## 3 Results and Discussion
Figure 6 shows the results of five second binned fits over time for the [110] and [200] peak, these peak positions are in good agreement with literature [17; 19]. The [200] peak occurs after about 200 s of deposition and the [110] peak after about 250 s. This is about 140 s after the appearance of the crystalline Yoneda peak. In the early crystallization stage, the amount of diffraction sources are to small too overcome the background and noise in the GIXRPD detector. With the simultaneous use of two detectors for GISAXS and GIXRPD we have followed the crystallization in detail through the whole evolution of the thin film growth. First, it appears after around 60 s in cuts along \(q_{z}\) at the Yoneda peak (see Figure 3c) as slowly growing peak at the density of crystalline BaTiO3. After a growing amount of scattering sources through the film thickness it leads to a broadening of the peak along \(q_{y}\) direction (see Figures 2b and 3a) at around 100 s. And finally it appears as a scattering signal in the GIXRPD detector (see Figures 5 and 6) after around 200 s.  
To confirm and refine these findings, we used the software BornAgain (BA) [20] to simulate and fit the evolution of the thin film. For the early formation, we get the best results for small sized hemispherical shaped particle (\(\delta=2.9\) g/cm\({}^{2}\)) with a small distribution on the height of 3 nm and the radius of 3 nm for the beginning and growing over time. In the early stage, there is only a cumulative disorder of inter-particle distances along the X-ray footprint. This corresponds to the early stage of nucleation, diffusion and adsorption (see Figure 7a-c). Over time, the mean particle diameter rises, forming small clusters growing in diameter and shrinking in distance. At this stage, the best model assumes uniform hemispherical clusters of radius R arranged in a 2D hexagonal lattice with the correlation distance D between the lattice points. Assuming that all BaTiO3 deposited on this triangular area is separated into three individual uniform clusters (see Figure 3d), this leads to the equation:  
\[R=\sqrt{\frac{2\pi}{f_{A}}}\frac{t}{3^{\frac{1}{4}}}\]  
With \(R\) the micro-domain size, \(f_{A}\) the form factor ratio \(\frac{A_{divide}}{A_{hexagon}}\), with \(A_{circle}=3\pi R^{2}\) and \(A_{hexagon}=\frac{3\sqrt{3}}{2}t^{2}\), where t is the side length of the hexagon, with the local maximum for close packing at \(t=2\tau\). This maximum emphasizes the fully covered surface as mentioned before (see Figure 3a). Since we were depositing on a substrate temperature higher than the known crystallization energy for BaTiO\({}_{3}\)[1, 2], a re-crystallization was expected to happen, and was shown in the cuts along the \(q_{y}\) direction at about 60 s. Following our evolution modeling, we have assumed a covered surface of hemispherical clusters (\(\delta\) = 2.9 g/cm\({}^{2}\)) with embedded particles (\(\delta\) = 5.8 g/cm\({}^{2}\)). For these particles with the density of crystalline  
100 s of deposition, the best fitting model is a 10 nm thick amorphous thin film with a high surface roughness and growing hemispherical poly-crystalline clusters (see Figure 7f). After the crystallization of the whole thin film the arriving amorphous particle coming from the sputtering target will quickly crystallize. The high surface roughness stays until the end of the deposition which is also confirmed by the missing interference pattern in the \(q_{z}\)-cuts at the \(q=0\) position shown in Figure 3c, which is in strong agreement with the literature [10]. A schematic of the thin film evolution during the deposition is shown in Figure 8.  
Figure 7: GISAXS patterns and fit of BaTiO3 after the deposition time marked in Figure 3 (a = 10, b = 20, c = 30, d = 60, e = 90, f = 150 s). Showing the early formation (nucleation, diffusion, adsorption) in a, b, c to the start of the crystallization in d and e to the stage of the domination of the poly-crystalline BaTiO3 in f as also marked in Figure 3.