Structural Properties of Vicsek-like Deterministic Multifractals
Abstract
:1. Introduction
2. Theoretical Background
2.1. Multifractals
2.2. Small-Angle Scattering
- if the particle’s length is scaled as ,
- if the particle is translated by the vector ,
- , if the particle can be decomposed as a union of two non-overlapping subsets I and .
3. Results and Discussions
3.1. Construction of the Multifractal Model
3.2. Dimension Spectra
3.3. Pair Distance Distribution Function
3.4. Small-Angle Scattering Form Factor
4. Conclusions
- If , the system is highly heterogeneous and structural parameters are more clearly visible in pddf (see Figure 3a), since the mass fractal region of the scattering intensity is very short Figure 4a). The scaling factor is extracted from the periodicity of large groups of distances, while can be extracted in a relatively good approximation, from the periodicity of smaller groups found inside larger ones. The number of fractal iterations coincide with the number of large distinct groups in pddf.
- If , separation of pddf in distinct groups of distance is not very clear since the values of distances arising from each of the scaling factors begin to mix with each other (see Figure 3b,c), and thus extracting exact values of the scaling factors can become a very difficult task. However, in the reciprocal space, the corresponding mass fractal region of scattering intensity is characterized by a succession of maxima and minima on a power-law decay (generalized power-law decay) and the value of the largest scaling factor can be clearly estimated from the periodicity of minima. In addition, the fractal dimension can be obtained from the scattering exponent of this power-law decay while the fractal iteration number can be obtained from the number of the minima.
- If , the system reduces to a single scale fractal. Structural properties of such systems have been studied elsewhere (see Reference [30]).
Author Contributions
Funding
Conflicts of Interest
References
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Anitas, E.M.; Marcelli, G.; Szakacs, Z.; Todoran, R.; Todoran, D. Structural Properties of Vicsek-like Deterministic Multifractals. Symmetry 2019, 11, 806. https://doi.org/10.3390/sym11060806
Anitas EM, Marcelli G, Szakacs Z, Todoran R, Todoran D. Structural Properties of Vicsek-like Deterministic Multifractals. Symmetry. 2019; 11(6):806. https://doi.org/10.3390/sym11060806
Chicago/Turabian StyleAnitas, Eugen Mircea, Giorgia Marcelli, Zsolt Szakacs, Radu Todoran, and Daniela Todoran. 2019. "Structural Properties of Vicsek-like Deterministic Multifractals" Symmetry 11, no. 6: 806. https://doi.org/10.3390/sym11060806