Editing Cross section (physics) (section)

=== Cross section and Mie theory ===

Cross sections commonly calculated using [[Mie scattering|Mie theory]] include efficiency coefficients for extinction <math display="inline">Q_\text{ext}</math>, scattering <math display="inline">Q_\text{sc}</math>, and Absorption <math display="inline">Q_\text{abs}</math> cross sections. These are normalized by the geometrical cross sections of the particle <math display="inline">\sigma_\text{geom} = \pi a^2</math> as <math display="block">
Q_\alpha = \frac{\sigma_\alpha}{\sigma_\text{geom}}, \qquad \alpha = \text{ext}, \text{sc}, \text{abs}.
</math>
The cross section is defined by 
: <math>
\sigma_\alpha = \frac{W_\alpha}{I_{\text{inc}}}
</math>
where <math>\left[W_\alpha \right] = \left[ \text{W} \right]</math> is the energy flow through the surrounding surface, and <math> \left[I_{\text{inc}}\right] = \left[ \frac{\text{W}}{\text{m}^2} \right]</math> is the intensity of the incident wave. For a [[plane wave]] the intensity is going to be <math>I_{\text{inc}} = |\mathbf{E}|^2 / (2 \eta)</math>, where <math>\eta = \sqrt{\mu \mu_0 / (\varepsilon \varepsilon_0)}</math> is the [[Impedance of free space|impedance of the host medium]].

The main approach is based on the following. Firstly, we construct an imaginary sphere of radius <math>r</math> (surface <math>A</math>) around the particle (the scatterer). The net rate of electromagnetic energy crosses the surface <math>A</math> is
: <math>
W_\text{a} = - \oint_A \mathbf{\Pi} \cdot \hat{\mathbf{r}} dA
</math>
where <math display="inline">\mathbf{\Pi} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}^* \times \mathbf{H} \right]</math> is the time averaged Poynting vector. If <math>W_\text{a} > 0</math> energy is absorbed within the sphere, otherwise energy is being created within the sphere. We will not consider this case here. If the host medium is non-absorbing, the energy must be absorbed by the particle. We decompose the total field into incident and scattered parts <math>\mathbf{E} = \mathbf{E}_\text{i} + \mathbf{E}_\text{s}</math>, and the same for the magnetic field <math>\mathbf{H}</math>. Thus, we can decompose <math>W_a</math> into the three terms <math> W_\text{a} = W_\text{i} - W_\text{s} + W_{\text{ext}} </math>, where
: <math>
W_\text{i} = - \oint_A \mathbf{\Pi}_\text{i} \cdot \hat{\mathbf{r}} dA \equiv 0, \qquad
W_\text{s} = \oint_A \mathbf{\Pi}_\text{s} \cdot \hat{\mathbf{r}} dA, \qquad
W_{\text{ext}} = \oint_A \mathbf{\Pi}_{\text{ext}} \cdot \hat{\mathbf{r}} dA.
</math>
where <math>\mathbf{\Pi}_\text{i} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_\text{i}^* \times \mathbf{H}_\text{i} \right] </math>, <math>\mathbf{\Pi}_\text{s} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_\text{s}^* \times \mathbf{H}_\text{s} \right] </math>, and <math>\mathbf{\Pi}_{\text{ext}} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_s^* \times \mathbf{H}_i + \mathbf{E}_i^* \times \mathbf{H}_s \right] </math>. 

All the field can be decomposed into the series of [[Vector spherical harmonics|vector spherical harmonics (VSH)]]. After that, all the integrals can be taken.
In the case of a '''uniform sphere''' of radius <math>a</math>, permittivity <math>\varepsilon</math>, and permeability <math>\mu</math>, the problem has a precise solution.<ref>Bohren, Craig F., and Donald R. Huffman. Absorption and scattering of light by small particles. John Wiley & Sons, 2008.</ref>  The scattering and extinction coefficients are <math display="block">
Q_\text{sc} = \frac{2}{k^2a^2}\sum_{n=1}^\infty (2n+1)(|a_{n}|^2+|b_{n}|^2)
</math> <math display="block">
Q_\text{ext} = \frac{2}{k^2a^2}\sum_{n=1}^\infty (2n+1)\Re(a_{n}+b_{n}) 
</math> Where <math display="inline">k = n_\text{host} k_0</math>. These are connected as <math display="block">
\sigma_\text{ext} = \sigma_\text{sc} + \sigma_\text{abs} \qquad \text{or} \qquad Q_\text{ext} = Q_\text{sc} + Q_\text{abs}
</math>