What is the physical interpretation of f(0) in the optical theorem?

[erwinscat]

Hello everyone. I have an understanding problem with the Optical theorem definition.

From Wikipedia :

"In physics, the optical theorem is a general law of wave scattering theory, which relates the forward scattering amplitude to the total cross section of the scatterer. It is usually written in the form:

\sigma _{tot}=\frac{4\pi}{k}Im f(0)

where f(0) is the scattering amplitude with an angle of zero, that is, the amplitude of the wave scattered to the center of a distant screen. Because the optical theorem is derived using only conservation of energy, or in quantum mechanics from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics, σtot includes both elastic and inelastic scattering. "

What does "f(0) is the scattering amplitude with an angle of zero, that is, the amplitude of the wave scattered to the center of a distant screen" exactly mean ? Are we talking about a head to head scattering ? Does angle zero mean there is no impact parameter ?

I do not understand the physical situation...any help is welcome !

Thanks a lot in advance,
Erwin

 

[torquil]

As far as I know, the impact parameter is undefined in this situation. The incoming particles are momentum eigenstates to a good approximation, and therefore spread out in space. It doesn't make sense to talk about an impact parameter unless you are considering the collision of localized wave packets that are significantly separated. The optical theorem may have consequences for that situation also, but I don't know anything about that.

Torquil

 

[humanino]

Zero angle means there is no scattering ! The physical situation is as follow : a plane wave is sent on a fixed object called the "scatterer" on the wikipedia page (in section Derivation). Conservation of probability means that we have a relation between the total probability for any scattering at non zero angle (given by \sigma_\text{tot}) and the non-scattered wave. f(0) gives the non-scattered wave which must interfere destructively with the incident wave for scattering to occur.