  This study examines the radiative capture of a neutron by $^{10}$Be using the
Coulomb dissociation approach within the FRDWBA theory. We analyze the elastic
Coulomb breakup of $^{11}$Be on a $^{208}$Pb target at 72 MeV/A to determine
the photodisintegration cross-section and radiative capture cross-section.
Utilizing the Maxwell-averaged velocity distribution, we calculate the
resulting radiative neutron capture reaction rate for the
$^{10}$Be(n,$\gamma$)$^{11}$Be reaction. Comparative analyses are conducted
with experimental data, theoretical results from direct radiative capture
methods, and transfer reaction calculations. Additionally, we contrast our
findings with the existing $^{10}$Be($\alpha$,$\gamma$)$^{14}$C reaction rate
and conclude the dominance of neutron capture over $\alpha$ capture by
$^{10}$Be.
