What happens to alpha particles in matter? Alpha radiation can be stopped by a piece of paper, but what happens to a helium nucleus after it loses all its energy in matter? Does it became part of the material or does it change nuclear properties?
 A: Helium irradiation is a key problem in fusion devices, as the plasma can implant helium ash into reactor components. These tend to form helium bubbles, which in turn can act as binding sites for hydrogen. It helps to embrittle materials like W, where this can cause flaking or cracking of the W surface.
The helium is pretty stable in there (W), and the bubbles are a few nm in size. This only forms under heavy irradiation conditions. See e.g. Figure 5 of this 2016 paper: the He is not mobile until ~1500K.
I do not have any affiliations with the above work, nor work in the field.
A: An alpha particle in matter will interact with electrons and nuclei by scattering.  Those scattering interactions will tend to redistribute the alpha particle's kinetic energy until it is in thermal equilibrium with its surroundings.
An alpha particle at room temperature is just the same as a twice-ionized helium nucleus.  Eventually it will steal two electrons from other atoms in its environment, which turns it into a neutral helium atom.  Helium is hard to trap, so it will generally escape from any solid material and become part of Earth's atmosphere.
At Earth's surface, helium makes up about five parts per million of Earth's atmosphere.  Because of helium has such low mass, it has a taller scale height than other gas species and is a major component of Earth's exosphere, where some of the more energetic helium atoms find themselves with enough energy to escape forever into space.  A commenter links to a 2012 reference which claims that about $10^{-6}$ of Earth's atmospheric helium is replaced every year by radioactively-produced helium escaping from Earth's interior; a back-of-envelope estimate gives $10^{6\text{-ish}}$ years for helium to diffuse from the surface to the top of the atmosphere, as well.  So your stopped alpha particle will hang around for a long while.
A: As far as I know, an alpha particle which penetrates into a material will just kind of hang out there, until it can get two electrons to become a stable helium atom. Along the way, it may have affected neighboring atoms/molecules via ionization.
The reason alpha radiation can't penetrate too far is that the alpha particle is (relatively) fairly massive and so interacts strongly with the material by 'bumping' into other atoms and interacting with them electromagnetically. In order for it to become part of the material or affect it chemically would require extra energy so I wouldn't expect that to happen. 
A: When a radioactive atom emits an alpha particle it loses two protons' worth of electric charge. It therefore becomes a negatively-charged ion and soon loses a couple of orbiting electrons too.
The charges soon redistribute themselves; the electrons find a couple of positive ions, the alpha particle picks up a couple of electrons to become a helium atom.
Helium is unreactive and will diffuse out of any solid that slowed it down. Once in the air, it hangs around for many years, but eventually it has an unfortunate habit of drifting off into space.
