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PHY2505 - Lecture 6

Scattering by particles

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Outline

• Refractive index• Mie scattering – “white clouds”

an exact solution for homogeneous spherical particles

• Geometric scattering – refractive effects: rainbows, halos..

• Real atmospheric particlesScattering effects due to shape and variations in composition

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Refractive indexLast 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, p529-532Refractive index,m, by definition is the ratio of speed of light in a medium to that in a vacuum (m2=o)

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 n=/4 and line strength S is Ne2/mec . Thus the absorption coeffiecient, k, is 4omi/c.

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Plots of refractive index components

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Mie scatteringAn exact scattering solution for homogeneous spherical particles.

Derivation not too difficult but very long…

won’t go into in detail here..just main points:

• 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

• Related to vector solutions by

• Series expansion for E and H of form

MNaM andrr

Hard part

Legendre Bessel Neumann

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Mie scattering fieldsConsider 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

Scattered field

Giving coefficients:

Field expressed by incoming wave, Bessel function (k1r)

Field expressed by superposition of incoming and outgoing waves

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Angular dependenceDefine functions

…from Bohren & Huffmann, 1986

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Scattering matrix formDefine scattering functions:

Then by considering parallel and perpendicular components of the field

Scattering matrix

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Extinction efficiencyFind extinction cross section by superimposing incident and scattered fields in the

forward direction S1(=0) and integrating over the sphere

Extinction efficiency

Approximations to Mie theory are based on a power series expansion of the Bessel functions

where

Rayleigh term

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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

4 3 2 1

1 Attenuation by extinction

2. Single scattering

3. Multiple scattering

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, = s/e

Asymmetry parameter, g = the “average” scattering angle

From expansion of the phase function

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SummaryWe 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