  We present a search for a new particle $T'$ decaying to top quark via
$T^{\prime} \rightarrow t + X$, where $X$ is an invisible particle. In a data
sample with 4.8 fb$^{-1}$ of integrated luminosity collected by the CDF II
detector at Fermilab in $p\bar{p}$ collisions with $\sqrt{s}=1.96$ TeV, we
search for pair production of $T'$ in the lepton+jets channel, $p\bar{p}
\rightarrow t\bar{t} +X+X \rightarrow \ell\nu b q q' b + X +X$. We interpret
our results primarily in terms of a model where $T'$ are exotic fourth
generation quarks and $X$ are dark matter particles. Current direct and
indirect bounds on such exotic quarks restrict their masses to be between 300
and 600 GeV$/c^2$, the dark matter particle mass being anywhere below
$m_{T^{\prime}}$. The data are consistent with standard model expectations, and
we set 95% confidence level limits on the generic production of $T'\bar{T'}
\rightarrow t\bar{t}+X+X$. We apply these limits to the dark matter model and
exclude the fourth generation exotic quarks $T'$ at 95% confidence level up to
$m_{T'}=360$ GeV$/c^{2}$ for $m_X\le 100$ GeV$/c^2$.
