= Fixed-target threshold for production of equal-mass particles
{title2=$E_{\min}=\frac{N^2-2}{2}mc^2$}
An incident particle of <rest mass> $m$ and energy $E$ striking a stationary identical target has total squared invariant energy $W^2=2mc^2(E+mc^2)$. Producing $N\geq2$ final particles of the same rest mass requires $W\geq Nmc^2$, giving the threshold. Equality means no final kinetic energy in the <centre-of-momentum frame>. From the <Lorentz factor>, the corresponding incident speed is at least $cN\sqrt{N^2-4}/(N^2-2)$.
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