a paracrystal with imperfections of the first kind, the long‐range lattice order is retained but there are fluctuations in position around the lattice nodes (Figure 1.11a), i.e. there is positional disorder. In the observed diffraction pattern, the peak width increases linearly with the peak order [31], and the intensity is modulated by a Debye‐Waller factor as described above [12, 31]. With lattice distortions of the second kind, the long‐range lattice order is disrupted as shown in Figure 1.11b, i.e. there is long‐range positional and bond orientational disorder. This leads to a quadratic increase in peak width with peak order [12, 31]. Analytical solutions are available for one‐dimensional paracrystal models of the first and second kind [12, 22].
1.6.4 The Phase Problem
The amplitude structure factor in Eq. (1.27) is a complex quantity and so can be written as the product of an amplitude and a phase:
(1.59)
where ϕhkl is the phase angle for reflection with indices hkl. The amplitude can be obtained from the square root of the intensity:
(1.60)
In general, the phase appearing in Eq. (1.59) cannot be determined in an x‐ray diffraction or SAXS experiment. There are methods in x‐ray crystallography to circumvent the problem (such as heavy atom replacement). In the study of SAXS by soft materials, the reconstruction of the electron density profile is usually only done for a limited subset of systems. In particular, it is performed for lamellar structures such as those of lipid bilayers, as discussed further in Section 4.12.
1.6.5 Fractal Structures
A number of structure factors have been proposed for fractal structures. For these systems, the separation of form and structure factors often does not make sense; therefore, we denote the scattering just by the intensity. One model commonly used in the analysis of SAS data is the Fischer–Burford fractal model, for which the structure factor is given by [32, 33]
(1.61)
where I0 is the forward scattering intensity, D is the fractal dimension, and Rg is the radius of gyration of the aggregate.
Another widely employed expression is that for the structure factor of a mass‐fractal: [33–36]
(1.62)
where I0 denotes forward scattering, D is the fractal dimension, and ξ is the correlation length.
Another widely used model was developed by Beaucage for a variety of systems with ordering on multiple length scales including fractal structures, and is based on an expression for the scattering from excluded volume polymer fractals [37, 38]. This gives a function that interpolates between a Guinier function at low q and a Porod function at high q. It is used for many types of systems, such as porous materials (Section 5.11), colloidal and gel aggregate structures, polymer foams, polymer nanocomposites, nanopowders, and other systems. The unified Beaucage expression for a system with one structural level is [39]:
(1.63)
Here G is a Guinier scaling factor, Rg is the radius of gyration of the mass fractal object, P is a Porod constant, D is the fractal dimension, and erf denotes the error function. The constant P is related to G via the expression [38]:
(1.64)
Here Γ(D/2) denotes a Gamma function. Analytical expressions are available for G and P for a variety of uniform particles with different shapes as well as types of polymers [40].
Figure 1.12 shows an example of a fit to USAXS data for a titania nanopowder, along with indicated contributions from the Guinier and Porod components of Eq. (1.64).
Figure 1.12 Example of fitting of SAS data using the unified Beaucage model, Eq. (1.63), to fit USAXS data for a titania nanopowder (inset: TEM image). The contributions from the Guinier and Porod components of Eq. (1.63) are shown. Also shown for comparison is the limiting Porod slope in a scattering profile from spheres of the same radius of gyration Rg as from the Beaucage model fit.
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