Astrophysical reaction rate for $\alpha(\alpha n,\gamma)$

We study the astrophysical reaction rate for the formation of $^{9}$Be through the three body reaction $\alpha(\alpha n,\gamma)$. This reaction is one of the key reactions which could bridge the mass gap at A = 8 nuclear systems to produce intermediate-to-heavy mass elements in alpha- and neutron-rich environments such as r-process nucleosynthesis in supernova explosions, s-process nucleosynthesis in asymptotic giant branch (AGB) stars, and primordial nucleosynthesis in baryon inhomogeneous cosmological models. To calculate the thermonuclear reaction rate in a wide range of temperatures, we numerically integrate the thermal average of cross sections assuming a two-steps formation through a metastable $^{8}$Be. Off-resonant and on-resonant contributions from the ground state in $^{8}$Be are taken into account. As input cross section, we adopt the latest experimental data by photodisintegration of $^{9}$Be with laser-electron photon beams, which covers all relevant resonances in $^{9}$Be. We provide the reaction rate for $\alpha(\alpha n,\gamma)^{9}$Be in the temperature range from T$_{9}$=10$^{-3}$ to T$_{9}$=10$^{1}$ both in the tabular form and in the analytical form. The calculated reaction rate is compared with the reaction rates of the CF88 and the NACRE compilations. The CF88 rate is valid at $T_{9} > 0.028$ due to lack of the off-resonant contribution. The CF88 rate differs from the present rate by a factor of two in a temperature range $T_{9} \geq 0.1$. The NACRE rate, which adopted different sources of experimental information on resonance states in $^{9}$Be, is 4--12 times larger than the present rate at $T_{9} \leq 0.028$, but is consistent with the present rate to within $\pm 20 %$ at $T_{9} \geq 0.1$