What happens after barium sulphate absorbs gamma rays? BaSO4 will happily absorb gamma-rays, presumably bumping electrons up into outer orbitals. What happens next?
I assume fluoresence as electrons drop back. Will the BaSO4 be safe to handle immediately after the gamma-rays stop or should I wait a while before using it in a subsequent experiment?
I read in "Sharaf, M.A., & Hassan, Gamal M. (2004). Radiation induced radical in barium sulphate for ESR dosimetry: a preliminary study" that the BaSO4 maintains an ESR signature for a year after getting zapped (!). I want to duplicate exactly this experiment, but I don't want to bring a gamma-ray emitting monster into my lab either.
 A: Dumping a lot of energy into a crystal lattice by, for example, absorbing a gamma photon then can lead to creation of crystal defects. This is well known in semiconductors (vacancy and interstitials in Si) and simple salts (F-centers for example). In BaSO4 it is no different - you can (and will) generate vacancies on either the cation or the anion sublattices, and the corresponding interstitials. These crystal defects will have localized electronic states (leading to the ESR signatures in the paper you reference). These may photoluminesce when pumped by an external light source. In addition, as the sample is heated, recombination of an interstitial with a vacancy releases energy, emitting a photon as thermoluminescence.
A fairly recent paper, admittedly on Eu-doped BaSO4, may give you a start on just what defects are created, which ones anneal out at which temperatures (the light released as the temperature of a thermoluminescent dosimeter, TLD, is called the glow curve), and how to grow more sensitive materials. The paper is The role of anion and cation vacancies in the thermoluminescence and phototuminescence processes of BaSO$_{4}$:Eu$^{2+}$, R. Sangeetha Rani and Arunachalem Lakshmanan, Journal of Luminescence 174 63-69 (2016). 
A: The simplest and most basic answer to this question is that it is very nearly impossible to make a substance measurably radioactive by exposing it to any reasonable amount of radioactivity, the exception being neutrons. To make something measurably radioactive by exposing it to gamma rays would typically require a gamma source so intense that you wouldn't be able to obtain or handle it.
