Why does not a free electron fall into the nucleus during elastic scattering? When a free electron finds a hole in the medium, they can re-combine and annihilate each other. So far so good.
A free electron can also scatter from positive atomic center (say it elastic scattering - Mott scattering). I imagine the electron is already heading to the nucleus and there is a strong attraction between them. But some how this electron makes a high angle scattering and does not recombine. Or does it?
My question is why does not this electron can't re-combine with the nucleus?
 A: 
My question is, why can a free electron scattered from the positive atomic center not recombine with the nucleus?

I rephrased your last sentence, so please tell me if this was not what you wanted.
Some (well known) remarks at the beginning.

*

*The ionization of atoms takes place when portions of energy combine with subatomic particles. For illustration purposes only, imagine the electric field lines between the nucleus and the electron. By absorbing photons, these field lines become longer and the distance to the nucleus increases.


*At some point of energy consumption the field lines are so long that the bounded state between the electron and the nucleus gets lost. The atom is ionized.
This well known facts are applicable to your question. Which scenarios for a free electron and an ion are known?

*

*An arriving electron with low kinetic energy is captured by the nucleus. The excess energy is released by photons.


*Electrons with higher kinetic energy are stopped by the electron-proton barrier.The kinetic energy of the electrons is emitted by photons and reunites immediately again in the above mentioned extending (imaginary or not, don't matter) field lines and the electron leaves the nucleus again.
The more interesting question is why there is an electron-proton barrier.
