= Nuclear elastic electron scattering kinematics
{title2=$E'=E/[1+2E\sin^2(\theta/2)/(Mc^2)]$}
An <elastic scattering> event leaves the nuclear internal <quantum state> unchanged while transferring <energy> to its motion. Let a target of <mass> $M$ initially be at rest. Neglect the <Electron> rest <mass>, but retain relativistic target recoil. <Conservation of momentum> and <conservation of relativistic energy> give
$$
\hbar^2c^2q^2=E^2+E'^2-2EE'\cos\theta,
\qquad
(Mc^2+E-E')^2=M^2c^4+\hbar^2c^2q^2.
$$
Eliminating $q$ yields $Mc^2(E-E')=EE'(1-\cos\theta)$ and hence
$$
E'=\frac{E}{1+\dfrac{2E}{Mc^2}\sin^2(\theta/2)},\qquad
T_R=E-E'.
$$
The static relation $q=2E\sin(\theta/2)/(\hbar c)$ follows only when $E/(Mc^2)$ is negligible. For small recoil, $T_R\simeq\hbar^2q^2/(2M)$; this last approximation is nonrelativistic for the recoiling <nucleus>, not for the incident <Electron>. The invariant spacelike transfer, expressed in energy-squared units, is
$$
Q^2=\hbar^2c^2q^2-(E-E')^2=4EE'\sin^2(\theta/2),
$$
so $Q/(\hbar c)$ and the spatial inverse length $q$ coincide only in the static limit. Recoil kinematics alone does not correct the static <differential scattering cross-section>: a finite-mass target also changes its kinematic prefactor and the appropriate current matrix elements.
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