Polydisperse Layers and ID Particles

For an ensembie of uncorrelated ID particles (cylinders, layers) with a Gaus-sian particle thickness distribution the ID scattering intensity is [197] 

Here dc is the average thickness and (T is the variance of the particle thickness distribution modeled by a Gaussian. Ap is the ID Porod asymptote (cf. p. 125, Table 8.3). The particle thickness distribution considers polydispersity (cf. Chap. 1). 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the LORENTZ correction yields the scattering curve of the isotropic material I (s) = h (s) / (2tri ). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity / i (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and Jc. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and Oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

We have chosen Gaussian thickness distributions, because structure visualization by means of IDF or CDF exhibits thickness distributions that frequently look very similar to Gaussians. The presented relations for the ID intensity and the IDF are the basic relations for many ID structure models, comprising the general analysis of materials made from layers, highly oriented microfibrillar materials, and the direction-dependent analysis of anisotropic materials. 

Imperfect Orientation. If the streak were fanned out (cf. p. 202, Fig. 9.6), the orientation smearing must first be extinguished (Sect. 9.7) before the scattering of the perfectly oriented structural entities is retrieved. 

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