= Nuclear alpha-decay recoil correction
{title2=$T_\alpha\simeq Q_\alpha M_d/(M_d+m_\alpha)$}
In two-body <alpha decay> of a parent at rest, <momentum conservation> makes daughter and alpha momenta equal and opposite. The nonrelativistic kinetic <energies> therefore satisfy $T_d/T_\alpha=m_\alpha/M_d$, giving
$$
T_\alpha=Q_\alpha\frac{M_d}{M_d+m_\alpha},\qquad T_d=Q_\alpha\frac{m_\alpha}{M_d+m_\alpha}.
$$
An excited daughter reduces the available <Q value>. To leading recoil order, an alpha line separation $\Delta T_\alpha$ corresponds to excitation $E_x=(1+m_\alpha/M_d)\Delta T_\alpha$. Thus ratios of excitation energies inferred from line separations do not need the common recoil factor. These formulas assume kinetic energies small compared with rest <masses>; an exact <relativistic two-body decay> calculation can replace them when necessary.
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