  We have developed ultra-low-background NaI(Tl) crystals to reproduce the DAMA
results with the ultimate goal of achieving purity levels that are comparable
to or better than those of the DAMA/LIBRA crystals. Even though the achieved
background level does not approach that of DAMA/LIBRA, it is crucial to have a
quantitative understanding of the backgrounds. We have studied background
simulations toward a deeper understanding of the backgrounds and developed
background models for a 9.16-kg NaI(Tl) crystal used in the test arrangement.
In this paper we describe the contributions of background sources
quantitatively by performing Geant4 Monte Carlo simulations that are fitted to
the measured data to quantify the unknown fractions of the background
compositions. In the fitted results, the overall simulated background spectrum
well describes the measured data with a 9.16-kg NaI(Tl) crystal and shows that
the background sources are dominated by surface $^{210}$Pb and internal
$^{40}$K in the 2 to 6-keV energy interval, which produce 2.4 counts/day/keV/kg
(dru) and 0.5 dru, respectively.
