# Concerning Scattering Intensity and Particle Concentration

I am trying to determine what governs my sensor output. I have an optical sensor that emits infrared radiation on a sample volume and gives me a voltage output from the scattering of (1 to 10 micron) particles through a 90 degree scattering angle. I introduce aerosol particles of various sizes and keep track of the scattering output voltage

Parallel to my sensor output I also record the Number, surface and mass concentration of my aerosol particles from a different sensor unit.

I am trying to determine what governs my sensor output voltage. I know that Mie scattering depends on Mie intensity parameters, which depends on particle size, i.e.

$$I(\theta) = \frac{I_0 \lambda^2(i_1 + i_2)}{8\pi R^2}$$

where $$i_1, i_2$$ depends on the particle size.

As I increase the number concentration, my signal increases, indicating a dependence on total particle surface area in the sample.

I have plotted my sensor output vs number, surface and mass concentration for different particle sizes.

Should I not be able to make my Sensor Output vs Number Concentration for the different sized particles collapse if I multiply the number concentration by the area of the individual particles? Or is the area dependence in Mie scattering more complicated than this?

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First of all, the amount of scattered light should be proportional (linearly) to the concentration of aerosol particles, regardless of their size. The dependence may be nonlinear if you put too much particles (the gas becomes opaques). It may also be the property of your light detector - it may be better to measure current instead of voltage.

For large particles particles, the scattering intensity does indeed scale linearly with the surface area. However, if particle size is of the same order of magnitude as the wavelength, the dependence is more complicated, see the Wikipedia article on Mie scattering.

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For a population of identical aerosol particles, the scattering intensity should be approximately proportional to the number concentration of particles. Under two approximations, this is exact. @giacyan's answer points to one approximation: the number concentration of the particles has to be low enough that you can neglect multiple scattering events. If each photon is hitting more than one particle before you detect it, then the scattering will cease to be proportional to the concentration (I believe in that case the scattering increases more slowly than proportional to the concentration). The other approximation is that the particles don't interact with each other. If there are $A_2$ effects, then the scattering can increase either faster or more slowly than linearly with concentration.

If you have a mixed population of different sizes of particle, then the above holds as long as you keep the relative mixture the same. Particles that scatter more will bring you into the multiple scattering regime at lower concentrations. Particles that interact with each other more will bring you into the $A_2$ regime at lower concentrations.

The dependence on total surface area is more complicated. The formula you give is the scattering intensity for a single particle. $R$ is the radius, so $R^2$ is proportional to the surface area. But as you note, the $i_1, i_2$ functions are also functions of the radius (and therefore surface area). So you have to account for those functions when calculating the scattering. The total scattering is the sum of the scattering from each individual particle. For the same total surface area, the scattering can change if you change the distribution of particle sizes.

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