# Mie scattering intensity and spherical particles on

I am running an experiment where I shine infrared light on (almost) spherical particles on the micron scale (PM2.5 - PM10). I then look at the (90 deg) scattering properties to try and size the particles.

I am looking for the theory of Mie scattering and its intensity, specifically how the scattered intensity depends on the surface area of the particles.

Looking for suggestions of good literature on this, or just simple equations.

-

If the spherical approximation is good enough, you should be able to convert the surface areas into radii. I mention that because in my work with light scattering, all the equations are usually written in terms of the radius of the scatterers. It also looks like all the Mie scattering tables are written in terms of the radius of the scatterers as well.

The best I can do in terms of actual formulas is that the scattering is proportional to $1/a^2$, times an intensity factor that you have to look up in a table. $a$ is the radius of the scatterer. (If it seems weird that scattering could go down as radius increases, see below for an explanation.) Knowing that you're only looking at one angle eliminates the angular portion of the intensity factor, but it's a complicated function of the size of the scatterer due to resonance effects when $a/\lambda \approx 1$. I found two tables of Mie coefficients. The first only has values for $n=1.40$. The second has more indexes of refraction, but might have access restrictions (it initially told me I had access thanks to my university library).

Section 10.4 of Jackson derives some equations for the scattering of electromagnetic radiation from spherical particles, beginning from Maxwell's equations. He eventually leaves off without discussing the full problem. That might be a useful starting point for the theory.

Wikipedia just pointed me to an English translation of the original paper by Mie, which I didn't know existed. I haven't yet had a chance to read it, so I don't know how useful it is.

Everyone seems to refer back to Kerker as the first textbook that contains a full treatment of the Mie problem, but it's difficult to find, and very expensive. I would consider it only if you can get it through your university's library. I have a copy of the Dover edition of van de Hulst, which focuses specifically on the light scattering problem. It appears that there is a Dover edition of Stratton, now, as well.

Why does the scattering intensity go down as the particle's radius increases? The first caveat is that the scattering isn't just proportional to $1/a^2$; the intensity factor is also a function of particle radius. I'm only marginally familiar with the general Mie problem, so I don't fully know all the complications that introduces.

The other issue, and one that I am comfortable with, can be explained with reference to the Rayleigh scattering problem. In the experimetns I'm used to, we plot $1/I$ on the y-axis, and $\sin^2(\theta/2)$ on the x-axis ($I$ is the scattering intensity, and $\theta$ is the scattering angle). Under a set of approximations, that gives a straight line. The y-intercept is proportional to 1 over the molecular weight of the particle. The slope is proportional to the square of the radius of the particle. So for a given molecular weight, increasing the particle's radius will increase the slope of that line. So for a given angle, you have to increase $1/I$, which means that $I$ must go down. I think the underlying explanation for that that the density of the particle decreases, because you have increased the volume while keeping the same mass.

-