Redoubled effect of a neutron spin rotation in deformed noncentrosymmetric crystal for the Bragg diffraction scheme

A doubled effect of a neutron spin rotation in the noncentrosymmetric quartz crystal for the Bragg reflected neutrons from the deformed exit crystal side is first observed. The effect arises due to a neutron Schwinger interaction with the crystal and depends on a value of a crystal deformation near its back exit face. Electric field acting on a neutron in the quartz crystal is about $\sim 10^8$ V/cm. This field affects the neutron during the whole time of its passage through the crystal both ways there and back. This time is limited only by the available size of crystal (14 and 27 cm in our case) or the neutron absorption length. Observation of such effects gives a real perspective to improve essentially the scheme and sensitivity of the experiment for a search for neutron electric dipole moment (EDM) by the crystal-diffraction technique. Moreover, the presented experimental scheme can be applied for neutron with energy close to the $P$ wave resonance one to search for $T$-odd part of a neutron-nuclei interaction, for example, because of relatively low requirements to a crystal quality.