Skip to content

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.

Lognormal Modes by Mass

A synthetic mode is usually known by its mass loading, not its number concentration. For a lognormal number mode with count median diameter \(CMD\) and geometric standard deviation \(\sigma_g\), the Hatch–Choate relations give the volume-mode median and the total volume:

\[VMD = CMD \cdot e^{3\ln^2\sigma_g}, \qquad V = N \cdot \frac{\pi}{6}\,CMD^3\, e^{4.5\ln^2\sigma_g}\]

so a mass concentration \(M\) with density \(\rho\) fixes \(N = M/(\rho V_1)\), where \(V_1\) is the volume of a unit-number mode. mie_lognormal(..., total_mass=, density=) and mie_multimodal(..., amount='mass', density=) do this conversion; hatch_choate() converts medians between weightings. The grid is built as \(dN/d\log_{10}D_p\), which is what the integration assumes.

In AeroViz

Usage lives in the guide: mie for the call and its mixing / distribution options, and Optical Closure for a worked comparison against measured extinction.

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.