= Inverse beta decay threshold
{title2=$E_{\nu,\mathrm{thr}}=\frac{(m_n+m_e)^2-m_p^2}{2m_p}c^2$}
The <neutron> and <Positron> together have more rest <mass> than the initial <proton>, so a finite incident <energy> is necessary even before recoil is considered. In the <proton>'s <rest frame>, use <four-momentum> conservation and a negligible <neutrino mass>. The initial squared center-of-mass <energy> is $m_p^2c^4+2m_pc^2E_\nu$; the least final squared center-of-mass <energy> is $(m_n+m_e)^2c^4$. Equating them gives the threshold above. At threshold the final particles share one center-of-mass <velocity>; they are not both at rest in the laboratory.
Writing $\Delta=m_n-m_p$, the result separates into
$$
E_{\nu,\mathrm{thr}}=(\Delta+m_e)c^2+\frac{(\Delta+m_e)^2}{2m_p}c^2.
$$
The positive second term is the laboratory recoil correction. With rounded rest <energies> $m_pc^2=938.272\,\mathrm{MeV}$, $m_nc^2=939.565\,\mathrm{MeV}$ and $m_ec^2=0.511\,\mathrm{MeV}$, the calculated threshold is about $1.806\,\mathrm{MeV}$. These rounded inputs follow the recommended particle <masses>.<Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>[2022 CODATA adjustment, proton, neutron and electron mass-energy equivalents in MeV]{ref}
As a further ideal <energy> balance, suppose the <Positron> stops and annihilates with an <Electron>, and all prompt annihilation <gamma rays> are contained. Its prompt visible <energy> is $T_{e^+}+2m_ec^2$. Neglecting <neutron> recoil gives $E_{\rm vis}\simeq E_\nu-(m_n-m_p)c^2+m_ec^2\simeq E_\nu-0.782\,\mathrm{MeV}$. Escaping <gamma rays>, detector response and recoil change that measured sum. The delayed capture signal is a separate energy deposit, not an extra contribution to the incident <electron antineutrino>'s prompt energy reconstruction.
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