  Purpose : The aim of this paper is to calculate the $^{15}$N($n,
\gamma$)$^{16}$N radiative capture cross section and its subsequent reaction
rate by an indirect method and in that process investigate the effects of
spectroscopic factors of different levels of $^{16}$N to the cross section.
  Method : A fully quantum mechanical Coulomb breakup theory under the aegis of
post-form distorted wave Born approximation is used to calculate the Coulomb
breakup of $^{16}$N on Pb at 100 MeV/u. This is then related to the
photodisintegration cross section of $^{16}$N($\gamma, n$)$^{15}$N and
subsequently invoking the principle of detailed balance, the $^{15}$N($n,
\gamma$)$^{16}$N capture cross section is calculated.
  Results : The non-resonant capture cross section is calculated with
spectroscopic factors from the shell model and those extracted (including
uncertainties) from two recent experiments. The data seems to favor a more
single particle nature for the low-lying states of $^{16}$N. The total neutron
capture rate is also calculated by summing up non-resonant and resonant
(significant only at temperatures greater than 1 GK) contributions and
comparison is made with other charged particle capture rates. In the typical
temperature range of $0.1-1.2$ GK, almost all the contribution to the reaction
rate comes from capture cross sections below 0.25 MeV.
  Conclusion : We have attempted to resolve the discrepancy in the
spectroscopic factors of low-lying $^{16}$N levels and conclude that it would
certainly be useful to perform a Coulomb dissociation experiment to find the
low energy capture cross section for the reaction, especially below 0.25 MeV.
