is the neutron energy that

is the neutron pkc inhibitor that corresponds to exact Bragg condition. The presence of a reflected wave with the amplitude equal to 1/ΔB leads to localization of neutron density in crystal on(or between) reflecting planes depending on the sign of ΔB:
The concentration of neutron density in the vicinity of maxima or minima of nuclear potential, as in the case of Laue diffraction, leads to additional changing the neutron kinetic energy and, respectively, the value of wave vector and refractive index n inside the crystal depending on the magnitude of this concentration, i.e. on deviation parameter ΔBfrom the Bragg condition. Notice that the neutron refraction index n is determined as usual:
Averaging the potential over the wave function (2), using (3), one gets
The last term in Eq. (4) increases infinitely approaching the Bragg condition (), so it becomes incorrect (and the perturbation theory is inapplicable as well) already for
The precise fulfilment of the Bragg condition means the equality of energies for two neutron states with momenta ℏk and , i.e. the neutron energy level becomes doubly degenerated. Amplitudes of these neutron states become comparable in value, and one should solve well-known two-level problem that corresponds to the two-wave approximation of the dynamic diffraction theory. Result is the following. Neutrons with the energies within the Bragg (Darwin) width B ≤/2 cannot penetrate into the working K3 crystal (Fig. 1), they will completely reflect from the crystal entrance face which is parallel to the crystallographic planes. So only the neutrons with B >/2 (ΔB >1) can pass through this crystal and can be accelerated. Using the expansion of the exact two wave dispersion equation over 1/ΔB, the following result for the kinetic energy of the neutron after its entrance into the crystal can be obtained:
The last term in Eq. (5) describes the additional potential neutron energy due to neutron localization. It significantly changes with small variation of neutron energy within the Bragg reflection width ΔB ≅ 1, i.e. in the narrow energy range
For thermal and cold neutrons /EB ≅ 10−5. The amplitude value itself is comparable to that of the mean crystal potential V0. Hence, by changing the incident neutron energy in the vicinity of EB, a well-defined resonance-type energy dependence of the neutron refraction index can be observed in the crystal. For example, for (110) plane of quartz =4·10eV, V0=10eV, and EB = 3.2·10eV for a diffraction angle close to π/2. It is worth to notice that in our case Eq. (4) can also be quite a good approximation, the infinities can be removed by overaging over the neutron spectrum within ΔB ∼ 2, because it is formed by two crystals K1 and K2. When ΔB=0 for the central part of the spectrum only the left and the right wings, having opposite potentials connected with the neutron concentration, can penetrate to the crystal so that averaged potential for this neutrons will be zero in accordance with Eq. (5).
Experimental setup If the neutron moves through the accelerating crystal, then the parameter of deviation from the Bragg condition and correspondingly the mean potential of neutron–crystal interaction will be time-dependent (see Eq. (5)). As a result, the refractive index will vary during a neutron travel in the crystal. Correspondingly, the changes in the neutron kinetic energies at the entrance and the exit surfaces of the crystal will differ. Therefore, one should observe either acceleration or deceleration of the neutron passing through such a crystal, because the kinetic energy of the neutron inside the crystal does not change because of the crystal homogeneity. It should be noticed that it does not matter in which way a change in the parameter of deviation from the Bragg condition occurs over a time interval of the neutron propagation through the crystal. For example, instead of the variation of the relative neutron–crystal velocity, the crystal temperature can be varied or the crystal can be deformed by squeezing. Both actions will cause a change in the crystal interplanar dimensions and so to the shift of the Bragg energy. The crystal movement was chosen due to the convenience of its realization (the above-mentioned accelerated medium effect [18] is negligible in this case). Numerical estimations show that for the quartz crystal plane (110) the Bragg width in the neutron-velocity units is equal to ∆νB ≅ 9mm/s, i.e., if the crystal velocity changes by 9mm/s over the time interval of neutron transit through the crystal, the deviation from the Bragg condition will vary by one Bragg width.