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 \( k_B T \) has the same de Broglie wavelength as that of this electron. If this temperature is given by \( T = \dfrac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B} \), (where \( h \) is the Planck’s constant, \( k_B \) is the Boltzmann constant, \( m_N \) is the mass of the neutron and \( a_0 \) is the first Bohr radius of hydrogen atom) then the value of \( \alpha \) is ________.
