Intrinsic Fano factor of nuclear recoils for dark matter searches

Nuclear recoils in germanium and silicon are shown to have much larger variance in electron-hole production than their electron-recoil counterparts for recoil energies between 10 and 200\,keV. This effect--owing primarily to deviations in the amount of energy given to the crystal lattice in response to a nuclear recoil of a given energy--has been predicted by the Lindhard model. We parameterize the variance in terms of an intrinsic nuclear recoil Fano factor which is 24.3$\pm$0.2 and 26$\pm$8 at around 25\,keV for silicon and germanium respectively. The variance has important effects on the expected signal shapes for experiments utilizing low-energy nuclear recoils such as direct dark matter searches and coherent neutrino-nucleus scattering measurements.