Way back on 9/3/00 -0400, I suggested a way to think about the physics of
the blue sky. I understand it a little better now, so here is a revised
and extended version.
A major goal is to understand why the scattering
depends on the fourth power of the wavelength.
## Assumption: We shall consider the limiting case of pure air without
clouds, dust, or pollution. This is "sometimes" a decent approximation in
real life, but alas it is not the whole story.
At 02:39 AM 9/2/00 -0400, Hugh Logan wrote:
>As for the density of a medium in the context of scattering, Stone
>emphatically means density in the optical sense. A dense medium in this
>sense is one in which the number of atoms in a wavelength cube is
>significant
Statements like that crop up a lot in discussions of the blue sky. However,
I believe any alleged explanation of that sort is wrong physics, and misses
the right physics, as will become clear below.
Consider the following diagram of the interaction:
1) Let's use this to identify a pattern in the index-deviations that will
result in strong scattering.
-- At each point P, a crest lines up with a crest. A positive
deviation in the index at this point will make a positive contribution to
the overall interaction.
-- At each point N, a crest lines up with a trough. A _negative_
deviation in the index at this point will make a positive contribution to
the overall interaction.
2) In fact, we can classify all the zones in the interaction region as to
whether a positive or negative index-deviation results in a positive
contribution to the desired interaction. Call them the "P zones" and "N
zones" respectively.
## Without loss of generality, we have made some mildly arbitrary
choices about the relative phases.
## We assume that the interaction region is transparent to zeroth
order. This is consistent with (and stronger than) previous assumptions.
3) Note that the P zones collectively cover half the interaction region, while
the N zones cover the other half.
## This assumes the interaction region is reasonably large relative to
lambda. This is consistent with previous assumptions.
## This assumes the scattered beam does not coincide with the
transmitted beam. In the other case (i.e. forward scattering) statement
(3) is not true, which is a good thing; the distinction allows us to
uphold the optical theorem, conservation of energy, and other good things.
4) Essentially we have figured out what sort of diffraction grating (or
hologram) would be ideal for creating the desired interaction. According to
this analysis, we are getting coherent contributions from throughout the
interaction region -- which is vastly larger than wavelength cubed! And
the total volume that contributes constructively (50% of the interaction
region) is in fact independent of wavelength.
5) Now the statistical question reduces to this: what is the chance that the
air will fluctuate into a configuration that has an extra-large number of
molecules in the P zones, and an extra-small number of molecules in the N
zones? Essentially we are talking about thermally-excited sound modes.
The amplitude of these excitations should be independent of frequency,
since the compressibility of air doesn't depend on wavelength.
## We are approximating the air as an ideal gas. This should be a very
good approximation.
Compressibility being independent of wavelength explains why I'm very
skeptical of any argument that says that we should consider fluctuations in
a volume on the order of wavelength cubed, implying that short-wavelength
fluctuations should be more prominent.
6) If you don't want to think of it in terms of compressibility, you can
instead use elementary notions of statistical mechanics. That is, we model
the distribution of air molecules as a random statistical
process. Distributing the air molecules into P and N regions is like
scattering sand particles _at random_ onto a checkerboard. It doesn't
matter how large are the black and white cells on the board. It doesn't
matter whether they are even square or not. As long as half of the board
is black and half of the board is white, and as long as the sand-grains are
randomly and independently distributed, about half of them will land on
black squares and about half will land on white squares.
7) The fluctuation in index of refraction is proportional to the
fluctuation in the number of atoms, according to the Clausius-Mossotti
relation.
## Assumption: This assumes the index is not too different from
unity. This is a good assumption for gasses under ordinary conditions. If
you care about liquids, you could easily relax this assumption and redo the
following derivation, to easily derive a more general result.
8) We can use a variant of the Born approximation to understand how a
fluctuation in refractive index produces scattering. Write the wave
equation as
v^2 (del / del x)^2 phi - (del / del t)^2 phi = 0 (Eq. 1)
and let the velocity (v) be given by
v = v0 / (1 + d) (Equation 2)
where (1 + d) is the index and (d) is the fractional deviation from the
average index, and is small compared to unity. And (v0) is the propagation
speed associated with the average index.
Then we have, to first order in d:
v0^2 (1-2d) (del / del x)^2 phi - (del / del t)^2 phi = 0
and by re-arrangement:
v0^2 (del / del x)^2 phi - (del / del t)^2 phi =
2d v0^2 (del / del x)^2 phi (Equation 4)
So, a term involving the index-deviation (2d) can be moved to the RHS and
viewed as a source term for the unperturbed wave equation.
## We are assuming that the scattering is not too strong, and the
interaction region is not too overly huge, so that we only need to worry
about _single_ scattering (from the incident beam to the scattered
beam). We are ignoring any possible secondary scattering (out of the
scattered beam). This is the _first_ Born approximation.
## For simplicity, we are ignoring polarization. It would introduce
some factors that depend on theta (the scattering angle). You can add them
in if you like.
## For clarity, I left out the (y) and (z) variables in equation
(4). But you get the idea.
9) But wait, (2d) is not the only factor on the RHS of equation (4). The
source term is not (2d) times the wavefunction, it is more like (2d) times
the second derivative of the wavefunction. For periodic waves, the
scattered amplitude picks up a factor of k^2 (or omega^2), and the
scattered power picks up a factor of k^4 (or omega^4).
10) This factor of k^4 is not "on top of" the factor of k^4 you would get
in the formula for scattering from a single molecule. It is the _same_
factor, derived in a slightly more general context. Specifically, you can
view the isolated atom as an isolated localized deviation in the index of
refraction of the vacuum.
======================
To summarize:
-- Divide the interaction region in to P zones and N zones. This is just
geometry. (The size, shape, and location of the zones will depend on
wavelength, but this is unimportant. The important thing is that all
molecules in the P zones contribute coherently and constructively to the
scattered wave we are trying to create, while all molecules in the N zones
contribute coherently and destructively.)
-- Distribute the air molecules into these zones at random, like sand on
a checkerboard.
-- If the distribution comes out even, there is no scattering. If there
is, by fluctuation, an imbalance between P and N, there will be
scattering. The chance of an imbalance is independent of wavelength.
-- The significance of an imbalance does depend on wavelength. In the
Born approximation, a index-deviation is a source term for the wave
equation. It is a stronger source when the wavelength is small. The
scattering amplitude is proportional to k^2, and the intensity is
proportional to k^4.