The energy dependence of neutron refraction index in a perfect crystal for neutron energy, close to the Bragg ones, was studied. The resonance shape of this dependence with approximately the Darwin width was found. As a result, the value of deviation from the exact Bragg condition can change during the neutron time of flight through the accelerated crystal and so the refraction index and the velocity of outgoing neutron can change as well. Such new mechanism of neutron acceleration in the accelerating perfect crystal was proposed and found experimentally. This mechanism is march more effective then known one concerning with the neutron acceleration in the accelerating usual media.