Editing Cross section (physics) (section)

== Scattering of light ==

For light, as in other settings, the scattering cross section for particles is generally different from the [[Cross section (geometry)|geometrical cross section]] of the particle, and it depends upon the [[wavelength]] of light and the [[permittivity]], shape, and size of the particle. The total amount of scattering in a sparse medium is proportional to the product of the scattering cross section and the number of particles present.

In the interaction of light with particles, many processes occur, each with their own cross sections, including [[Absorption cross section|absorption]], [[scattering]], and [[photoluminescence]]. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section.
: <math>\sigma = \sigma_\text{abs} + \sigma_\text{sc} + \sigma_\text{lum}.</math>
The total extinction cross section is related to the attenuation of the light intensity through the [[Beer–Lambert law]], which says that attenuation is proportional to particle concentration:
: <math>A_\lambda = C l \sigma,</math>
where {{math|''A<sub>λ</sub>''}} is the attenuation at a given [[wavelength]] {{math|''λ''}}, {{math|''C''}} is the particle concentration as a number density, and {{math|''l''}} is the [[Distance|path length]]. The absorbance of the radiation is the [[logarithm]] ([[Common logarithm|decadic]] or, more usually, [[Natural logarithm|natural]]) of the reciprocal of the [[transmittance]] {{mathcal|T}}:<ref name="Bajpai">{{Cite book|title=Biological instrumentation and methodology|last=Bajpai, P. K.|date=2008|publisher=S. Chand & Company Ltd|isbn=9788121926331|edition= Revised 2nd|location=Ram Nagar, New Delhi|oclc=943495167}}</ref>
: <math>A_\lambda = -\log \mathcal{T}.</math>

Combining the scattering and absorption cross sections in this manner is often necessitated by the inability to distinguish them experimentally, and much research effort has been put into developing models that allow them to be distinguished, the Kubelka-Munk theory being one of the most important in this area.

=== 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>

=== Dipole approximation for the scattering cross section ===

Let us assume that a particle supports only electric and magnetic dipole modes with polarizabilities <math display="inline">\mathbf{p} = \alpha^e \mathbf{E}</math> and <math display="inline">\mathbf{m} = (\mu \mu_0)^{-1}\alpha^m \mathbf{H}</math> (here we use the notation of magnetic polarizability in the manner of Bekshaev et al.<ref name="Bekshaev2013">{{cite journal | last=Bekshaev | first=A Ya | title=Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows | journal=Journal of Optics | volume=15 | issue=4 | date=2013-04-01 | issn=2040-8978 | doi=10.1088/2040-8978/15/4/044004 | page=044004| arxiv=1210.5730 | bibcode=2013JOpt...15d4004B | s2cid=119234614 }}</ref><ref name="Bliokh2014">{{cite journal | last1=Bliokh | first1=Konstantin Y. | last2=Bekshaev | first2=Aleksandr Y. | last3=Nori | first3=Franco | title=Extraordinary momentum and spin in evanescent waves | journal=Nature Communications | publisher=Springer Science and Business Media LLC | volume=5 | issue=1 | date=2014-03-06 | issn=2041-1723 | doi=10.1038/ncomms4300 | page=3300| pmid=24598730 | arxiv=1308.0547 | bibcode=2014NatCo...5.3300B | s2cid=15832637 | doi-access=free }}</ref> rather than the notation of Nieto-Vesperinas et al.<ref name="Nieto-Vesperinas2010">{{cite journal | last1=Nieto-Vesperinas | first1=M. | last2=Sáenz | first2=J. J. | last3=Gómez-Medina | first3=R. | last4=Chantada | first4=L. | title=Optical forces on small magnetodielectric particle | journal=Optics Express | publisher=The Optical Society | volume=18 | issue=11 | date=2010-05-14 | pages=11428–11443 | issn=1094-4087 | doi=10.1364/oe.18.011428 | pmid=20589003 | bibcode=2010OExpr..1811428N | doi-access=free }}</ref>) expressed through the Mie coefficients as 
<math display="block">
\alpha^e = 4 \pi \varepsilon_0 \cdot i \frac{3 \varepsilon}{2 k^3} a_1, \qquad  \alpha^m = 4 \pi \mu_0 \cdot i \frac{3 \mu}{2 k^3} b_1.
</math> 
Then the cross sections are given by <math display="block">
\sigma_{\text{ext}} = \sigma_{\text{ext}}^{\text{(e)}} + \sigma_{\text{ext}}^{\text{(m)}} =
\frac{1}{4\pi \varepsilon \varepsilon_0} \cdot 4\pi k \Im(\alpha^e) + 
\frac{1}{4\pi \mu \mu_0} \cdot 4\pi k \Im(\alpha^m)
</math> <math display="block">
\sigma_{\text{sc}} = \sigma_{\text{sc}}^{\text{(e)}} + \sigma_{\text{sc}}^{\text{(m)}} = 
\frac{1}{(4\pi \varepsilon \varepsilon_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^e|^2 +
\frac{1}{(4\pi \mu \mu_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^m|^2
</math> and, finally, the electric and magnetic absorption cross sections <math display="inline">\sigma_{\text{abs}} = \sigma_{\text{abs}}^{\text{(e)}} + \sigma_{\text{abs}}^{\text{(m)}}</math> are <math display="block">
\sigma_{\text{abs}}^{\text{(e)}} = \frac{1}{4 \pi \varepsilon \varepsilon_0} \cdot 4\pi k \left[ \Im(\alpha^e) - \frac{k^3}{6 \pi \varepsilon \varepsilon_0}  |\alpha^e|^2\right]
</math> and <math display="block">
\sigma_{\text{abs}}^{\text{(m)}} = \frac{1}{4 \pi \mu \mu_0} \cdot 4\pi k \left[ \Im(\alpha^m) - \frac{k^3}{6 \pi \mu \mu_0}  |\alpha^m|^2\right]
</math>

For the case of a no-inside-gain particle, i.e. no energy is emitted by the particle internally (<math display="inline">\sigma_{\text{abs}} > 0</math>), we have a particular case of the [[Optical theorem]] <math display="block">
\frac{1}{4\pi \varepsilon \varepsilon_0} \Im(\alpha^e) + \frac{1}{4\pi \mu \mu_0} \Im(\alpha^m) \geq \frac{2 k^3}{3} \left[ \frac{|\alpha^e|^2}{(4\pi \varepsilon \varepsilon_0)^2}  + \frac{|\alpha^m|^2}{(4\pi \mu \mu_0)^2} \right]
</math> Equality occurs for non-absorbing particles, i.e. for <math display="inline">\Im(\varepsilon) = \Im(\mu) = 0</math>.

=== Scattering of light on extended bodies ===

In the context of scattering light on extended bodies, the scattering cross section, {{math|''σ''<sub>sc</sub>}}, describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from the [[cross section (geometry)|geometrical cross section]] of a particle, as it depends upon the wavelength of light and the [[permittivity]] in addition to the shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross section and the number of particles present. In terms of area, the ''total cross section'' ({{math|''σ''}}) is the sum of the cross sections due to [[absorption cross section|absorption]], scattering, and [[luminescence]]:
:<math>\sigma = \sigma_\text{abs} + \sigma_\text{sc} + \sigma_\text{lum}.</math>

The total cross section is related to the [[absorbance]] of the light intensity through the [[Beer–Lambert law]], which says that absorbance is proportional to concentration: {{math|''A<sub>λ</sub>'' {{=}} ''Clσ''}}, where {{math|''A<sub>λ</sub>''}} is the absorbance at a given [[wavelength]] {{math|''λ''}}, {{math|''C''}} is the concentration as a [[number density]], and {{math|''l''}} is the [[Distance|path length]]. The extinction or [[absorbance]] of the radiation is the [[logarithm]] ([[decadic logarithm|decadic]] or, more usually, [[natural logarithm|natural]]) of the reciprocal of the [[transmittance]] {{mathcal|T}}:<ref name="Bajpai" />
: <math>A_\lambda = - \log \mathcal{T}.</math>

==== Relation to physical size ====
There is no simple relationship between the scattering cross section and the physical size of the particles, as the scattering cross section depends on the wavelength of radiation used. This can be seen when looking at a halo surrounding the Moon on a decently foggy evening: Red light photons experience a larger cross sectional area of water droplets than photons of higher energy.  The halo around the Moon thus has a perimeter of red light due to lower energy photons being scattering further from the center of the Moon. Photons from the rest of the visible spectrum are left within the center of the halo and perceived as white light.

=== Meteorological range ===
The scattering cross section is related to the [[visibility|meteorological range]] {{math|''L''<sub>V</sub>}}:
: <math>L_\text{V} = \frac{3.9}{C \sigma_\text{scat}}.</math>

The quantity {{math|''Cσ''<sub>scat</sub>}} is sometimes denoted {{math|''b''<sub>scat</sub>}}, the scattering coefficient per unit length.<ref>{{GoldBookRef|title=Scattering cross section, {{math|''σ''<sub>scat</sub>}}|file=S05490}}</ref>