Lecture #11a-- Optical properties of dust


1) The Thomson cross section per gram of a free electron, where the "per gram" part is attributed to the obligatory nearby proton, is about 0.4 cm^2/g. If we assume that a 1 micron spec of dust has a cross section of roughly its physical dimensions for any light with a wavelength much smaller than those dimensions, then it has a cross section per gram about ten thousand times that of a free electron plus proton combination. But isn't the opacity in the dust due to electrons, and aren't those electrons less free to move about than free electrons? So why is the cross section per gram so much higher? The answer is the difference between a laser and a lightbulb: coherence effects. Relative to the Thomson cross section, forces that bind the charge can reduce the total cross section, but coherence effects can recover and even vastly exceed the Thomson cross section.
2) When restrictions on charge movement reduce the scattering
In the limit of low frequency, the cross section from each scattering charge exhibits the Rayleigh result that the induced dipole moment is proportional to the electric field and independent of frequency, so the total scattered power scales like frequency to the 4th power. Thus at low frequency, the total cross section can be much less than the Thomson result. So there will always be some frequency where the product of N times the frequency yields a cross section that is the same as N incoherent free electrons, and below that frequency, the total cross section is less than N free electrons.
3) When coherence giveth
In this Rayleigh limit, the dust particle interacts with light that has a much larger wavelength than the size of the dust particle, so the entire dust grain experiences an induced dipole that is nearly in phase. Thus all the individual charges that participate in the induced dipole moment are producing scattering amplitudes that are in phase with each other also, and will add coherently, even at large distance. That means N amplitudes create a total amplitude that is N times larger than a single charge, and the intensity of the scattered light scales like N squared. Since this is the Rayleigh limit, the amplitudes are themselves proportional to frequency squared (to give an intensity proportional to frequency to the 4th power). Thus the total amplitude scales like the square of N times frequency, and at high enough N, this can exceed the incoherent sum of N Thomson cross sections. Thus the cross section per gram of a large enough dust particle can exceed 0.4 square cm per g. This excess over N single charges is due to coherence in the response of the entire dust grain to the field, and if N is large enough it can overcome the tendency for the binding forces to weaken the scattering response of each charge.
<4>When coherence saturates
In the opposite limit when the wavelength is much smaller than the radius of the dust particle, we know the cross section becomes the geometric cross section. That means the cross section per gram scales like the geometric cross section divided by the volume (for fixed mass density in the dust particle), so like inverse radius. So the cross section per gram must make a transition from scaling like N^2/V, which is like R^3, to scaling like 1/R, somewhere around where R (or really 2 pi times R) is of order the wavelength. This is also where the coherence is lost-- the entire dust grain is no longer responding like a single in-phase dipole, there are becoming multiple wavelengths across the particle so multiple phases in the dipole oscillations. The coherence has saturated, and we get no further benefit from it, indeed we begin to lose benefit as destructive interference creeps in and gives the falling 1/R dependence in the oross section per gram.

This loss of coherence can be described easily in the limit of many wavelengths across the radius of the dust particle. Then all N amplitudes contributing to the total scattering will add with effectively random phase, so the N amplitudes will cancel up to the Poisson variance of 1 over the square root of N. That means the total amplitude sum will have an expectation value of the square root of N, rather than the N, and the intensity will scale like the square of that-- so like N rather than N^2. That is the telltale sign of incoherence (as opposed to the N^2 signature of constructive interference, or the zero of destructive interference). In that limit, the total cross section per gram scales like N/V, which is independent of R, it is just a constant, and this is the Thomson cross section of a collection of incoherent scatterers in the limit of fast vibration.

However, that is not what the cross section per gram of a large dust grain is, the actual cross section per gram will be less than that, just as will be the cross section per gram of a dust grain much smaller than the wavelength. The small grain has a small cross section per gram because of the binding of the electrons that restrain their oscillation, and the large grain will have a small cross section because of a different effect-- self-shadowing. A large just grain is not an arbitrary group of effectively free electrons within a homogeneous mass distribution, it is a clump of effectively free electrons, surrounded by near vacuum, so the radiative flux can be diminished in the clump, and most of the charges in the dust grain might not receive any radiative flux to scatter. This is the effect that limits the cross section of the dust grain to its geometric cross section.

5) Summary of cross-section per gram of a dust grain
In summary, the scatterers of light in a dust grain are electrons, so one can imagine that the Thomson cross section per gram of an electron-proton pair is a benchmark for the cross section per gram of a dust grain. However, the dust grain shows no tendency to follow the cross section per gram of such an incoherently summed collection of effectively free electrons, and can instead be much less than that for very small grains (relative to the wavelength), can be much greater than that for medium sized grains, and can again fall far below that benchmark for very large grains. The first effect is that the electron binding forces can suppress scattering to sub-Thomson levels, the second effect is the enhancing effects of the coherence of the N electrons in the dust grain, and the third effect is the self-shadowing within the dust grain.

The net result of all this is that if you fix the wavelength of interest, and vary over the dust-grain size, you find the peak contribution for the cross section per gram comes when the radius R obeys 2*pi*R equals the wavelength of the light. So if you have a distribution dN/dR of dust grains over R, you will typically find that the cross section per gram is set by the particles within about a factor of 2 or so of that ideal size.