section.

      Alternatively, the scaling behaviour can be obtained from the behaviours of the autocorrelation functions at large r (relating to low q behaviour of the intensity). For cylinders (radius R), [56]

(1.83)

For flat particles (thickness T) [56]

(1.84)

Substitution of these expressions