Small-Angle Scattering. Ian W. Hamley

Eq. (1.8) leads to the following equations: [56]

(1.85)

      (1.86)

      For monodisperse spherical particles of radius R, the minima in the form factor are located at qR = 4.493, 7.725... [10]. For an isotropic system of long cylindrical rods (radius R), the form factor minima are located at qR = 3.83, 7.01… [10] and for flat particles (thickness T) at qT/2 = 3.14, 6.28… These values can be obtained from the minima in the corresponding Bessel or sine functions as in the equations in Section 1.7.3.

      It should be noted that the form factors and the position of the minima in them depend on products of q and an appropriate particle dimension; therefore, the form factors have the same appearance for different pairs of reciprocal units (i.e. calculations with q in Å⁻¹ with dimensions in Å or q in nm⁻¹ with dimensions in nm give the same result).

1.7.3 Factoring Scattering from the Particle Cross‐Section

For a long cylindrical particle the scattering intensity (form factor) can be computed from Eq. (1.6) by calculating the average exp[−iq.r] in polar coordinates (l, z, γ), where the q vector and vector r to a point in the cylinder are related by the polar angles (ϕ, γ) (Figure 1.18):

      (1.87)

Figure 1.18 Inter‐relationship between vectors q and r in cylindrical co‐ordinates.

      For a uniform cylinder (radius R, length L) this leads to the expression [57]

      (1.88)

      (1.89)

      Performing the integral over z (using the same formula as in Eq. (1.12)) we have

      (1.90)

For L > > R we can factor this as the product of intensity associated with the length of the particle IL(q) and that of the cross‐section Ic(q) (i.e. the intensity is written as a convolution product):

      (1.91)

      where

      (1.92)

      The integral over ϕ extends to infinity to make use of the Dirichlet integral, this is valid when q ≥ 2π/L, since the integrand is negligibly small for x > 1 [7]. This leads to the factorization [5, 6]

(1.93)

where Ic(q) is the form factor of the cross‐section. As mentioned above, this is valid when L > > R. Eq. (1.93) shows the I(q) ∼ q⁻¹ scaling for a cylindrical particle at low q (where IL(q) dominates). This derivation is for a uniform cylinder for which Ic(q) can be evaluated as in the following, however Eq. (1.93) applies in general for other rod‐like particles.

      The cross‐section intensity is related to the distance distribution function of the cross‐section, γc(r),