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In this paper the diffraction of neutrons by the lattice of a vibrating, piezoelectrically excited quartz bar is investigated. The dependence of the observed significant increase of the integrated intensity of diffracted neutrons, due to the vibrations of a single-crystal, on the resonator current, on the bar thickness and on the wavelength of neutrons was studied. In the time spectrum of the diffracted neutrons the modulation components with both even and odd multiples of the resonance frequency of the quartz bar were observed. The experimental data are interpreted on the basis of the kinematical theory of neutron diffraction, using the `layer' structure of the vibrating single-crystal.