1. Crystal acceleration effect for cold neutrons
in vicinity of Bragg resonance
V.V. Fedorov1,2,3, Yu.P. Braginetz1,2, Ya.A. Berdnikov2, I.A. Kuznetsov1, M.V. Lasitsa1,2, S.Yu. Semenikhin1, E.O. Vezhlev1,2, V.V. Voronin1,2,3
1) Petersburg Nuclear Physics Institute NRC KI, Gatchina, Leningrad district, Russia
2) Peter the Great St. Petersburg Polytechnic University, Saint-Petersburg, Russia
3) Saint-Petersburg State University, Saint-Petersburg, Russia
Physics of fundamental Symmetries and Interactions – PSI 2016, October 2016

2. A new mechanism of neutron acceleration in the accelerated perfect crystal is proposed and found experimentally.
The effect arises due to the resonance energy dependence of neutron refraction index in a perfect crystal for neutron energies, close to the Bragg ones.
As a result during the neutron time-of-flight through the crystal the value of deviation from the exact Bragg condition changes and so the refraction index and the velocity of outgoing neutron changes as well.

3. Short hystory.
Yu.V. Petrov, The acceleration of neutrons scattered by exited isomeric nuclei. Sov. Phys. JETP 10(4) (1960) 833)
Yu. Petrov, 1972, paid attention, acceleration of neutrons in an inversely populated medium can be very important in processes of stellar nucleosynthesis. Sov. Phys. JETP 36 (1972) 395, Yu.V. Pertov and A.I. Shlyalchter, Nucl. Phys. A292(1977)88.
1980 the effect was discovered experimentally (I.A. Kondurov, E.M. Korotkikh, and Yu.V. Petrov, JETP Lett.31(4) (1980) 232, Phys. Lett. B 106(5) (1981)383)
1988. Acceleration of neutrons in the uniform magnetic field due to spin flip by radio-frequency flipper is well known and successfully used in physical experiments. See, e.g., H. Weinfurter, G. Badurek, H. Rauch, and D. Schwahn, Z. Phys. B: Condens. Matter 72(2) (1988) 195.
1989 A neutron acceleration by vibrationally excited nitrogen molecules was observed. G.A. Petrov, Yu.S. Pleva, and M.A. Yamshchikov. Sov. J. Nucl. Phys. 49 (1989) 204.
1989. N. acceleration in a strong alternating magnetic field (of amplitude 0.4 T) (L. Niel and H. Rauch, Z. Phys. B: Condens. Matter 74(1) (1989) 133.)
2012. N acceleration in a weak alternating magnetic field (of 1–10 G) was measured using anomalous behavior of the velocity dispersion for
neutrons, moving in a crystal close to the Bragg directions. V.V. Voronin, Yu.V. Borisov, A.V. Ivanyuta, et al., JETP Lett. 96(10) (2012) 613.

4. Also acceleration and deceleration of neutrons are well known
Basic principles of the neutron acceleration in a laser radiation field were considered in L.A. Rivlin, Quantum Electron. 40(5) (2010) 460–463.
Also acceleration and deceleration of neutrons are well known
by reflection from moving mirror (A.V. Antonov, D.E. Vul’, M.V. Kazarnovskii, JETP Lett. 9(5) (1969) 180, A.Z. Andreev, A.G. Glushkov, P Geltenbort, et al., Tech. Phys. Lett. 39(4) (2013) 370)
Steyerl turbine in ILL and FRM, super-mirror UCN turbine in KUR.
or by Doppler-shifted Bragg diffraction from moving crystal T.W. Dombeck, J.W. Lynn, S.A. Werner, et al., Nucl. Instr. Meth. 165(2) (1979) S. Mayer, H. Rauch, P. Geltenbort, et al., Nucl. Intr. Meth. A 608(3) (2009) 434.
Those are widely used in experiments with ultra cold neutrons.
Not long ago an interest arised to the acceleration of neutron, passing through accelerating media (F.V. Kowalski, Phys. Lett. A 182 (1-2) (1993) 335, V.G. Nosov and A.I. Frank, Phys. At. Nucl. 61(4) (1998) 613.
This effect was first observed by A.I. Frank, P. Geltenbort, G.V. Kulin, et al., JETP Lett. 84(7) (2006) 363 and described in detail by A.I. Frank, P. Geltenbort, M. Jentshel, et al., Phys. At. Nucl. 71(10) (2008) 1656–1674.
It was noticed that "the observed effect was a manifestation of quite a general phenomenon – the accelerated medium effect (AME) – inherent to waves and particles of different nature".

5. The acceleration of the samples in these experiments reached
several tens of g units
The energy transfer En to a neutron with energy En is
v is a value of a relative neutron-matter velocity variation during the neutron time of flight through the sample, n is the refraction index for neutron
It is (2–6)10-10 eV for UCN, so up to now AME was observed only for UCN and by only one research group.
For cold neutron eV
so AME in that case
It will be negligible in further consideration (V0 is averaged crystal potential).

6. In present report a new much more effective mechanism of acceleration is discussed, which is observed experimentally for cold neutrons passed through the accelerating perfect crystal.
Energy transfer to a neutron in this case is at the level of 410–8 eV. This value in contrast to AME is determined by the amplitude Vg of the corresponding harmonics of the nuclear neutron-crystal periodic potential, but not by a relative neutron-crystal velocity variation during the neutron time of flight through the crystal.
The essence of the crystal acceleration effect is the following. The crystal refraction index for neutron in vicinity of Bragg resonance
sharply depends on the crystal-neutron relative velocity. The neutrons enter into accelerated crystal with one potential of a neutron-crystal interaction and exit with the other, so the energy change at the crystal boundaries will be different and neutrons will be accelerated or decelerated after passage trough such a crystal, energy transfer to a neutron being at the level of Vg ~ 410–8 eV.

7. of nuclear ponential an so will have different mean potential energy.
The neutron-crystal interaction potential can be written as a sum of harmonics corresponding to all nuclear plane systems described by reciprocal lattice vectors g normal to the plane system, |g| = 2/d, d is interplanar spacing:
Neutron wave function will be significantly modified in the crystal in conditions close to the Bragg ones. As a result neutrons concentrate on maxima (B< 0) or minima (B > 0)
of nuclear ponential an so will have different mean potential energy.
max|| (B < 0)
max|| (B > 0)

8. Some more accurate consideration gives
So neutron kinetic energy in the crystal after passing through the crystal boundary will be
Some more accurate consideration gives
Here
is the dimensionless parameter of deviation from exact Bragg condition for some system of planes g
is the neutron energy that corresponds to exact Bragg condition.

9. Changing the deviation parameter B, by variation of inerplanar spacing or relative neutron-crystal velosity, we can change mean crystal potential and so the energy (velocity) changes at the crystal boundaries.
If neutron moves through accelerated crystal, the parameter B
will vary during neutron travel in crystal, as well as the mean potential (the refractive index).
So the boundary changes of the neutron kinetic energy at the
entrance and exit surfaces of the crystal will be different.
But the kinetic energy of neutron inside the crystal can not change due to the mean crystal potential homogeneity.
Therefore, we should observe acceleration or deceleration
of the neutron after passage through such a crystal,

10. Scheme of the experiment
Pyrolytic graphite mosaic crystals PG are used to direct incoming and outgoing beams.
The experiment was carried out at VVR-M reactor (PNPI,Gatchina).
Crystals K1 and K2 form a double-crystal scheme. K1 is monochromator, K2 is analyzer. The crystal K3, moving along g harmonically by piezoelectric motor, changes neutron energy, which measured by analizer.
Crystals K1, K2 and K3 have the same working planes (110) with parallel orientation. The diffraction angle was close to the right one: B = 890 (  4.9Å).
Scanning over the Bragg wave length performed by varying temperature difference T21 = T2 - T1 between crystals K2 and K1, the temperature T3 of K3 being a reference one.

11. Example of two crystal scanning curve, when the crystal K3 is absent.
The curve reaches its maximum when interplanar distances or K1, K2 crystals coincide (T21 = 0). Width in units of interplanar spacing is equal to
d/d 1.810-5.

12. We could vary the temperature of the crystal K3 and so its interplanar spacing too.
Deviations from the Bragg condition caused by the temperature difference and by crystal movement for quartz crystal (110) plane (d =2.456), which correspond appr. to Bragg width, are
ΔT=1K  Δv =1 cm/s  neV.
Velocity amplitudes of crystal K3 was 3 or 1.5 mm/s. The frequency of crystal vibration was c = 4.5 kHz (the period c = 222 μs), vibration amplitude μm, L = 5 cm; i.e., the time of flight of the neutron through the crystal was n = 62 μs ≈ c /4.
If the velocity of the working crystal K3 is depend on time as
the deviation from the Bragg condition for neutrons moving through that crystal will depend on time in the same way
Deviation B0 (for rest K3) is controlled by the temperatures difference (interplanar distances) T13 = T1 - T3.

13.  is the thermal expansion coefficient
The relationship between parameters B0 and T13:
 is the thermal expansion coefficient
for a quartz crystal in the direction perpendicular to crystallographic
planes.
The effect of the neutron energy change after passage through the crystal boundaries is determined by variation of the crystal velocity (and so averaged potential) during the neutron time-of-ight through the accelerated crystal
where v(t0) = v(t0) - v(t0+n), t0 is the time of neutron entrance into the crystal,
n is the neutron time-of-ight through the crystal.
Thus the effect is proportional to derivative of mean crystal potential!

14. Exact Bragg positions for K3 crystal are indicated by arrows.
Two-crystal scanning curves for neutrons passed through immovable working crystal with different deviationsfrom the Bragg energy.
Exact Bragg positions for K3 crystal are indicated by arrows.

15. The position of the maximum Es(t0) and the maximum intensity N(t0) of the scanning curve identically depend on the deviation parameter (T13) and so on the crystal velocity v(t0). Therefore
in case of the absence of the crystal acceleration effect it should be observed a bijection between the maximum position and intensity (dotted line)
Line position and amplitude correlation
Crystal acceleration effect E(t0) contains a phase-shifted term, which will result in closed curve like Lissajous figure

16. Effect of neutron acceleration after
passage through a vibrating crystal
Magnitude of energy variation for a neutron passed through the accelerating crystal as function of the deviation from the Bragg condition for incident beam.

17. Experimental dependence of neutron-crystal interaction potential on the deviation from exact Bragg condition

18. Thank you for attention !
One should take this phenomenon into account in precision neutron optical experiments, because the neutron refraction index is determined not only by averaged crystal potential, but also by its harmonics, which have the same order of value as the average potential itself.
Thank you for attention !