1. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals M. Vysotskyy and V. Vysotskii National Taras Shevchenko University of Kyiv

2. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 2 In the report the methods of optimization of nuclear processes at interaction of low energy proton or deuteron beams with oriented crystal targets are considered. These methods are connected with the peculiarities of coherent correlation effects at oriental motion and channeling of charged particles in crystals. Similar processes of optimization of controlled fusion in crystals (but without the use of coherent correlated states) were examined for 7-8 years before historical experiments of Fleischmann and Pons. [Vysotskii V.I., Kuzmin R.N. Soviet Phys. -J.T.P. Letters), v.7, #16, 1981, p. 981-985; Vysotskii V.I., Kuzmin R.N. Soviet Phys. -J.T.P.), v. 53, № 9, 1983, p. 1861-1863] In last time a lot of experiments on investigation of low- energy beam-target interaction were conducted (e.g. [A. Kitamura, Y.Awa, T.Minary, M.Kubota, A.Tamiika, Y.Furuyama. Proc. 10th ICCF, p. 623-634; A.Huke, K.Szerski, p.Heide, Proc. 11th ICCF, p.194-209]).

3. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 3 Channeling effect If the direction of a charged particle incident upon the surface of a monocrystal lies close to a major crystal direction, the particle with high probability will suffer a small-angle scattering as it passes through the several layers of atoms in the crystal. If the direction of the particle's momentum is close to the crystalling plane, but it is not close to major crystalling axes, this phenomenon is called "plane channelling".

4. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 4 Avaraged potential of crystal plane

5. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 5 Positive particles (positrons, protons) Negative particles (electrons) Negatively-charged particles like antiprotons and electrons are attracted towards the positively-charged nuclei of the plane, and after passing the center of the plane, they will be attracted again, so negatively-charged particles tend to follow the direction of one crystalline plane. Positively-charged particles like protons and positrons are repulsed from the nuclei of the plane, so they tend to follow the direction between two neighboring crystalline planes, at the largest possible distance from each of them. The positively-charged particles have a smaller probability of interacting with the nuclei and electrons of the planes.

6. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 6 Is it possible to increase the possibility of proton interaction with crystal nuclei? (for example for the optimization of the nuclear reactions using the scheme of nuclear synthesis based on a beam of accelerated particles, incident on the LiD or LiT crystals).

7. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 7 Studies of particle channeling process in a crystal are based on a supposition concerning the mutual independence of particle quantum states at each level of transverse motion. This supposition is incorrect for the initial region of channels and for the total length of nanochannels, to which hollow channels in zeolites, asbestos filaments and, to a lesser extent, in carbon nanotubes and fullerenes correspond. In these areas the wave function of a particle is formed during the process of the joining of its states prior to entry into a channel and the superposition of possible states in the crystal channel, and corresponds to mutually coherent states. The special nature of these states is related to the possibility for the manifestation of a different type of interference effects. The most interesting among these is the process of coherent correlated state formation

8. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 8 The quantitative characteristics of this correlation are determined by the correlation coefficient r The unlimited growth of dispersions at r → 1 leads to the possibility of much more efficient penetration of the particle with a small transverse motion energy in the region below the barrier than for the same particle in the uncorrelated state (at r = 0).

9. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 9 How coherent correlated state can be formed? The possible method for the formation of the correlated state with |r| → 1 is the deformation of the potential well structure The method for formation of the correlated state with |r| → 1 which does not require considerable deformation of the potential well parameters is needed.

10. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 10 The simplest method for excitation of the particle correlated state is related to nonstationary deformation of the harmonic potential V(x) in the field of which this particle is situated, i.e., virtually to the deformation of the harmonic oscillator.

11. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 11 In this case the final expression for r Let us present the solution as a complex function: The system of equations makes it possible to find the exponent for the amplitude of oscillator fluctuations according to the given law for the change in ω(t), and to find r(t) on the basis of obtained expression α(t). The equation for the harmonic oscillator and the corresponding initial conditions in the dimensionless form

12. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 12 It is possible to find r(t) for any law of the change in ω(t), independent of what causes this change, either due to the change in the potential well parameters V(t) and α(t) or even due to the change in the particle mass M(t). We can determine the nonstationary dynamics of correlated state formation under different modes of change in the frequency ω(t).

13. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 13 1. T 1. The case of a monotonic change of channel parameters 50 60 70 80 90 0 102030 40 0t0t 1.0 0.5 0.6 0.8 0.9 0.7 0.4 0.3 0.1 0.2 0.0 1 2 3 4 5 6 7 8 d)d) |r(t)| A monotonic decrease in the harmonic oscillator frequency ω(t) leads to an increase in α(t), which leads to an increase in the amplitude of its oscillations and the correlation coefficient up to its limiting value |r| → 1

14. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 14 The impracticability of this method for formation of the coherent correlated state, for example, during particle motion in the interplane crystal channel is obvious. 1. T 1. The case of a monotonic change of channel parameters

15. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 15 2. T 2. The case of CCS formation at the harmonic law of channel parameters change limited range Such a mode can be achieved, e.g., for the constant potential well depth V in which the particle is situated and the periodic change in its width a in the range Or for the constant well width a and small change in its depth in the range

16. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 16 0.6 0.8 0.4 0.2 0.0 1.0 120 160 0t0t 0 40 8080 a)a) |r| 0.6 0.8 0.4 0.0 1.0 60 80 0t0t 0 2040 |r| 0.2 b) 0.6 0.8 0.4 0.2 0.0 1.0 60 8080 0t0t 0 20204040 |r| c) Time dependence of the correlation coefficient r(t) for the case of the limited periodic change in the oscillator frequency ω(t) = ω 0 (1 + g Ω cosΩt) at Ω = ω 0 for different values of the frequency modulation index: (a) g Ω = 0.1; (b) g Ω = 0.2; (c) g Ω = 0.3. 2. T 2. The case of CCS formation at the harmonic law of channel parameters change

17. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 17 It is seen that when g increases, the rate of change in the correlation coefficient with time increases and its value approaches that of the limiting |r| → 1 faster, providing complete translucence of any potential barrier. An increase in the modulation frequency from Ω = ω to Ω = 2ω leads to sharp growth in the rate of increase in the correlation coefficient’s (|r(t)|) maximum value, which approaches the limiting value very fast even at a small g = 0.01 value 2. T 2. The case of CCS formation at the harmonic law of channel parameters change

18. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 18 120 160 0t0t 0 40408080 0.6 0.8 0.4 0.2 0.0 1.0 |r| a) 120 160 0t0t 0 4040 8080 0.6 0.8 0.4 0.2 0.0 1.0 |r| b) 120 160 0t0t 0 40408080 0.6 0.8 0.4 0.2 0.0 1.0 |r| c) 120 160 0t0t 0 40408080 0.6 0.8 0.4 0.2 0.0 1.0 |r| d) 39.5 40 0t0t 38 38.539 40.5 0.6 0.8 0.4 0.2 0.0 1.0 |r| a) 40.1 0t0t 40.05 40.15 0.6 0.8 0.4 0.2 0.0 1.0 |r| b) Time dependence of the correlation coefficient r(t) for a change in the oscillator frequency at Ω = 2ω for different values of the frequency modulation index: (a) g = 0.01; (b) g = 0.025; (c) g = 0.05; (d) g = 0.1. The fragment marked in fig. d is shown in a larger form in below

19. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 19 The structure of the expression |r(t)| is characterized by a fast increase in the amplitude |r(t)| to the maximum value |r| → 1. Very narrow hollows are observed in the gaps between the extended maxima for |r| ≥ 0.99, caused by the interference phenomena. The width of these hollows sharply decreases when time increases. It is seen from these results that the relative width of these hollows is extremely small, not exceeding 1–2% at t ≈ 40/ω, and decreases to zero for further increases in t. The remaining part of the dependence |r(t)| corresponds to the almost constant and maximum possible value |r| → 1. Let us consider the possibility for implementing this effect during channeling. 2. T 2. The case of CCS formation at the harmonic law of channel parameters change

20. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 20 The nonstationary modulation of the well parameters corresponds to a periodic spatial modulation of the well parameters. Taking into account that the velocity of the motion of particles in the channel is always much higher than the velocity of an ultrasound (US) wave, this modulation can be achieved during the excitation of the longitudinal traveling US wave in the crystal along the channel direction, the effect of which leads to periodic changes in the longitudinal distance a between the atoms, and also the height of the channel wall.

21. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 21 The analysis performed above shows that the relation between Ω and ω close to optimal is Ω = 2ω, which corresponds to the necessary spatial modulation period of the channel walls (equal to the wave length of the longitudinal US oscillations)

22. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 22 Using a typical crystal with V max (0) = 20 eV a0 = 1.5 × 10 –8 cm Proton beam with the energy E = 100 keV v = 4.5 × 10 8 cm/s The length of the US wave necessary for forming the correlated state is Λ ≈ 0.4 μm. The length of the US wave is the same for a deuteron beam with the same energy. This wave should have an amplitude that ensures the necessary maximum relative change in the interatomic distance Δa/a = g = 0.1–0.01, leading to an analogous dynamic change in the potential barrier height. All necessary parameters determined for this example, are quite real.

23. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals Slide. № 23 When these conditions are met, the correlation coefficient for a particle (proton or deuteron),moving in the channel, even at a small value of the modulation index g, increases very quickly. Since for these particles the correlation coefficient can reach values close to |r| = 1, one should expect manifestations of the anomalous character of their interactions with crystal nuclei, including optimization of the nuclear reactions. In particular using a deuteron beam, a sharp increase in the efficiency of nuclear synthesis should be expected when it passes through such a modulated crystal, containing deuterium or tritium nuclei (for example, when using the scheme of nuclear synthesis based on a beam of accelerated particles, incident on the LiD or LiT crystals).