  The neutron emission rates in thermal excited nuclei are conventionally
described by statistical models with a phenomenological level density parameter
that depends on excitation energies, deformations and mass regions. In the
microscopic view of hot nuclei, the neutron emission rates can be determined by
the external neutron gas densities without any free parameters. Therefore the
microscopic description of thermal neutron emissions is desirable that can
impact several understandings such as survival probabilities of superheavy
compound nuclei and neutron emissivity in reactors.
  To describe the neutron emission rates microscopically, the external thermal
neutron gases are self-consistently obtained based on the Finite-Temperature
Hartree-Fock-Bogoliubov (FT-HFB) approach. The results are compared with the
statistical model to explore the connections between the FT-HFB approach and
the statistical model.
  The Skyrme FT-HFB equation is solved by HFB-AX in deformed coordinate spaces.
Based on the FT-HFB approach, the thermal properties and external neutron gas
are properly described with the self-consistent gas substraction procedure.
Then neutron emission rates can be obtained based on the densities of external
neutron gases.
  The thermal statistical properties of $^{238}$U and $^{258}$U are studied in
detail in terms of excitation energies. The thermal neutron emission rates in
$^{238, 258}$U and superheavy compound nuclei $_{112}^{278}$Cn and
$_{114}^{292}$Fl are calculated, which agree well with the statistical model by
adopting an excitation-energy-dependent level density parameter.
  The coordinate-space FT-HFB approach can provide reliable microscopic
descriptions of neutron emission rates in hot nuclei, as well as microscopic
constraints on the excitation energy dependence of level density parameters for
statistical models.
