Small-Angle Scattering. Ian W. Hamley

a particle (maximum distance from the geometric centre).

      In terms of the isotropically averaged autocorrelation function γ(r) this can be written as

      (1.16)

      Or in terms of the Debye correlation function Γ(r):

      (1.17)

      This then leads to the expression

(1.18)

Here p(r) = Γ(r)r² is the pair distance distribution function (PDDF). This is an important quantity in SAS data analysis since as can be seen from Eq. (1.18), it is related to the intensity via an indirect Fourier transform:

      (1.19)

The PDDF provides information on the shape of particles, as well as their maximum dimension D_max. Figure 1.3 compares the PDDF of different shaped objects.

Figure 1.3 Sketches of pair distance distribution functions for the colour‐coded particle shapes shown with a D_max = 10 nm.

      Source: Adapted from Ref. [9].

      Many SAS data analysis software packages such as ATSAS and others (Table 2.2) and software on synchrotron beamlines is able to compute PDDFs from measured data. Methods to obtain PDDFs by indirect Fourier transform methods are discussed further in Section 4.6.1.

The radius of gyration can be obtained from p(r) via the second moment [4, 7, 10, 11]:

(1.20)

1.4 GUINIER APPROXIMATION

      The Guinier equation is used to obtain the radius of gyration from a simple analysis of the scattering at very low q (from the first part of the measured SAS intensity profile). The Guinier approximation can be obtained from Eq. (1.18), substituting the expansion [6, 7, 11]:

      (1.21)

      gives at sufficiently low q (such that the expansion can be truncated at the second term)

      (1.22)

Considering the expression for the radius of gyration as the second moment of p(r) (Eq. (1.20)) we obtain [6, 11, 12]

(1.23)

      Using the series expansion with , and truncating at the second term (valid if q is small), this can be rewritten as an exponential

(1.24)

This is the Guinier equation. A Guinier plot of lnI(q) vs q² has slope at low q. Figure 2.8 shows representative Guinier plots. The Guinier equation (Eq. (1.23)) can also be obtained starting from Eq. (1.10), using the same series expansion for the exponential exp[−iq.r].

      For a homogeneous sphere of radius R, the radius of gyration is given by [6, 11] whereas for a homogeneous infinite cylinder of radius R it is given by and for a thin disc of thickness T, [6, 7, 10, 13]. For an ellipse