Light Scattering: Gases

E = E₀ cos (omega t) = E₀ cos (2 pi ct/ lambda)

p = alphaE = alphaE₀ cos (2 pi ct/ lambda)

This oscillating dipole is itself the source of electromagnetic radiation that is emitted in all directions. Its amplitude depends on

• The frequency of the oscillation, through d² p/d t²
• The angle from the transmitted beam, through sin theta_z
• The distance from the scattering center, through 1/r

E = (alphaE₀/r)(2 pi r/lambda)² sin theta_z

i_z' = (16 pi ⁴/r²lambda⁴) I_0,z alpha² sin² theta_z

Conversion to unpolarized light is achieved by summing the contribution from two polarized incident beams of the same intensity, with polarizations 90^o different from one another.

i' = (8 pi⁴/r²lambda⁴) I₀ alpha² (sin² theta_z + sin² theta_y)

i' = (8 pi⁴/r²lambda⁴) I₀ alpha² (1 + cos² theta)

For N molecules in volume V, with each molecule acting independently,

i = (N/V)(8 pi⁴I₀/lambda⁴) alpha² [(1 + cos² theta)/r²}

where the four factors have the following origins:

• N/V: Concentration
• 8 pi⁴I₀/lambda⁴: Incident beam
• alpha²: Molecular property
• (1 + cos² theta)/r²: Instrument geometry

alpha = (1 / 2 pi)(d n/ d c)(M/L)

where L denotes Avogadro's number. This expression yields

i_x'r²/I₀ = R_theta = 2 pi² (d n/ d c)² (1/lambda⁴) (1 + cos² theta) (N/V) (M/L)²

R_theta, the Rayleigh ratio, depends on M².

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