 Fixed-target production threshold

For an incident particle of rest mass $m_1$ and total energy $E_1$ hitting a stationary target of mass $m_2>0$, four-momentum conservation gives $W^2=m_1^2{c}^4+m_2^2{c}^4+2m_2{c}^2{E}_1$. Applying the relativistic collision mass threshold gives the displayed necessary condition. The incoming energy must also satisfy $E_1\ge m_1{c}^2$; other conserved quantum numbers or reaction dynamics can impose further restrictions.