Small-Angle Scattering. Ian W. Hamley

preceding derivation (Eqs. (1.26)–(1.32)) applies for the case of monodisperse particles with spherical symmetry. Other expressions have been introduced for other cases. For particles that are slightly anisotropic, the ‘decoupling’ approximation [16] is often used. The intensity for monodisperse particles is written as

      (1.33)

      where

(1.34)

Here, F(q) is the ‘amplitude’ form factor, given by Eq. (1.27). Note that there is confusion in the literature, and the term structure factor can be used to refer to intensity or amplitude according to the context. Here, S(q) is used for intensity structure factor and F(q) for form amplitude factor. In Eq. (1.34), the structure factor is calculated for the average particle size defined as R_av = [3 V/(4π)]^1/3 [17].

For systems with small polydispersities, a decoupling approximation can also be used [17] according to the expression:

      (1.35)

      where

      (1.36)

      Here, D(R) is the dispersity distribution function (Section 1.7.4), which may have a Gaussian or log‐normal function form, for example. The structure factor S(q) is evaluated for the average particle size.

      An alternative approximation is the local monodisperse approximation, which treats the system as an ensemble of locally monodisperse components (i.e. a particle of a given size is surrounded by particles of the same size). The intensity is then [17]

      (1.37)

      Other approximations for the factoring of form and structure factors have been proposed [17].

1.6 STRUCTURE FACTORS

      1.6.1 Analytical Expressions

      Analytical expressions for structure factors are available for a few simple systems including spherical and cylindrical particles [17, 18] and lamellar structures. For spheres, the hard sphere structure factor is the simplest model, as the name suggests this structure factor is derived based on purely steric interactions between solid packed spheres (volume fraction ϕ and hard sphere radius RHS). It is written

(1.38)

      Here

      (1.39)

with

      (1.40)

      At q = 0, the Carnahan‐Starling closure to the hard sphere structure factor gives the expression [11, 19]

      (1.41)

      The sticky hard sphere potential allows for a simple attractive potential between spheres (the equation for the structure factor is presented elsewhere [17, 18]). For charged spherical particles, the screened Coulomb potential may be employed.

      For cylinders, a random phase approximation (RPA, Section 1.8) equation may be used or the PRISM (polymer reference interaction site model, Section 5.8) structure factor. The RPA expression is

      (1.42)

      Here n is the number density of cylinders, ν is usually treated as a fit parameter and Pcyl(q) is the form factor of a cylinder of radius R and length L (see also Table 1.2):

      (1.43)

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