Understanding internal backgrounds of NaI(Tl) crystals toward a 200~kg array for the KIMS-NaI experiment

The Korea Invisible Mass Search (KIMS) collaboration has developed low-background NaI(Tl) crystals that are suitable for the direct detection of WIMP dark matter. With experience built on the KIMS-CsI programs, the KIMS-NaI experiment will consist of a 200~kg NaI(Tl) crystal array surrounded by layers of shielding structures and will be operated at the Yangyang underground laboratory. The goal is to provide an unambiguous test of the DAMA/LIBRA's annual modulation signature. Measurements of six prototype crystals show progress in the reduction of internal contaminations of radioisotopes. Based on our understanding of these measurements, we expect to achieve a background level in the final detector configuration that is less than 1~count/day/keV/kg for recoil energies around 2~keV. The annual modulation sensitivity for the KIMS-NaI experiment shows that an unambiguous 7$\sigma$ test of the DAMA/LIBRA signature would be possible with a 600~kg$\cdot$year exposure with this system.