# Is scattering cross section determined by mass?

This is just for visible light/photon scattering. Let's say I have small spherical particles in water, if I keep the weight concentration of glass the same but change the diameter of glass sphere from 0.5$\mu$m to 1$\mu$m, how will the scattering power, or the total scattering cross section change for visible light, e.g., at 632nm?

I was told as long as the mass doesn't change, the scattering cross section won't change, the size of diameter only modifies the phase function of the scatter. is that true?

• Who told you that? What were the assumptions for this statement? – CuriousOne Sep 11 '14 at 0:43

At around the $1\mu$m particle size you're in the Mie scattering regime, and this makes life hard because there isn't a simple analytic formula for the cross section due to Mie scattering. However if you're prepared to consider particle sizes significantly smaller than the wavelength of light, say $0.1\mu$m and smaller, then it's easy to show there is an effect of particle size.

If the mass of particles per unit voume is $M$, then the number of partices per unit volume, $N$, is just $M$ divided by the mass of a single particle, $\tfrac{4}{3}\pi r^3\rho$. The total scattering approximately (ignoring multiple scattering) $N\sigma$, where $\sigma$ is the scattering cross section, so:

$$S = \frac{M}{\tfrac{4}{3}\pi\rho} \frac{1}{r^3}\sigma$$

We'll ignore the constant numerical factors and just write:

$$S \propto \frac{1}{r^3}\sigma \tag{1}$$

Now, in the small particle size regime the cross section is given by the Rayleigh scattering equation. Again ignoring the constant factors (because there are lots of them):

$$\sigma \propto r^6$$

and substituting in our expression for the total scattering (1) gives:

$$S \propto \frac{1}{r^3}r^6$$

so:

$$S \propto r^3$$

So the total scattering does depend on the particle size, and indeed depends quite strongly on it. Our result applies only in the single scattering limit and small particle size, but since Rayleigh scattering evolves smoothly into Mie scattering as the size increases it does at suggest that you'd expect a dependance on particle size even in the Mie region.