# Mathematical explanation of the extinction paradox

I am trying to learn properly about scattering. For this I was pointed to Wave Propagation and Scattering in Random Media by Ishimaru.

I got a bit stuck in section 2-2 General properties of the Cross section. In general I don't understand the extinction paradox and even if I've watched some youtube videos and read other references I still find this obscure.

I'll quote the bits I don't get but a full mathematical explanation would be appreciated.

If the size of a particle is much greater than a wavelength the total cross section $$\sigma_t$$ approaches twice the geometric cross section $$\sigma_g$$.

Further down in the same page we have

The total flux $$S_i \sigma_g$$ within the geometric cross section $$\sigma_g$$ is either scattered out or absorbed by the particle. Behind the particle there should be a shadow region where practically no wave exists.

Question: What is this shadow region? and why we have "practically no wave exists"

In this shadow region the scattered wave from the particle is exactly equal to incident wave but 180 degrees out of phase abd this scattered flux is equal to $$S_i \sigma_g$$ in magnitude. The total scattered and absorbed flux, therefore, approaches $$S_i \sigma_g + S_i \sigma_g$$ and the total cross section $$\sigma_t$$ approaches $$\sigma_t \to 2S_i\frac{\sigma_g}{S_i} = 2\sigma_g$$

I get a bit confused because I don't understand the formulation, mathematical, of this phenomenon. Everything is stated but I cannot put this together using mathematical formulation.

There's also a reference (Van de Hulst - 1957) where the explanation is very similar to the one in this book with very little mathematical insight.

Can anyone clarify?

Note: The main reference is Scattering and Propagation in Random media - Ishimaru

Update : I think maybe this is consequence of the fact that at any point in space (where we have a scattering object) the electromagneting field can be written as:

$$E(x) = E_{incident}(x) + E_{scattered}(x)$$

If the wave is assumed to planar and assuming that shadow area means that for any $$x$$ in this area we have $$E(x) = 0$$ then we have

$$E_{incident}(x) + E_{scattered}(x) = 0 \Rightarrow E_{incident}(x) = -E_{scattered}(x)$$

This idea is taken from Comment on The Extinction Paradox.

Note: To be precise I should actually be talking about electromagnetic field I believe.

Please can anyone confirm my understanding is correct?

• Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. Apr 14 at 16:41
• Scattering is complicated in EM due to the polarization. Most of the basic features including the extinction paradox can be recovered in scalar theories like the Schrödinger equation quantum mechanics. Did you check out section 6.5 of Sakurai’s Modern Quantum Mechanics for example?
– LPZ
Apr 18 at 23:44
• No, I'll check. Thank you Apr 19 at 22:45