I've been wondering about this question for a while. If you have alpha and beta particles released from a radioactive core, how do they ionise surrounding particles?
When alpha particles (helium nuclei) or beta particles (electrons) are released in a radioactive decay they carry significant kinetic energy. As they go through the surrounding material they bump here and there occasionally kicking off some electrons from the surrounding atoms which are then ionised, until they finally stop.
Alpha particles are the nuclei of He, i.e. lacking the two electrons they carry a charge of $+2e$, while beta particles are either electrons or positrons and thus carrying a charge of $\mp 1e$. Positrons meeting other atoms' electrons annihilate thus leaving a positive ionization. The other particles may just carry enough momentum to transfer the energy required for the ionization of shell electrons.