Consider an electron in the n=3 orbit of a hydrogen-like atom with...

Consider an electron in the $n=3$ orbit of a hydrogen-like atom with atomic number $Z$. At absolute temperature $T$, a neutron having thermal energy $k_{\mathrm{B}} T$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $T=\frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}$, (where $h$ is the Planck's constant, $k_B$ is the Boltzmann constant, $m_{\mathrm{N}}$ is the mass of the neutron and $a_0$ is the first Bohr radius of hydrogen atom) then the value of $\alpha$ is ________