Rayleigh Scattering

The scattering of light by gases was first treated quantitatively by Lord Rayleigh in 1871 in an effort to explain the blue colour of the sky and the red colour of the sunset. There are numerous ways of arriving at the equation which governs Rayleigh scattering. Classical EM theory can be employed using the far-field solution while retaining only the dipole term (e.g.: Jackson, 1962). Quantum mechanical perturbation theory for two-photon elastic processes provides both a powerful and elegant means (e.g.: Craig and Thirunamachandran, 1984).

In order for Rayleigh scattering to be valid, the size of the particle must be much smaller than the wavelength of the incident radiation, both inside and outside of the particle. These conditions can be expressed as,

  

$$r \ll \lambda {\rm or} \alpha \ll 1$$ (3.46)

$$r \ll \frac{\lambda}{\vert m\vert} {\rm or} \vert m\vert\alpha \ll 1$$ (3.47)

where m is the refractive index. Rayleigh scattering theory is applicable to scattering of UV and visible radiation by air molecules, infra-red radiation by small aerosols, and microwave radiation by cloud and rain drops. Practical applications of Rayleigh scattering include lidar, weather radar, and remote sounding of cloud water.

The route adopted herein is to consider the limiting case of Mie scattering. When equation (2.46) is valid, only the n=1 term in the Mie scattering functions need be retained so that,

$$S_1(\Theta) \simeq \frac{3}{2} [a_1 \pi_1(\Theta) + b_1 \tau_1(\Theta)]$$ (3.48)

$$S_2(\Theta) \simeq \frac{3}{2} [b_1 \pi_1(\Theta) + a_1 \tau_1(\Theta)]$$ (3.49)

where $\pi_1(\Theta)=1$, $\tau_1(\Theta)=\cos\Theta$ and $P^1_1(\cos \Theta)=\sin \Theta$. Furthermore, if equation (2.47) is valid, then $\vert b_1\vert\ll\vert a_1\vert$ and hence,

 

$$S_1(\Theta) \frac{3}{2}a_1$$ = (3.50)

$$S_2(\Theta) \frac{3}{2}a_1 \cos\Theta$$ = (3.51)

$$-\frac{i2\alpha^3}{3} \frac{(m^2-1)}{(m^2+2)}.$$ a₁ = (3.52)

Using equation (2.32), the Rayleigh scattering matrix has the form

$${\b P}(\Theta) = \frac{3}{2} \left[ \begin{array}{c c c c} \f... ...os\Theta & 0 \\ 0 & 0 & 0 & \cos\Theta \\ \end{array}\right]$$ (3.53)

and from equation (2.39), the Rayleigh scattering cross-section is,

$$\sigma_R = \frac{8 \pi^3}{3} \frac{(m^2-1)^2}{\lambda^4 N^2}$$ (3.54)

where the approximation has been made, $m+2\simeq3$, and N is the number of molecules per unit volume at standard temperature and pressure. The $\lambda^{-4}$ dependence is a very important property of molecular scattering and is the reason why optical depth varies by a factor of 10 throughout the visible spectral region. The Rayleigh asymmetry factor is zero as forward and backward scattering are equally probable. Note that if the index of refraction has an imaginary component, then the Rayleigh absorption cross-section will be non-zero and can be determined in a similar manner using equation (2.39). While air consists of many species which absorb in the visible and near-UV, it is convenient to treat this separately and so the refractive index of air is taken to be real.