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lecture notes2

2 2 at long wavelengths a we therefore see that

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Unformatted text preview: the cross section is € 2 8πa 6 ε − 1 σ sc = . 3 4 ε + 2 At long wavelengths, we see that the scattering cross section decreases rapidly as ~1/λ4 (aside from possible issues with the dielectric constant) – this is the same € physics as the Rayleigh blue ­sky law. We may also consider absorption. The power dissipation (in erg/cm3/s) is given by the work done by the electric field on the dipoles: 1 ω Imε ˙ W = E ⋅ P = ω (Im χ ) | E |2 = | E |2 . 2 8π Integrating this over the sphere, we find an absorbed power € 2 3 ωa 3 Imε 2 ωa Imε 3E ext Pabs = |E | = . 6 6 ε +2 The absorption cross section is then € 12πa 3 Imε σ abs = . | ε + 2 |2 At long wavelengths, λ>>a, we therefore see that the absorption will dominate unless the medium is lossless (ε nearly real). € B. ABSORPTION & SCATTERING: SHORT WAVELENGTH LIMIT For materials with modest dielectric constants (i.e. χ of order unity), one may estimate the large ­angle scattering of radiation using geometric optics. Radiation falling within the geometric cross section σ=πa2 is refracted or reflected at the grain surface, and is either scattered or absorbed depending on whether the optical depth through the grain (~ka Im ε) is small or large. An additional contribution to the cross section occurs due to diffractive effects. The presence of an obstruction causes small ­angle scattering of radiation with a power equal to that that would have passed through the obstruction. This 3 radiation is deflected through a typical angle θ~λ/a, and the corresponding cross section is πa2. Thus the total cross section is twice the geometrical value: σ tot = 2πa 2 . At λ<<a the diffracted radiation can result in a “halo”...
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