  We calculate the next-to-leading order QCD corrections to the excited
charmonium $\chi_{cJ}$ associated with $ W^+ b $ production from top-quark
decay. Our results show that detecting the $\chi_{c0}$ production from
top-quark decay is very difficult, but the $\chi_{c1}$ and $\chi_{c2}$
productions have the potential to be detected at the LHC. If the prompt
$\chi_{cJ}$ production from top-quark decay is really detected at the LHC, it
will be useful not only for investigating $J/\psi$ production from top-quark
decay but also for understanding the heavy quarkonium production mechanism.
