target="_blank" rel="nofollow" href="#fb3_img_img_528ce98c-1e62-548b-9853-51f6d3c2aac9.png" alt="q Subscript italic h k Baseline equals StartFraction 4 pi left-parenthesis h squared plus k squared plus italic h k right-parenthesis Superscript 1 slash 2 Baseline Over StartRoot 3 EndRoot a EndFraction"/>
where qhk is the position of the Bragg peak with indices hk and a is the lattice constant.
For a cubic structure [26, 29]
(1.54)
where qhkl is the position of the Bragg peak with indices hkl and a is the lattice constant.
The general expression for all crystal systems is [12, 26]
(1.55)
with
(1.56)
Here a, b, c are the unit cell lengths and α, β, γ are the unit cell angles.
Explicit expressions for qhkl for other lattices can be found elsewhere (see e.g. [26, 29]). Figure 1.9 shows an example of the determination of the lattice constant a by use of Eq. (1.54) by indexing the observed reflections of a cubic structure (lipid bicontinuous cubic structure). Similar methods can be used to determine d for lamellar structures from Eq. (1.52) or a for hexagonal structures from Eq. (1.53).
Figure 1.9 Indexation of SAXS reflections observed for the lipid monoolein forming a bicontinuous
Source: From Ref. [15].
1.6.3 Partially Ordered Systems and Paracrystals
The number of observed reflections as well as the peak width for an ordered structure gives an indication of the extent of order. Figure 1.10 illustrates this schematically for a one‐dimensional lattice with variable degrees of short‐ or long‐ranged order [20, 30].
Figure 1.10 Long‐range versus short‐range order showing schematics of real space distribution functions g(r) (left), and scattered intensity profiles I(q) (right). (a) Long-range order, (b) Quasi-long-range order, (c) Short range order.
For the case of a crystalline sample with long‐range positional order, the scattering pattern will consist of a function of sharp resolution‐limited diffraction peaks, as shown in Figure 1.10a. Thermal disorder leads to peak attenuation described (in the case of isotropic disorder) by a Debye‐Waller factor,
For a sample with quasi‐long‐range order, the scattering profile consists of a sharp peak with power‐law tails (Figure 1.10b). More commonly observed is short‐range order. This produces a scattering profile containing a Lorentzian peak (Figure 1.10c) given by
(1.57)
Here ξ is a correlation length and q0 = 2π/d is the peak position (d is the preferred spacing in the system). If the decay of the peaks in the correlation function is steeper, g(r) ∼ exp(−r2/ξ2), then the intensity profile shows a Gaussian peak:
(1.58)
For more ordered or partly crystalline materials, paracrystal models can be employed in the analysis of SAS data. Paracrystals are imperfect crystals in which a degree of lattice order is retained. Figure 1.11 illustrates paracrystals of the first and second kind, for the case of a two‐dimensional lattice.
Figure 1.11 Schematic of lattice distortions in paracrystals with lattice distortions of (a) first and (b) second kind.
In
