Nuclear-recoil kinematics of WIMP scattering ($E_{R,\max }=2{\mu }_{\chi N}^2{v}^2/m_N$)

For nonrelativistic elastic scattering of a WIMP of mass $m_{\chi }$ from a stationary nucleus of mass $m_N$, the reduced mass is ${\mu }_{\chi N}=m_{\chi }{m}_N/(m_{\chi }+m_N)$. In the centre-of-momentum description, a scattering angle $\vartheta$ gives momentum transfer squared $q^2=2{\mu }_{\chi N}^2{v}^2(1-\cos \vartheta )$. The target kinetic energy is $E_R=q^2/(2m_N)$, giving the displayed maximum at backscattering. In SI units, using masses in kilograms and velocity in metres per second gives joules. Low recoil energy and a falling recoil spectrum make threshold and response calibration important for dark-matter direct detection.