= Nuclear-recoil kinematics of WIMP scattering
{title2=$E_{R,\max}=2\mu_{\chi N}^2v^2/m_N$}
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 $q^2=2\mu_{\chi N}^2v^2(1-\cos\vartheta)$. 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>.
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