Maximum Nuclear-Recoil Energy in Elastic Dark MatterNucleus Scattering

Abstract. Multiple independent observations indicate that most of the matter in the universe is nonluminous and nonbaryonic. A leading experimental strategy is direct detection, in which a dark matter particle scatters elastically from a target nucleus, producing a nuclear recoil. In this paper, we derive the kinematic maximum recoil energy expected from elastic scattering as a function of the dark matter mass, the target-nucleus mass, and the incident speed (taking a representative galactic speed 𝑣𝑣~10−3𝑐𝑐). Using representative target materials found in current experiments (cryogenic crystals, scintillators, and liquid noble gases), we evaluate how the maximum recoil energy scales across the low-mass and high-mass regimes. The results show a rapid rise in recoil energy when the dark matter mass is below the target-nucleus mass, followed by saturation in the heavy dark matter limit; for (𝑣𝑣~10−3𝑐𝑐), the saturation values are ~245 keV for xenon and ~52.3 keV for silicon. This simple kinematic analysis helps clarify how target choice constrains energy thresholds and, as a result, influences experimental sensitivity to different dark matter mass ranges without computing event rates or experimental limits.