Prediction of Underground Argon Content for Dark Matter Experiments

In this paper, we demonstrate the use of physical models to evaluate the production of $^{39}$Ar and $^{40}$Ar underground. Considering both cosmogenic $^{39}$Ar production and radiogenic $^{40}$Ar production in situ and from external sources, we can derive the ratio of $^{39}$Ar to $^{40}$Ar in underground sources. We show for the first time that the $^{39}$Ar production underground is dominated by stopping negative muon capture on $^{39}$K and ($alpha,n)$ induced subsequent $^{39}$K(n,p)$^{39}$Ar reactions. The production of $^{39}$Ar is shown as a function of depth. We demonstrate that argon depleted in $^{39}$Ar can be obtained only if the depth of the underground resources is greater than 500 m.w.e. below the surface. Stopping negative muon capture on $^{39}$K dominates over radiogenic production at depths of less than 2000 m.w.e., and that production by muon-induced neutrons is subdominant at any depth. The depletion factor depends strongly on both radioactivity level and potassium content in the rock. We measure the radioactivity concentration and potassium concentration in the rock for a potential site of an underground argon source in South Dakota. Depending on the probability of $^{39}$Ar and $^{40}$Ar produced underground being dissolved in the water, the upper limit of the concentration of $^{39}$Ar in the underground water at this site is estimated to be in a range of a factor of 1.6 to 155 less than the $^{39}$Ar concentration in the atmosphere. The calculation tools presented in this paper are also critical to the dating method with $^{39}$Ar.