  We report a novel correlated background in the antineutrino detection using
the inverse beta decay reaction. Spontaneous fissions and $(\alpha,n)$
reactions in peripheral materials of the antineutrino detector, such as
borosilicate glass of photomultipliers, produce fast neutrons and prompt gamma
rays. If the shielding from the material to the detector target were not thick
enough, neutrons and gammas could enter the target volume and mimic
antineutrino signals. This paper revisits the yields and energy spectra of
neutrons produced in B$(\alpha,n)$N and F$(\alpha,n)$Na reactions. A Geant4
based simulation has been carried out using a simplified detector geometry for
the present generation reactor neutrino experiments. The background rates in
these experiments are estimated. If this background was not taken into account,
the value of the neutrino mixing angle $\sin^22\theta_{13}$ would be
underestimated. We recommend that Daya Bay, RENO, Double Chooz, and JUNO,
carefully examine the masses and radiopurity levels of detector materials that
are close to the target and rich in boron and fluorine.
