PHY2505 - Lecture 6
Scattering by particles
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Outline
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Refractive index
Mie scattering – “white clouds”
an exact solution for homogeneous spherical particles
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Geometric scattering – refractive effects: rainbows,
halos..
Real atmospheric particles
Scattering effects due to shape and variations in composition
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Refractive index
Last time: Rayleigh scattering – what is the origin of the real and imaginery components of refractive
index? – Fundamentally it is the dispersion relationship for modes of oscillation in the Lorenz atom:
Liou: see Appendix D, p529-532
Refractive index,m, by definition is the ratio of speed of light in a medium to that in a vacuum (m2=eo)
We have an expression for refractive index in terms of “polarizability” – how do we relate this to EM
frequency…
Definition of polarizability is the separation of charges in the dipole induced by the electric field..
Relationship of polarizability to frequency is found by solving the equation for displacement r
generated by the Lorenz force on a charged particle.
The solution is in terms of a resonant frequency of oscillation of the dipole, and dispersion of
wave frequencies induced by the medium about this frequency
The half width of the natural broadening depends on the damping an=g/4p and line strength S is
pNe2/mec . Thus the absorption coeffiecient, k, is 4pnomi/c.
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Plots of refractive index components
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Mie scattering
An exact scattering solution for homogeneous spherical particles.
Derivation not too difficult but very long…
won’t go into in detail here..just main points:
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Based on wave equation in spherical polar co-ordinates, origin centre of
particle….Most of the mathematical complexity in this theory is due to expressing
a plane wave as an expansion in spherical polar functions
Scalar solutions to the wave equation are
Legendre
Bessel
Neumann
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Related to vector solutions by
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Series expansion for E and H of form
Hard part
M    ar r and N    M
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Mie scattering fields
Consider particle sphere radus r, refractive index m, surrounded by vacuum, m=1
To find the coefficients defining the scattered wave, use boundary condition at surface
of sphere
Incident field
Field expressed by incoming wave,
Bessel function  (k1r)
Scattered field
Field expressed by superposition of
incoming and outgoing waves
Giving coefficients:
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Angular dependence
Define functions
…from Bohren & Huffmann, 1986
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Scattering matrix form
Define scattering functions:
Then by considering parallel and perpendicular components of the field
Scattering
matrix
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Extinction efficiency
Find extinction cross section by superimposing incident and scattered fields in the
forward direction S1(q=0) and integrating over the sphere
Extinction efficiency
Approximations to Mie theory are based on a power series expansion of the Bessel
functions
Rayleigh
term
where
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Extinction efficiency for a sphere
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Geometric optics
http://www.sundog.clara.co.uk/atoptics/phenom.htm
A glory
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Geometric optics
In the geometric limit light can be thought of as a collection of individual rays.
This approximation is increasingly bad as the size parameter gets smaller, where phase
effects become important and the effect of a wavefront is smeared out over the
particle
To express the scattered wave field in the geometric scattering limit we must
superimpose the fields due to effects of diffraction, reflection and refraction
governed by fixed phase relations
Far field= Fraunhofer diffraction:
Bessel function solution:
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Geometric optics
Reflection and refraction:
Responsible for rainbows and glories:
Deviation due to multiple reflection and
refraction
p is number of internal reflections
Differenciating Snell’s law,
we get a minimum which governs the
angle the ray exits:
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Geometric optics
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Comparison of geometric and Mie scattering
Geometric optics is poor approximation for small x, asymptotic improvement for large x:
Computation of exact solutions for spheres,
spheroids and cyllinders possible from
Mie approach
(any shape where boundary can be expressed
on a surface of the co-ordinate system)
And for coated inhomogenous
particles with spherical symmetry..
BUT..
Computationally expensive:
a rough rule of thumb is that x
terms must be retained in Bessel expansion,
so for a raindrop – implies 12,000
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Real particles
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Real particles: thought expt
Smoothing out of features as size
parameter decreases…also observable
with “geometric effects”..
Inner rainbow observable only for
raindrops above a certain size
parameter….with small raindrops, it is
smeared out and disappears
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Real particles: refractive index
Smoothing out of features as size parameter
decreases…also observable with “geometric
effects”:
Inner rainbow observable only for raindrops
above a certain size parameter….with small
raindrops, it is smeared out and disappears
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Equation of radiative transfer through a scattering layer
1 Attenuation by extinction
2. Single scattering
3. Multiple scattering
4 3 2 1
4. Emission
Coefficients:
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RTE scattering parameters
To account for shape variation and inhomogenuity in real
particles, we introduce two parameters to characterize particles:
Single scattering albedo, w = bs/be
Asymmetry parameter, g = the “average” scattering angle
From expansion of the phase function
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Summary
We have looked at Mie scattering, geometric scattering and the
refractive index of real particles
…and related this to the radiative transfer equation
Next time we are going to look at thermal radiation…(Liou, chapter 4)
..radiative transfer models & computational techniques (practical 1)
..then look at the radiative transfer problem through a cloudy
atmosphere
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