As a consequence of the Pauli exclusion principle, nuclei with an excess of protons or neutrons have a higher average energy per nucleon.
One such nuclide is 9Be; its photodisintegration is significant in nuclear astrophysics, pertaining to the abundance of beryllium and the consequences of the instability of 8Be.
The neutron emission process itself is controlled by the nuclear force and therefore is extremely fast, sometimes referred to as "nearly instantaneous".
Thus, the delay in neutron emission is not from the neutron-production process, but rather its precursor beta decay, which is controlled by the weak force, and thus requires a far longer time.
Induced fission happens only when a nucleus is bombarded with neutrons, gamma rays, or other carriers of energy.