# Light-Scattering Properties for Aggregates of Atmospheric Ice Crystals within the Physical Optics Approximation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{min}is the radius of the inscribed sphere of the single particle, and L

_{avg}is the average distance between the center of each particle and the center of the aggregate, defined as follows:

_{i}is the distance between the geometrical aggregate’s center and the center of a single particle i; N is the number of particles in the aggregate. Coordinates of the center of the aggregate and single particle are calculated by summing the coordinates of all vertices of the aggregate and single particle, respectively, and dividing it by the number of vertices. Based on Equation (1), the most compact aggregate will have C = 1.

_{avg}and C on N in the aggregate are shown in Figure 3.

## 3. Calculation Results within the Geometrical Optics Approximation

_{11}and M

_{22}elements of the light-scattering matrix are shown in Figure 4 and Figure 5. The value of M

_{11}for the scattering angle of 0° is removed from the plots because it is too high to display. The element M

_{22}is normalized over M

_{11}:m

_{22}= M

_{22}/M

_{11}.

_{11}, which defines the intensity of light in the scattering angle (θ) for unpolarized incident light. Variability of the other elements is not critical.

_{11}divided by the average geometric shadow area (G

_{A}). This area can be calculated using the geometry of the aggregate without solving the light-scattering problem. The result shows that the M

_{11}/G

_{A}very slightly changes with the number of particles in the aggregate, except in the case of an aggregate of compact plates.

_{11}/G

_{A}for the forward scattering direction for these aggregates vs. N (M

_{11}*/G

_{A}in Figure 7). In this case, the peak of intensity in the forward scattering direction is re-scattered by another plate appearing right behind the first one with an increasing number of particles. That peak is created by the forward transmission of light through the large plane-parallel facets of a plate particle. While in the case of a non-compact aggregate of plates, the particles do not overlap each other, so the light moves freely in the forward scattering direction (see Figure 8).

_{11}/G

_{A}in Figure 11. The normalized elements m

_{12}, m

_{22,}and m

_{44}are also shown in Figure 11. The results show that the M

_{11}/G

_{A}and normalized elements m

_{12}, m

_{22,}and m

_{44}almost do not change with the number of particles in the bullet-rosette. It means that the optical properties of the bullet-rosette can be evaluated from the optical properties of one bullet.

## 4. Calculation Results within the Physical Optics Approximation

_{0}, y

_{0}, z

_{0}). Then, they are scaled to obtain a proper dimension for an aggregate in calculation. The orientation of the single hexagonal column is specified by three Euler angles α

^{0}, β

^{0}, and γ

^{0}, where α

^{0}defines rotation of the column about the vertical direction; β

^{0}is the angle between the vertical direction and the crystal main axis; and γ

^{0}describes the column rotation about the main axis. The main axis of the hexagonal column is assumed to pass through the centers of the hexagonal facets. Table 1 lists the values of initial geometric parameters for an aggregate composed of eight hexagonal columns. Note that these characteristics are slightly different from the characteristics presented in the paper by P. Yang [39], in order to avoid self-intersections. For convenience, we define the aggregate size through its maximal size D

_{max}.

^{0.63}. Sizes are given in µm.

_{max}= 670 µm for the wavelength of 0.355 µm takes about 18 days on a modern server with 2 Xeon E5-2660 v2 processors (40 threads), we can precisely calculate only a few of them.

_{11}for bullet-rosette (6 bullets) and for a single bullet using the existing database. Since D

_{max}for aggregate and for a single particle is different, we use the dependence of M

_{11}on the length of a single bullet (L

^{bul}). Then, M

_{11}for a single bullet was multiplied by the total scattering cross-section for bullet-rosette. The result is presented in Figure 14.

## 5. Discussion

_{11}) on the number of particles (N) in the scattering-angle range of 20–180° (Figure 6). The scattering matrix can be obtained by multiplying the scattering efficiency of a single particle by the geometrical cross-section of an aggregate. As far as the physical optics solution is obtained from the geometrical optics solution, it should also be slightly changed with increasing N. However, this effect does not work with compactly packed-plate aggregates because of their specific geometry. This is a very important conclusion that allows us to extend the light-scattering database of a single particle to the case of aggregates of particles.

_{11}for column aggregates shows an unpredictable distribution at angles of 0–20°. This fact can be explained by the decreasing energy at the angle of 0° (forward-scattering direction intensity peak) and redistribution of it to different directions. This energy peak is caused by light that falls orthogonal to the surface of facets and propagates through the particle without refraction. However, it can be refracted in the case of an aggregate of two or more particles when the light that leaves one particle is redirected by falling on another particle. In the case of plate aggregates, this effect is insignificant because of the similar, spatial orientation of plate particles in aggregates.

_{11}in the angular range of 0–20° for a column aggregate with a different arrangement may be different. However, the main dependencies are consistent with the initial assumptions. Further studies should consider more examples of aggregates to obtain satisfactory statistics. It is also necessary to calculate the backscattering matrices in the physical optics approximation with the absorption coefficient.

_{11}for bullet-rosette shows a more predictable dependence on the number of particles. It can be obtained by multiplying M

_{11}for a single bullet of the same size by the total scattering cross-section for bullet-rosette both within the geometrical and the physical optics approximation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometrical shapes of the particles for aggregates: (

**a**) hexagonal column; (

**b**) hexagonal plate.

**Figure 2.**Models of aggregates of 9 particles: (

**a**) compact columns; (

**b**) non-compact columns; (

**c**) compact plates; (

**d**) non-compact plates.

**Figure 4.**The M

_{11}element vs. scattering angle (θ) for the following aggregates: (

**a**) non-compact columns; (

**b**) compact columns; (

**c**) non-compact plates; (

**d**) compact plates.

**Figure 5.**The m

_{22}element vs. scattering angle (θ) for the following aggregates: (

**a**) non-compact columns; (

**b**) compact columns; (

**c**) non-compact plates; (

**d**) compact plates.

**Figure 6.**M

_{11}/G

_{A}for the following aggregates: (

**a**) non-compact columns; (

**b**) compact columns; (

**c**) non-compact plates; (

**d**) compact plates.

**Figure 10.**Dependences of elements of the light-backscattering matrix on scattering angle (θ): (

**a**) M

_{11}; (

**b**) m

_{12}; (

**c**) m

_{22}; (

**d**) m

_{44}.

**Figure 13.**M

_{11}and M

_{22}vs. D

_{max}at two wavelengths of incident light: (

**a**) for aggregate of 8 columns; (

**b**) for bullet-rosette.

No. | D, [μm] | L, [μm] | γ^{0} | β^{0} | α^{0} | x_{0} | y_{0} | z_{0} |
---|---|---|---|---|---|---|---|---|

1 | 92 | 158 | 23 | 50 | −54 | 0 | 0 | 0 |

2 | 80 | 124 | 16 | 81 | 156 | 15.808 | 107.189 | −60.108 |

3 | 56 | 78 | 5 | 57 | 94 | −26.691 | 73.005 | 49 |

4 | 96 | 126 | 13 | 76 | 130 | −88 | −39.19 | −11.643 |

5 | 106 | 144 | 11 | 29 | −21 | 106.532 | 33.08 | 27.801 |

6 | 38 | 54 | 8 | 62 | −164 | 35.923 | −51.5 | −37.533 |

7 | 68 | 102 | 29 | 41 | 60 | 40.11 | −57.227 | 112.5 |

8 | 86 | 138 | 19 | 23 | −122 | −9.7524 | −132.57 | 57.131 |

Particle | λ = 0.355 μm | λ = 1.064 μm | |
---|---|---|---|

Aggregate of 8 columns, $30\hspace{0.17em}\mathsf{\mu}\mathrm{m}\le {D}_{max}\le 2500\hspace{0.17em}\mathsf{\mu}\mathrm{m}$ | M_{11} | 0.00089∙D_{max}^{3.022} | 0.00022∙D_{max}^{3.025} |

M_{22} | 0.00055∙D_{max}^{3.008} | 0.00014∙D_{max} ^{3.017} | |

Bullet rosettes, $40\hspace{0.17em}\mathsf{\mu}\mathrm{m}\le {D}_{max}\le 2500\hspace{0.17em}\mathsf{\mu}\mathrm{m}$ | M_{11} | 0.0146∙D_{max}^{2.184} | 0.00587∙D_{max}^{2.129} |

M_{22} | 0.0068∙D_{max}^{2.201} | 0.0028∙D_{max}^{2.155} |

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**MDPI and ACS Style**

Timofeev, D.; Kustova, N.; Shishko, V.; Konoshonkin, A.
Light-Scattering Properties for Aggregates of Atmospheric Ice Crystals within the Physical Optics Approximation. *Atmosphere* **2023**, *14*, 933.
https://doi.org/10.3390/atmos14060933

**AMA Style**

Timofeev D, Kustova N, Shishko V, Konoshonkin A.
Light-Scattering Properties for Aggregates of Atmospheric Ice Crystals within the Physical Optics Approximation. *Atmosphere*. 2023; 14(6):933.
https://doi.org/10.3390/atmos14060933

**Chicago/Turabian Style**

Timofeev, Dmitriy, Natalia Kustova, Victor Shishko, and Alexander Konoshonkin.
2023. "Light-Scattering Properties for Aggregates of Atmospheric Ice Crystals within the Physical Optics Approximation" *Atmosphere* 14, no. 6: 933.
https://doi.org/10.3390/atmos14060933