= Maximum laboratory deflection in relativistic elastic scattering
{title2=$\sin\theta_{\max}=m_2/m_1\quad(m_1>m_2>0)$}
A projectile of <mass> $m_1$ elastically scattering from a stationary target of <mass> $m_2<m_1$ has the displayed maximum laboratory angle. In $c=1$ units, let its incident <energy> be $E>m_1$ and $s=m_1^2+m_2^2+2m_2E$. Boosting the outgoing <four-momentum> from the <center-of-momentum frame> gives $\tan\theta=\sin\varphi/[\Gamma(\cos\varphi+B)]$, with $\Gamma=(E+m_2)/\sqrt s$ and $B=(m_1^2+m_2E)/[m_2(E+m_2)]>1$. The maximum at $\cos\varphi=-1/B$ has $\tan\theta_{\max}=1/[\Gamma\sqrt{B^2-1}]=m_2/\sqrt{m_1^2-m_2^2}$. Equality is allowed; a strict upper bound would exclude a realizable outgoing direction. The angular cap is independent of incident <energy>.
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