form and structure factors

Form Factor F

In the following figures simulated form factors |F(qy,qz)|2 for different shapes of objects are presented (log scale intensity). The scattering distribution in the patterns show, that one is able to distinguish between different form factors. By measuring the angle between the horizon and the scattering maxima in the case of, for example, a pyramidal shape one is able to calculate the angle of the side walls of the objects (Figure 7). Due to the reciprocal space pattern, the real shape of all scattering objects is rotated by 90 degrees.

Figure 4: simulated cylindrical form factor (R=5nm; H/R=1)

Figure 5: simulated spherical form factor (R=5nm; H/R=2)

Figure 6: simulated half-spherical form factor (R=5nm; H/R=1)

Figure 7: simulated pyramidal form factor (R=5nm; H/R=1; φ= 54.73°)

2D Interference Function S (structure factor)

In the case of arranged objects in a 2D-lattice, an interference function S just along qy (parallel to surface) appears. Most commonly, such 2D-arrangements appear in monolayers of nanoparticles or other very thin films, where a reapting unit of objects in the third dimension is missing. For the case of upright standing hexagonal arranged cylinders, the interference pattern (Figure 8) and the resulting combined form factor and interference pattern is presented in Figure 9. Only the combination of both functions F and S (Figure 4 and Figure 8) results in a valid scattering pattern:

Figure 8: simulated interference function (2Dhex,a=20 nm)

Figure 9: hexagonal arranged cylinders (R=5nm; H/R=1; 2Dhex,a=20 nm)

The same is observable in the following figures, which show the resulting scattering patterns for the same lattice S but different form factors F:

Figure 10: hexagonal arranged spheres (R=5nm; H/R=2; 2Dhex,a=20 nm)

Figure 11: hexagonal arranged half spheres (R=5nm; H/R=1; 2Dhex,a=20 nm)

3D Interference Function S (structure factor)

When the investigated material is thicker (more "bulky") the function of the lattice becomes three-dimensional. Repeating distances in all spatial directions can take place and one has to calculate with lattice functions like in a normal tramsmission scattering experiment. The only difference is, that all 4 terms of the DWBA have to take into account, depending on the absorption of the material and the reflectance of the substrate. Normally, when the material is thick enough, only the incoming beam causes scattering and calculations are much easier and it is possible to the use simpler SAXS-equations with common S-functions for example like FCC, BCC, SC for spherical close-packed particles.