Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures
Abstract
:1. Introduction
- The fractal dimension , from the generalized power-law decay:
- The fractal scaling factor , from the period of the scattering curve on a double logarithmic scale,
- The number of fractal iterations m, equal to the number of the main minima,
- The inner and outer fractal cutoffs from the beginning and the end of periodicity region, i.e., from fractal regime,
- The total number of basic objects composing the fractal, from the relation .
2. Theoretical Background
2.1. Fractals
2.2. Small-Angle Scattering
2.2.1. General Background
2.2.2. Two-Phase Fractal Systems
2.2.3. Form Factor
2.2.4. Structure Factor
2.2.5. Polydispersity
- when the length is scaled as ,
- when the particle is translated ,
- , when the particle consists of two non-overlapping subsets I and .
2.3. Monte Carlo Simulations
3. Small-Angle Scattering from Fractals
3.1. Random Mass Fractals
- When ,
- When ,
3.2. Random Surface Fractals
3.3. Deterministic Mass Factals
3.4. Deterministic Surface Fractals
3.5. Deterministic Multifractals
4. Conclusions
- Mass and surface fractals (Figure 9 upper row, left). The differentiation is made through the value of the scattering exponent in the fractal region that is for random mass fractals, and for surface fractals. Here, d is the Euclidean dimension of the space in which the fractal is embedded.
- Random and deterministic fractals (Figure 9 upper row, right). The differentiation is made based on the type of power-law decay, i.e., a simple power-law decay for random fractals, and a generalized power-law decay (a complex superposition of minima and maxima on a simple power-law decay) for deterministic fractals.
- Mono and multifractals (Figure 9 middle row, left). The differentiation is made through the presence of one or more power-decays, either simple or generalized. For monofractals, there is a single power-law decay, while for two-scale multifractals, there is a succession of a mass fractal followed by a surface fractal. When the two scaling factors have similar values, the length of the surface fractal region is very short, and vice-versa.
- Thin and fat fractals (Figure 9 middle row, right). The differentiation is made in a similar way as in the previous case. However, the main difference is that the surface fractal region is replaced by another mass fractal region with the exponent smaller than the one of the first mass fractal region.
- and fractals (Figure 9 lower row). The differentiation is made through the presence of an additional region of constant intensity between the fractal and Porod regions. For fractal in which the ratio of the size of basic units to the minimal characteristic distances between them is about unity, the length of this constant region is very short. However, for fractals with , the length of the constant region is much bigger.
Funding
Conflicts of Interest
References
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Fractal Type | Parameters | Source | Fractal Power-Law Decays |
---|---|---|---|
Random mass fractals | Exponent of power-law decay | A single simple power-law decay with exponent . | |
Random surface fractals | Exponent of power-law decay | A single simple power-law decay with exponent . | |
Deterministic mass fractals | Exponent of power-law decay | A single generalized power-law decay with exponent . | |
Period on the logarithmic scale | |||
m | Number of periods in logarithmic scale | ||
Deterministic surface fractals | Exponent of power-law decay | A single generalized power-law decay with exponent . | |
Period on the logarithmic scale | |||
m | Number of periods in logarithmic scale | ||
Deterministic fat fractals | Exponents of power-law decays at each structural level | A succession of generalized power-law decay with exponents | |
Periods on the logarithmic scale at each structural level | |||
Number of periods in logarithmic scale at each structural level | |||
As for deterministic mass fractals, but at each structural level | |||
Deterministic multifractals with two scaling factors | Exponents of power-law decays in each fractal region | A succession of mass-to-surface fractal generalized power-law decays, with exponents , and respectively . | |
Periods on the logarithmic scale from mass, and surface fractal regions | |||
m | Number of periods in logarithmic scale from mass or surface fractal regions | ||
Deterministic mass fractals with | Exponents of the power-law decay | A region with constant intensity occurs after the fractal region. | |
Periods on the logarithmic scale from mass fractal region | |||
m | Number of periods in logarithmic scale from mass fractal regions | ||
h | End of the constant region | ||
As for deterministic mass fractals |
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Anitas, E.M. Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures. Symmetry 2020, 12, 65. https://doi.org/10.3390/sym12010065
Anitas EM. Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures. Symmetry. 2020; 12(1):65. https://doi.org/10.3390/sym12010065
Chicago/Turabian StyleAnitas, Eugen Mircea. 2020. "Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures" Symmetry 12, no. 1: 65. https://doi.org/10.3390/sym12010065