  The path towards the production of \textit{r}-process seed nuclei follows a
course where the neutron-rich light and medium mass nuclei play a crucial role.
The neutron capture rates for these exotic nuclei could dominate over their
$\alpha$-capture rates, thereby enhancing their abundances at or near the drip
line. We calculate the radiative neutron capture cross-section for the
$^{33}$Na(n,$\gamma)^{34}$Na reaction via the Coulomb dissociation of $^{34}$Na
as it undergoes elastic breakup on $^{208}$Pb when directed at a beam energy of
100 MeV/u using the entirely quantum mechanical theory of finite range
distorted wave Born approximation upgraded to incorporate deformation effects.
The non-resonant one neutron radiative capture cross-section for
$^{33}$Na(n,$\gamma)^{34}$Na is calculated and is found to increase with
increasing deformation of $^{34}$Na. An analytic scrutiny of the capture
cross-section with neutron separation energy as a parameter is also done at
different energy ranges. The calculated reaction rate is compared with the rate
of the $^{33}$Na($\alpha$,n)$^{36}$Al reaction (deduced from the
Hauser-Feshbach theory), and is found to be significantly higher below a
temperature of $T_9 = 2$. Further, at the equilibrium temperature of $T_9 =
0.62$, the rate for the neutron capture had a small but non-negligible
dependence on the structural parameters of $^{34}$Na. In addition, this neutron
capture rate exceeded that of the $\alpha$-capture reaction by orders of
magnitude, indicating that the $\alpha$-process should not break the
(n,$\gamma$) \textit{r-}process path at the $^{33}$Na isotope, thus,
effectively pushing the abundance of sodium isotopes towards the neutron drip
line.
