Question

Do unstable atoms give off smaller particles as they fall apart?

Answer 1

In radioactive decay, the nucleus of an unstable isotope decays and sometimes releases radioactive particles.

Explanation:

If by fall apart you mean radioactively decay then yes and no.

In radioactive decay, an unstable nucleus of an isotope begins to decay - releasing energy (gamma rays) in the process and sometimes alpha particles (made up of two protons and two neutrons) or Beta particles (a high energy electron for beta minus decay and a high energy positron for beta plus decay).