The scattering matrix assumes the following form in the small-particle, i.e. Rayleigh approximation for Mie theory:
|
|
| |
½(1 + cos2θ) |
½(cos2θ - 1) |
0 |
0 |
|
| |
½(cos2θ - 1) |
½(1 + cos2θ) |
0 |
0 |
|
| |
0 |
0 |
cosθ |
0 |
|
| |
0 |
0 |
0 |
cosθ |
|
|
(1)
|
where, a1, the first of the an coefficients of the Mie series (see Mie theory: Overview), can be approximated as follows
where
The optical efficiencies: attenuation efficiency, Qc, scattering efficiency, Qb, and absorption efficiency, Qa, can be expressed as follows (for example, Bohren and Huffman 1983):
|
Qc = 4x Im { P [1 +
|
|
P
|
|
]} +
|
|
x4 Re(P2)
|
(2)
|
|
Qa = 4x Im { P [1 +
|
|
Im P
|
]}
|
(4)
|
If (4x3 / 3) Im P << 1, which applies in the small-particle limit, then the absorption efficiency can be expressed approximately as follows:
See Fig. 1 in Mie theory: Particle size-dependent patterns for a comparison of the Rayleigh approximation of the attenuation efficiency with the results of Mie theory.
