What happens to a radioactive element or isotope's electrons when it undergoes alpha decay? It seems to make sense that when an atom loses two protons, it would lose two electrons as well, but I don't actually know what happens.
 A: After alpha decay, the alpha particle can be thought of as a doubly positively ionized helium atom, and the parent atom is now a doubly negatively charged ion.  Under normal circumstances, the two ions will eventually neutralize their charges.  In a near vacuum, this may take some time.
In a crystalline solid, the nuclei share an electron cloud, so the two "extra" electrons are added to the cloud; if the alpha particle stops within the solid, it will collect two electrons and become a normal atom of helium.
A: It is complicated and we ignore it, but your intuition is right.  When the nucleus loses an alpha particle its charge decreases by two.  The atomic physicists now claim their job is done and don't care.  The solid state physicists don't consider radioactivity, so they don't care either.  If it is an atom floating freely in space, two electrons will move off in some direction, but who cares?  If it is an atom in a solid crystal, you should ask how much the recoil moves the atom and whether it dislocates the crystal.
