The dispersion (resonance) behavior of the refractive index of a neutron moving in a crystal with energies close to Bragg values has been studied. It has been shown that a small change in the energy of the neutron by about the Bragg width in this case (ΔE/E ∼ 10⁻⁵) results in a significant (several tens of percent) change in the potential of the interaction of the neutron with the crystal. A new phenomenon—the acceleration of the neutron passing through a perfect crystal moving at a variable velocity near a Bragg resonance—has been observed. The effect appears because the parameter of deviation from the Bragg condition and, therefore, the neutron-crystal interaction potential change during the time of flight of the neuron through the accelerated crystal. As a result, the kinetic energy of the neutron leaving the crystal changes.