I am doing some simulations with the Meep code to study some properties of a metal nanoparticle.
The situation is this: an incident electromagnetic wave on a metal nanoparticle. By now I know how to calculate the scattering, absorption and extinction cross sections and their efficiencies but I need the scattering angular distribution, that's why I need to calculate the differential scattering cross section and now I am a bit lost.
Is there a way to relate the cross section to the differential cross section?
Best Answer
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 element: $d\Omega\equiv \sin\theta\ d\theta\ d\phi $, with $\theta$ and $\phi$ the usual angles defined on the sphere, which parametrize the outgoing direction of the scattering object. So, we have $$\sigma=\oint d\Omega \frac{d\sigma}{d\Omega} $$ In many cases, we have cylindrical symmetry so that we can simplify: $$\sigma =b\oint \frac{d\Omega}{\sin\theta}\left|\frac{d b}{d\theta}\right|=2\pi b\int_0^\pi d\theta\left|\frac{d b}{d\theta}\right| $$ Here, $b$ is the impact parameter. The absolute value signs are there to assure that we get a positive outcome, even though one usually has lower deflection with larger impact parameters. You may also benefit from reading the relevant Wikipedia article, although it will not tell you what to do in your specific situation .