with the prime superscript denoting a first derivative with respect to the argument of a function, x = kr and y = mx, r is the radius of the particle, m its refractive index. In the above Mie solution, the medium surrounding the scattering particle was taken to be vacuum (m=1), however, if the surrounding medium has a real refractive index m2 (no absorption), then we obtain the solution by replacing m by m/m2 and A (in vacuum) by X/m2. Also
In the above, P^ (cos 0) is the associated Legendre polynomial, ^n and are the Ricatti-Bessel functions that are related to the Bessel functions through
where jn(z) an(i hn (z) are the spherical Bessel functions, while Jn^i(z) and (2)
are the fractional-order Bessel functions, and 2 is a complex number. The scattering function for forward scattering can be computed from
We can define a dimensionless efficiency factor for extinction by Qext = ^ext/nr2, and similarly for scattering and absorption. We can then compute Qext from
and the corresponding efficiency factor for scattering from
We can compute, for example, the scattering cross-section for a distribution of particle sizes from
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