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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 ...
I don't know specifically what you're looking for, but I can give you the basic idea of the relation between the cross section and the differential cross section. Generally, the cross section $\sigma$ is defined as the integral of the differential cross section $\frac{d\sigma}{d\Omega}$ over the entire solid angle. Here, $d\Omega$ is the spherical surface ...