Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil

We present a systematic derivation and discussion of the practical formulae needed to design and interpret direct searches for nuclear recoil events caused by hypothetical weakly interacting dark matter particles. Modifications to the differential energy spectrum arise from the Earth's motion, recoil detection efficiency, instrumental resolution and threshold, multiple target elements, spin-dependent and coherent factors, and nuclear form factor. We discuss the normalization and presentation of results to allow comparison between different target elements and with theoretical predictions. Equations relating to future directional detectors are also included.