Depth profile study of $^{210}$Pb in the surface of an NaI(Tl) crystal

The surface $^{210}$Pb is one of the main background sources for dark-matter-search experiments using NaI(Tl) crystals, and its spectral features associated with the beta-decay events for energies less than 60 keV depends on the depth distribution of $^{210}$Pb in the surface of an NaI(Tl) crystal. Therefore, we must understand the profile of surface $^{210}$Pb to precisely model the background measurement in the low-energy region for the low-background experiment using NaI(Tl) crystals. We estimate the depth profile of the surface $^{210}$Pb contamination by modeling the measured spectrum of the alpha emission from the decay of $^{210}$Po at the decay sequence of the surface $^{210}$Pb contamination that is obtained using an $^{222}$Rn-contaminated crystal. In order to describe the energy spectra of the surface contamination we perform a log-likelihood fit of the measured data to a sum of Geant4 Mote Carlo simulations, weighted by an exponential curve as a function of the surface depth. The low- and high-energy events from the beta decay of surface $^{210}$Pb are also modeled to improve the depth profile for shallow depths. We simulate the energy spectra from beta decays of $^{210}$Pb that are exponentially distributed in the surface by following two exponential functions where the mean-depth coefficients are free parameters in the data fitting; we observed that the energy spectra are in good agreement with the measured data.