What is forward scattering? If it is equivalent to no scattering, then why not call it "no scattering"?
Answer
Forward scattering need not be equivalent to "no scattering" - and, indeed, will only rarely be indistinguishable from it.
In the usual scattering-theory setup, you have an electron coming in in a plane wave $$\psi(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}=e^{ikz}$$ and impinging on some short-range potential. This will add to the wavefunction a scattered wave $$\psi_\text{scattered}(\mathbf{r})=F(\theta,\phi)\frac 1r e^{ikr}.$$ The form factor $F(\theta,\phi)$ governs the angular structure of the scattered wave, and the case where $\theta=0$ is called forward scattering.
Note that:
The forward-scattered wave is part of a spherical wave and its amplitude decays with the distance from the scattering centre in a different way to the incoming wave. In practice, the incoming beam will also suffer from wavepacket spreading, but in general the forward-scattered wave will be weaker unless special scattering conditions are at play.
The form factor in general includes a phase. This means that the forward-scattered wave will interfere nontrivially with the incoming beam, providing a delay in the phase of the final wave.
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