## 2 Materials and Methods
### Fractal Analysis  
Based on SEM and scanning transmission electron microscopy (STEM) images, which are characterizing the microstructure morphology, we have applied the fractal nature analysis on the same structure with four different magnifications (60 k, 100 k, 150 k and 250 k). We have successfully reconstructed the grain shape and presented it in Figure 1.  
For further analysis and discussions about the influence of fractal correction on dielectric constant in Curie-Weiss law and capacitance, we obtained Hausdorff fractal dimension D\({}_{\rm H}\) (Figure 2) and fractalcorrection \(\alpha_{\rm s}\) based on grain surface influence [22; 23]. Figure 2 shows the dependence of number of boxes, N, that cover the surface, on a box of size h. This dependence is shown on a logarithmic scale.  
Regarding the reconstruction of the pore morphology, we must emphasize that it is a very complex procedure, more complex than for the grains, and it is based on sampling the measured dimensions inside of the pore, which become data for fractal dimensions and reconstructions of pore shape and corresponding dimensions. Finally, the pore shape is reconstructed and presented in Figure 3, with the coordinates designation omitted, as they are not significant for determining the scale invariant fractal dimension.  
Coordinates designation is not significant for determining the fractal dimension, as it is scale invariant.  
As for the grain surface influence, we also obtained pore Hausdorff fractal dimension \(\rm D_{H}\) and corresponding fractal correction \(\alpha_{\rm p}\) based on pore shape influence [22; 23]. Figure 4 shows contour length (L) dependence on ruler size (r) on a logarithmic scale.  
Figure 1: Grains surface reconstruction based on scanning electron microscopy (SEM) and scanning transmission electron microscopy (STEM) microstructure characterization under 60 k times magnification.  
Figure 2: Grain surface fractal dimension \(\rm D_{H}=2.263\), \(\alpha_{\rm s}=0.263\).  
Regarding the third influence factor \(\alpha_{\text{M}}\), it is neglected.  
The implementation of fractal correction was developed within the experimental research based on a few thousand different electronic ceramics samples (BaTiO\({}_{3}\) with different additives) which are natural data confirming this approach with additional microstructural analysis and characterization concretely on the surface of the grains and pores. As we explained earlier, here we did not include the influence of particles' Brownian motion because this phenomenon is not dominant in ferroelectrics and dielectric materials and processes. Regarding the influence of grain and pore surfaces, it is definitely clear that this fractal level of microstructure characteristics directly and more precisely extend the structure influence on electrophysical properties, in this paper, specifically, ferroelectric and dielectric properties. Here, in this paper we applied previous knowledge to samples based on BaTiO\({}_{3}\) with Ytrium additives. Fractal correction is based on grains' and pores' surfaces fractal dimension which has been calculated by using well known methods of Richardson plot and box-counting (Figures 2 and 4).  
Problems associated with determining the fractal dimension and its correctness are multiple and require detailed analysis. Our results up to now could be a useful contribution to a precise determination of D\({}_{\text{H}}\) and problems related to it. As this is not the main topic of this paper, we can analyze this problems in future research in which results from this paper could be a contribution to further research and further clarification of this matter.