= Mass threshold after photon absorption
{title2=$M^2=m^2+2mE/c^2$}
A stationary particle of <rest mass> $m$ absorbing a <photon> of <energy> $E$ acquires total <invariant mass>
$$
M=\sqrt{m^2+2mE/c^2}.
$$
This follows from <four-momentum conservation>, since the incoming <photon> has <momentum> magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter <rest mass> is $M/2$, attained when both are at rest in the <centre-of-momentum frame>. Their common laboratory <speed> at threshold is $cE/(mc^2+E)$. Above this daughter-mass threshold there is insufficient <centre-of-momentum energy>; below it, relative <kinetic energy> is available.
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