Rayleigh Scattering Outline Electric fields by charges Electric potential of charges Dipoles The electric potential of a dipole far away The “retarded potential” (yes, that’s right) and the speed of light in a vacuum. Polarization (induced dipole) of neutral atoms by an external electric field Oscillating polarization of matter by a polarized E+M wave – the Rayleigh approximation Polarization of bulk matter The radiation component of of an oscillating dipole using the retarded potential The definition of polarized Rayleigh scattering The intensity and phase function of polarized Rayleigh scattering The polarization of the scattered wave The intensity and phase function of unpolarized Rayeligh scattering Examples from the blue sky Examples from radar meteorology Back-scatter of a radar signal by a particle The importance of l-4 and R6.

Electric Fields by Charges All molecules are made of protons and electrons, which are positively and negatively charged particles having the fundamental unit of charge e. Static (not moving) charges fill the space around them with an electric field that follows Gauss’ Law If there is another charge, q2, at location r2, it will feel a force equal to q’s electric field times its own charge. F = E(r2)q2 (This second charge will also create its own electric field that will exert a force on the first one…) q2 r2 q

Electric Potentials by Charges Because E depends ONLY on distance from the particle, we can simplify its representation by defining an electric potenial, V We just made the math way simpler by describing all the vector information in E within a scalar V. E has the magnitude and direction of the downhill gradient in V. If you take a 2-D slice through space centered on a point charge, and then plot V as a third dimension, you end up with “volcanos” around positive charges, and funnels below negative ones. You can visualize the downhill gradient as being the force on a positive charge. A negative charge will “fall” uphill.

Dipoles r+ A “dipole” is simply two opposite charges separated by some distance s Both E and V are additive Very far from two oppositely charged particles, their fields tend to cancel (same magnitude, opposite direction) Between the particles, the fields reinforce one another In the plane perpendicular to s, the part of the field directed away from the particles cancels. What’s left behind is a field that’s parallel to their separation vector, s. r- s

The Electric Potential far from a Dipole We simplify things by defining a “center” of the dipole for our coordinate system Far away from the particle, we can do a Taylor expansion to get Putting this into our dipole potential and doing another Taylor expansion yields We define a “dipole moment”, which captures the relevant properties of the dipole far from the charges Note that the dipole potential is zero in the plane z = 0 (i.e. q = 90º).This does NOT mean the electric field is zero, since there is still a gradient. Note V also falls off like s/r2 , which is faster than for the 1/r dependence of the point charges alone, due to the cancellation r+ r- r q s

The Retarded Potential This simply means that if a charge is moving (i.e. r = r(t)), the potential far from the particle does not instantaneously reflect the new position of the particle Instead, the “information” travels at the speed of light (through a vacuum). Thus the electric potential (NOT field) at a location far from the particle is given by The main point here is that if the two sides of a dipole are moving (oscillating, in our case), then: r(t-r/c) r(t) v V(t) V(t) ! The point “sees” the + and – sides of the dipole at different times in its history! This matters!

Polarization of Neutral Atoms by an Electric field Atoms are normally electrically neutral, because the centers of charge of the positive nucleus and negative electron shells are collocated. When a uniform external electric field Eext surrounds an atom, the nucleus is pulled in the direction of Eext and the electron shell is pulled oppositely. Now that the charges are not collocated, a new electric field is set up such that the attraction between the nucleus and the electrons balances the effect of Eext. Far from the atom, this is perceived as an induced dipole moment, p. That is, the atom has become polarized by Eext. The potential created by the new dipole, Vd, is governed by its moment, and added to the potential by the external field: No electric field Nucleus (+) Electron Shell (-) + Induced Dipole Eext + s -

Atomic Rayleigh Scattering by E+M Waves E+M waves are simply self-propagating electric (and magnetic) fields If an atom is small compared to the wavelength of radiation passing by it, it experiences an effectively uniform field around it This field instantaneously causes a dipole moment in the atom This implies that the dipole potential, Vd, will also oscillate, creating, in effect, a wave of its own. This is Rayleigh scattering. Induced Dipole

Particulate Rayleigh Scattering by E+M Waves A bulk medium has an induced dipole moment per unit volume P. Instead of thinking about each individual atom becoming a dipole, it’s more convenient to think of two continuous clouds of charge – one positive, and one negative that get shifted by an external field. The total dipole moment of a polarized sphere is The degree to which they shift is quantified by the electrical susceptibility of the medium, c0, which will be related to the polarizability, a, of the atoms within, and how tightly packed together these atoms are, N. The field within the medium feeds back on the atoms, and so you can’t just sum up all the Vps if the atoms were alone. + E0 - + -

Particulate Rayleigh Scattering by E+M Waves Because of this polarization, the electric field is reduced below that of the external field 1 + ce is also called the dielectric constant, K, where K = n2 and n is the index of refraction we’ve already introduced. These are all different sides of the same die. We often don’t know the polarizability of an individual atom, a, and instead compute it from that of the bulk medium.

Radiation of an oscillating dipole Now let’s go back to the retarded potential of an atomic dipole We simulate the dipole oscillation as Where, you recall, The particle now sees the potential from the far side of the dipole later than it sees the near side. We won’t work out the details (because it gets cumbersome), but how does this work out? r+ r- r q s

Radiation of an oscillating dipole In the end, there’s a large term that looks simply like the 1/r2 dipole potential. We don’t care about this, because for it to radiate energy indefinitely, we need a 1/r term. A small term does show up with 1/r dependence… Remember, r- r q s