= Solution
{c}
Take the initial ground-state <chemical atom> at rest in the laboratory. The incident <photon> has <energy> $h\nu$ and <momentum> $h\nu/c$. The excited <chemical atom> has <rest mass> $M+\Delta M$ and must carry that <momentum>. <Conservation of energy> and <conservation of momentum> give
$$
Mc^2+h\nu=\sqrt{(M+\Delta M)^2c^4+(h\nu)^2}.
$$
Squaring cancels the photon-energy square:
$$
M^2c^4+2Mc^2h\nu=(M+\Delta M)^2c^4.
$$
Consequently
$$
\boxed{\nu=\frac{\Delta Mc^2}{h}\left(1+\frac{\Delta M}{2M}\right)
=\nu_{\rm int}+\frac{h\nu_{\rm int}^2}{2Mc^2},\qquad
h\nu_{\rm int}=\Delta Mc^2.}
$$
The extra positive term is the <photon absorption recoil correction> and supplies atomic <recoil energy>. The formula is exact for the initially stationary isolated <chemical atom>; it does not assume $\Delta M/M\ll1$. Initial and final <chemical atoms> cannot both remain at rest during single-<photon> absorption without another body taking <momentum>.
Back to article page
