Small-Angle Scattering. Ian W. Hamley

      Here D is the cross‐section diameter. For a uniform cylinder this may be evaluated to give [6]

(1.95)

      In these equations J₀(qR) and J₁(qR) denote Bessel functions of integral order. The cross‐section radius is given by [58] (see also the discussion in Section 1.4).

      The pair distribution function of the cross‐section can be obtained from the cross‐section intensity via an inverse Hankel transform [7]

      (1.96)

      For flat particles (discs of area A) the intensity can be factored, via an equation analogous to Eq. (1.93) as [4, 6, 10]

(1.97)

      where It(q) is the cross‐section scattering that depends on the thickness T, which is related to the cross‐section radius via [58].

Expressions (1.93) and (1.97) indicate that as an alternative to the Guinier equation which provides Rg, the cylinder cross‐section radius can be obtained from a plot of ln[qI(q)] vs. q² and for discs/planar structures Rc can be obtained a plot of ln[q²I(q)] vs. q² [58, 59].

If the intensity is measured on an absolute scale it is possible to determine the mass per unit length for rod‐like particles or the mass per unit area for flat (lamellar) particles. The general expression for molar mass determination for an arbitrary particle from the differential scattering cross‐section dσ/dΩ in absolute units (cm⁻¹) is discussed in Section 2.9. For rod‐like particles in a solution of concentration c (in g cm⁻³), the mass per unit length Mc (in g mol⁻¹ cm⁻¹) can be obtained from the expression (containing a π/q factor from Eq. (1.95))

      (1.98)

      where NA is Avogadro's number, vp is the specific volume in cm³ g⁻¹, Δρ is the contrast in cm⁻² and q is in cm⁻¹.

      For flat particles, the area per unit length Mt (in g mol⁻¹ cm⁻²) is obtained from the analogous equation (with q² dependence cf. Eq. (1.93))

      (1.99)

      The derivations of these equations can be found elsewhere [60].

1.7.4 Effect of Polydispersity

      Particle polydispersity has a considerable influence on the shape of the form factor. It is included via integration over a particle size distribution D(R′):

      (1.100)

      Here the subscript 0 has been added to the form factor, in order to emphasize that this is the term for the monodisperse system.

The effect of polydispersity for the example of the form factor of a uniform sphere of radius R is illustrated in Figure 1.19. The polydispersity in this case is represented by a Gaussian function:

      (1.101)

Figure 1.19 Influence of polydispersity on the form factor of a sphere with R = 30 Å in terms of a Gaussian standard deviation with width σ (in Å) indicated.

      Here σ is the standard deviation, which is related to the full width at half maximum by , and c is a normalization constant. Other forms of distribution can be used and may be motivated by known dispersity distributions (based on the system synthesis, for example), for example log‐normal functions, Schulz‐Zimm functions etc.

      It is clear from the example of calculated form factors for a uniform sphere in Figure 1.19 that increasing σ causes the form factor oscillations to get progressively washed out such that they are largely