# The Analysis of Periodic Order in Monolayers of Colloidal Superballs
## 4 Results and Discussion
Figure 3e shows this process for the positions of the 10, 20 and 30 peaks at different \(\phi\) plotted on top of the calculated spherical form factor \(P(q)\). It can be seen that the positions of the \(hk\) peaks shifted to higher \(q_{y}\)-values with increasing \(\phi\), thus resulting in changes in their relative intensities. This example shows that if a large enough range in the scattering vector \(q_{y}\) can be probed to shift the positions of these \(hk\) peaks away from the \(P(q)\) minima, all peaks can be identified correctly. Similar analyses can be performed to determine the positions of the other \(hk\) peaks present in the scattering pattern. The 01 peak,  
Figure 3: (**a**–**c**) 2D SAXS patterns of a monolayer prepared from colloidal silica spheres with an average diameter of \(D=1.10\,\upmu\text{m}\pm 3.2\%\) imaged at rotations around the \(x\)-axis of \(\phi\). The Bragg peaks are labelled on their left. (**d**) Cross sections of the diffraction patterns at different \(\phi\) taken at \(q_{x}=0\,\upmu\text{m}^{-1}\) versus \(q_{y}\text{cos}(\phi)\), revealing that the positions of the identified \(hk\) peaks occur at specific \(q_{y}\text{cos}(\phi)\) positions. (**e**) Positions of the 10, 20 and 30 peaks for different \(\phi\) overlayed on top of the calculated spherical form factor \(P(q)\). for example, is visible in each of the diffraction patterns depicted in Figure 3a-c, whereas both the 11 and 21 peaks are only visible up to a rotation of \(\phi=20^{\circ}\) (Figure 3a,b). Next, we discuss the diffraction patterns of a monolayer prepared from hollow colloidal silica superballs with \(m=2.87\pm 13\%\) and \(D=0.96\,\upmu\mathrm{m}\pm 4.5\%\) that have a semi-cubic shape and, in addition, are now hollow shells, which will influence the form factor. Figure 4a-c show the 2D SAXS patterns for different rotations \(\phi\). Here, distinct Bragg peaks can again be observed, which we can identify as the different \(hk\) peaks as indicated in Figure 4a-c. Similar to the spherical colloids, we observed that different peaks are visible at different rotations. At \(\phi=0^{\circ}\), the 10 peak is only barely visible while both the 20 and 30 peaks are clearly visible. Interestingly, and perhaps coincidentally, at \(\phi=20^{\circ}\) no peaks are visible all along the vertical axis at \(q_{x}=0\,\upmu\mathrm{m}^{-1}\). At \(\phi=40^{\circ}\), the 10 and 20 peaks appear again while the 30 peak is still invisible. Figure 4d shows the cross sections taken along the vertical axis at \(q_{x}=0\,\upmu\mathrm{m}^{-1}\). Again, in these cross sections the observed peak behavior can be more clearly visualized. The 10 peak is only visible at high \(\phi\) values, while the 30 is only visible at low \(\phi\) values and no peaks are observed at rotations in the \(x\)-axis of \(\phi=20\) and \(30^{\circ}\). The appearance and disappearance of the peaks in the scattering patterns upon rotation for these superball monolayers is distinctly different from the behavior observed for the monolayers consisting of spheres. The explanation of this behavior is that the particle form factor, now from a hollow and anisotropic particle, clearly influenced the appearance of structure factor peaks  
Figure 4: **(a–c)** 2D SAXS patterns of a monolayer prepared from hollow colloidal silica superballs with an average shape parameter of \(m=2.87\pm 13\%\) and diameter of \(D=0.96\,\upmu\mathrm{m}\pm 4.5\%\) imaged at different rotations around the \(x\)-axis of \(\phi\). The Bragg peaks are labelled on their left. **(d–e)** Vertical cross sections of the diffraction patterns at **(d)**\(q_{x}=0\,\upmu\mathrm{m}^{-1}\) and **(e)**\(q_{x}=6.12\,\upmu\mathrm{m}^{-1}\). The vertical lines indicate the \(q_{y}\mathrm{cos}(\phi)\) positions of the identified peaks. considerably, as expected. To further characterize and understand these non-trivial SAXS patterns, it can be useful to also consider vertical cross sections at different \(q_{x}\) positions.