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Mie Scattering Theory

Mie theory provides the exact solution for the interaction of electromagnetic waves with homogeneous spherical particles, proposed by Gustav Mie in 1908.

Basic Principles

When electromagnetic waves encounter particles, scattering and absorption occur. Mie theory provides the mathematical framework for calculating these processes.

Efficiency Factors

For a particle with diameter \(D_p\), the efficiency factors are defined as:

\[Q_{ext} = Q_{sca} + Q_{abs}\]

Where: - \(Q_{ext}\) = Extinction efficiency factor - \(Q_{sca}\) = Scattering efficiency factor - \(Q_{abs}\) = Absorption efficiency factor

Size Parameter

\[x = \frac{\pi D_p}{\lambda}\]

Where \(\lambda\) is the wavelength of incident light.

Complex Refractive Index

\[m = n + ik\]
  • \(n\) = Real part (refraction)
  • \(k\) = Imaginary part (absorption)

Extinction Coefficient Calculation

Calculate total extinction coefficient from size distribution:

\[b_{ext} = \int_0^\infty Q_{ext}(D_p, m, \lambda) \cdot \frac{\pi D_p^2}{4} \cdot n(D_p) \, dD_p\]

Where \(n(D_p)\) is the particle number size distribution.

Mixing Modes

Internal Mixing

All components are uniformly mixed within a single particle, using volume-weighted average refractive index:

\[m_{mix} = \sum_i f_i \cdot m_i\]

Where \(f_i\) is the volume fraction of component \(i\).

External Mixing

Each component forms independent particles, calculated separately and summed:

\[b_{ext} = \sum_i b_{ext,i}\]

Core-Shell Structure

EC as the core with other components as the shell. Suitable for aged aerosols.

AeroViz Implementation

from AeroViz.dataProcess import DataProcess
from pathlib import Path

dp = DataProcess('Optical', Path('./output'))

# Calculate extinction distribution
result = dp.Mie(
    df_pnsd,           # Particle number size distribution
    df_RI,             # Refractive index DataFrame (n, k columns)
    wave_length=550    # Wavelength (nm)
)

# Output
result['extinction']   # Extinction coefficient (Mm-1)
result['scattering']   # Scattering coefficient (Mm-1)
result['absorption']   # Absorption coefficient (Mm-1)

References

  1. Mie, G. (1908). Beitrage zur Optik truber Medien. Annalen der Physik, 330(3), 377-445.
  2. Bohren, C. F., & Huffman, D. R. (1983). Absorption and Scattering of Light by Small Particles. Wiley.