# Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

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## Abstract

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## 1. Introduction

- The fractal dimension ${D}_{\mathrm{m}}$, from the generalized power-law decay:
- The fractal scaling factor ${\beta}_{\mathrm{s}}$, 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 ${k}_{m}$ composing the fractal, from the relation ${k}_{m}={\left(1/{\beta}_{\mathrm{s}}\right)}^{m{D}_{\mathrm{m}}}$.

## 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

- $F\left(\mathit{q}\right)\to F\left({\beta}_{\mathrm{s}}\mathit{q}\right)$ when the length is scaled as $L\to {\beta}_{\mathrm{s}}L$,
- $F\left(\mathit{q}\right)\to F\left(\mathit{q}\right){e}^{-i\mathit{q}\xb7\mathit{a}}$ when the particle is translated $\mathit{r}\to \mathit{r}+\mathit{a}$,
- $F\left(\mathit{q}\right)=\left[{V}_{I}{F}_{I}\left(\mathit{q}\right)+{V}_{II}{F}_{II}\left(\mathit{q}\right)\right]/\left({V}_{I}+{V}_{II}\right)$, when the particle consists of two non-overlapping subsets I and $II$.

#### 2.3. Monte Carlo Simulations

## 3. Small-Angle Scattering from Fractals

#### 3.1. Random Mass Fractals

- When $q\to 0$,$$S\left(q\right)\simeq 1+\Gamma \left({D}_{\mathrm{m}}+1\right){\left(\xi /a\right)}^{{D}_{\mathrm{m}}}\left(1-{q}^{2}{\xi}^{2}{D}_{\mathrm{m}}\left({D}_{\mathrm{m}}+1\right)/6\right),$$
- When ${\xi}^{-1}\lesssim q\lesssim {a}^{-1}$,$$S\left(q\right)\propto {q}^{-{D}_{\mathrm{m}}},$$
- When $q\to \infty $,$$S\left(q\right)\simeq 1+\frac{{D}_{\mathrm{m}}\Gamma \left({D}_{\mathrm{m}}-1\right)sin\left(\left({D}_{\mathrm{m}}-1\right)\pi /2\right)}{{q}^{{D}_{\mathrm{m}}}{a}^{{D}_{\mathrm{m}}}},$$

#### 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 $\tau $ in the fractal region that is $\tau ={D}_{\mathrm{m}}$ for random mass fractals, and $\tau =d-{D}_{\mathrm{s}}$ 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.
- $r\simeq 1$and$r\ll 1$ 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 $r\ll 1$, the length of the constant region is much bigger.

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Overall configuration of a SAS experiment. First, a beam of particles is emitted from a radiation source, and a small fraction pass through the collimator. Then, they hit the sample, and the scattered particles around the beam-stop are recorded. The quantities $\mathit{k}$ and ${\mathit{k}}^{\prime}$ denote the incident and, respectively the scattered wave vectors, $\Omega $ is the solid angle, and $\widehat{\mathit{r}}$ is the unit vector along the beam scattered at angles $\varphi $ and $\theta $ [72].

**Figure 2.**(Color online) Schematic representation of the sample’s structure investigated in this work. The irradiated macroscopic volume ${V}^{\prime}$ (gray region) consists from a matrix with SLD ${\rho}_{p}$ in which are dispersed fractals with SLD ${\rho}_{m}$, overall size $\xi $, and size of the basic objects $2{r}_{0}$. The orientations and positions of the fractals are uncorrelated. Their number is enough large such that an observable signal is recorded at detector, and enough small such that the distances between fractals are larger than their overall size $\xi $.

**Figure 3.**(Color online) Upper left: Approximating a disk of radius a with a set of randomly distributed points (blue). Upper-right: the corresponding pddf $p\left(r\right)$. Lower part: A comparison between the analytic expression (black curve) and Monte Carlo simulations (red curve) of SAS intensities.

**Figure 4.**(Color online) Upper row: Left—2D DLA cluster of overall size $\xi \simeq 234$ nm consisting from about 500 particles of radius $a\simeq 1$ nm, Right—the corresponding approximation with a set of randomly distributed points (blue). Lower row: Left—pddf of the 2D DLA (black points) and of an equivalent disk (red; see text for additional explanations), Right—A comparison between the analytic expression of SAS intensity (black curve) with Monte Carlo simulations (red curve).

**Figure 5.**(Color online) Upper row: Left—random surface fractal of overall size $\xi \simeq 234$ nm consisting from disks of different radii, Right—the corresponding approximation with a set of randomly distributed points (blue). Lower row: Left—pddf of the random surface fractal (black points) and of an equivalent disk (red), Right—A comparison between the analytic expression (black curve) of SAS intensity (Equation (34)) and Monte Carlo simulations (red curve).

**Figure 6.**(Color online) Upper row: Iterations $m=1,2$ and 3 of the deterministic “Cross” mass fractal of overall size $\xi \simeq 125$ nm. Middle row: the corresponding approximation with a set of randomly distributed points (blue). Lower row: Left—pddf of the “Cross” mass fractal (black points) and of an equivalent disk (red points), Right—A comparison between the mono- and polydisperse analytic expression (black, and respectively blue curve) of SAS intensity given by Equation (26) with the form factor given by Equation (65), and Monte Carlo simulations (red curve). The polydisperse curve is calculated with Equation (38), and the relative variance used is ${\sigma}_{\mathrm{r}}=0.5$.

**Figure 7.**(Color online) Upper row: Iterations $m=1,2$ and 3 of the deterministic “Cross” surface fractal of overall size $\xi \simeq 195$ nm (at $m=3$). Middle row: the corresponding approximation with a set of randomly distributed points (blue). Lower row: Left—pddf of the random surface fractal (black points) and of an equivalent disk (red points), Right—A comparison between the approximation of independent units (black curve) of SAS intensity (Equation (61)) and Monte Carlo simulations (red curve).

**Figure 8.**(Color online) Upper row: The second iteration ($m=2$) of the deterministic “Cross” multifractal of overall size $\xi =100$ nm at various scaling factors: Left—Model MI (${\beta}_{\mathrm{s}1}=0.15$ and ${\beta}_{\mathrm{s}2}=0.70$), Middle—Model MII (${\beta}_{\mathrm{s}1}=0.20$ and ${\beta}_{\mathrm{s}2}=0.60$), Right—Model MIII (${\beta}_{\mathrm{s}1}=0.33$ and ${\beta}_{\mathrm{s}2}=0.34$). For each fractal, the disks of the same radius have the same color. Middle row: the corresponding approximation with a set of randomly distributed points (blue). Lower row: Left—pddf of the multifractal models and of an equivalent disk (red points), Right—A comparison between the analytic expression (black curve) of SAS intensity (Equation (70)) and Monte Carlo simulations (red curve). The curves for models MII and MIII are shifted vertically by a factor of ${10}^{3}$, and respectively of ${10}^{6}$, for clarity.

**Figure 9.**(Color online) Schematic representation of SAS from different classes of 2D fractals Upper row: Left—mass and surface fractals, Right—random and deterministic fractals. Middle row: Left—mono and multifractals, Right—Thin and fat fractals. Lower row: $r\ll 1$ and $r\simeq 1$ fractals (see below). Here $\xi $ is the overall size of mass fractals, and respectively the size of the largest disk in a surface fractal, ${D}_{m}$ (including ${D}_{\mathrm{m}1}$ and ${D}_{\mathrm{m}2}$) and ${D}_{\mathrm{s}}$ are the mass and, respectively the surface fractal dimensions, l is the size of disks in a mass fractal, and respectively the size of smallest disk in a surface fractal, m (including ${m}_{1}$ and ${m}_{2}$) are the fractal iteration numbers, ${\beta}_{\mathrm{s}}$ (including ${\beta}_{\mathrm{s}}1$ and ${\beta}_{\mathrm{s}}2$) are the scaling factors, h is the minimal distance between the disks, and $r=l/h$.

**Table 1.**Fractal specific parameters which can be obtained from a SAS experiment. Here ${D}_{\mathrm{m}}$ and ${D}_{\mathrm{s}}$ are the mass, and respectively the surface fractal dimension, ${\beta}_{\mathrm{s}}$ is the mass and surface fractals scaling factor, m is the mass fractal iteration number, ${\beta}_{\mathrm{s}}^{\left(i\right)}$, ${k}_{m}^{\left(i\right)}$, ${m}^{\left(i\right)}$ and ${D}_{\mathrm{m}}^{\left(i\right)}$ are the scaling factors, the number of basic units, iteration number and respectively the fractal dimensions at i-th structural level ($i=1,2,\cdots $) in a fat fractal, ${\beta}_{\mathrm{s}1}$ and ${\beta}_{\mathrm{s}2}$, with ${\beta}_{\mathrm{s}1}<{\beta}_{\mathrm{s}2}$ are the multifractal scaling factors, ${k}_{m}$ is the number of basic units in a mass fractal, h is the characteristic minimal distance between basic units in a mass fractal, and $r\equiv l/h$, with l the size of basic units composing the mass fractal. The exponents of the scaling factors and number of units occurring for fat fractals denote an index (over the structural levels) and not a power.

Fractal Type | Parameters | Source | Fractal Power-Law Decays |
---|---|---|---|

Random mass fractals | ${D}_{\mathrm{m}}$ | Exponent of power-law decay | A single simple power-law decay with exponent ${D}_{\mathrm{m}}$. |

Random surface fractals | ${D}_{\mathrm{s}}$ | Exponent of power-law decay | A single simple power-law decay with exponent $d-{D}_{\mathrm{s}}$. |

Deterministic mass fractals | ${D}_{\mathrm{m}}$ | Exponent of power-law decay | A single generalized power-law decay with exponent ${D}_{\mathrm{m}}$. |

${\beta}_{\mathrm{s}}$ | Period on the logarithmic scale | ||

m | Number of periods in logarithmic scale | ||

${k}_{m}$ | ${k}_{m}={\left(1/{\beta}_{\mathrm{s}}\right)}^{m{D}_{\mathrm{m}}}$ | ||

Deterministic surface fractals | ${D}_{\mathrm{s}}$ | Exponent of power-law decay | A single generalized power-law decay with exponent ${D}_{\mathrm{s}}$. |

${\beta}_{\mathrm{s}}$ | Period on the logarithmic scale | ||

m | Number of periods in logarithmic scale | ||

Deterministic fat fractals | ${D}_{\mathrm{m}}^{\left(i\right)},{D}_{\mathrm{m}}^{\left(i\right)},\cdots $ | Exponents of power-law decays at each structural level | A succession of generalized power-law decay with exponents ${D}_{\mathrm{m}1}<{D}_{\mathrm{m}2}<\cdots .$ |

${\beta}_{\mathrm{s}}^{\left(i\right)},\cdots $ | Periods on the logarithmic scale at each structural level | ||

${m}^{\left(i\right)}$ | Number of periods in logarithmic scale at each structural level | ||

${k}_{m}^{\left(i\right)}$ | As for deterministic mass fractals, but at each structural level | ||

Deterministic multifractals with two scaling factors | ${D}_{\mathrm{m}},{D}_{\mathrm{s}}$ | Exponents of power-law decays in each fractal region | A succession of mass-to-surface fractal generalized power-law decays, with exponents ${D}_{\mathrm{m}}$, and respectively ${D}_{\mathrm{s}}$. |

${\beta}_{\mathrm{s}1},{\beta}_{\mathrm{s}2}$ | Periods on the logarithmic scale from mass, and surface fractal regions | ||

m | Number of periods in logarithmic scale from mass or surface fractal regions | ||

${k}_{m1}$ | ${k}_{m1}={\left(1/{\beta}_{\mathrm{s}1}\right)}^{m1{D}_{\mathrm{m}1}}$ | ||

Deterministic mass fractals with $r\gg 1$ | ${D}_{\mathrm{m}}$ | Exponents of the power-law decay | A region with constant intensity occurs after the fractal region. |

${\beta}_{\mathrm{s}}$ | 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 | ||

${k}_{m}$ | As for deterministic mass fractals |

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**MDPI and ACS Style**

Anitas, E.M.
Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures. *Symmetry* **2020**, *12*, 65.
https://doi.org/10.3390/sym12010065

**AMA Style**

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 Style**

Anitas, 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