Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals M. Vysotskyy and.

Subbarier Nuclear Interaction Subbarier Nuclear Interaction of Channeling Particles at of Channeling Particles at Self-Similar Formation of Self-Similar Formation of

Correlated States in Correlated States in Periodically Strained CrystalsPeriodically Strained Crystals

M. VysotskyyM. Vysotskyy and V. Vysotskii and V. Vysotskii

National Taras Shevchenko University of KyivNational Taras Shevchenko University of Kyiv

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]).


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".


Slide. № 4

Avaraged potential of crystal plane


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.


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).


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


Slide. № 8

1 1r

The quantitative characteristics of this correlation aredetermined by the correlation coefficient r

2 2

2

2

qp pq p qr

p q

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).


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

max ( )V t

( )a tThe method for formation of the correlated state with

|r| → 1 which does not require considerable deformation of the potential well parameters is needed.


Slide. № 10

2 2( , ) ( ) / 2V x t M t x

2max( ) 8 ( ) / ( ) ( )t V t M t a t

max ( )V t

( )a t

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.


Slide. № 11

In this case the final expression for r

( ) exp( ( )), ( ) ( ) ( )t t t t i t Let us present the solution as a complex function:

2

/ exp( 4 )d d

rdt dt

222

2exp( 4 ) ( )

d dt

dt dt

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

22

2( ) 0

dt

dt


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).


Slide. № 13

1. T1. The case of a monotonic change of channel parameters

0( ) exp( / )t t T

20 0 0 0 0 0 0 010 / ;0.25/ ;0.5 / ;1.0 / ;1.33/ ;2.0 / ;5.0 / ;10 / .T

50 60 70 80 900 10 20 30 40

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)

|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


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.

0( ) exp( / )t t T

( ) ~ exp( / )a t t T

1. T1. The case of a monotonic change of channel parameters

( ) ~ ( ) ~ exp( / )V t d t t T


Slide. № 15

2. T2. The case of CCS formation at the harmonic law of channel parameters change

0( ) (1 cos ),| | 1t g t g

limited range

0 0(1 ) ( ) (1 )g t g

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

20 0 0 max 0/(1 | |) /(1 | | / 2), 8 (0)a g a a g a V M

2 2max max max max 0 0/(1 2 | |) /(1 2 | |), (0) /8V g V V g V Ma


Slide. № 16

0.6

0.8

0.4

0.2

0.0

1.0

120 1600t

040

80

a)

|r|

0.6

0.8

0.4

0.0

1.0

60 800t

0 20 40

|r|

0.2

b)

0.6

0.8

0.4

0.2

0.0

1.0

60 800t

0 20 40

|r| c) Time dependence of the correlation coefficient r(t) for the case of the limited periodic change in the oscillator frequency

ω(t) = ω0(1 + gΩcosΩt) at Ω = ω0 for different values of the frequency modulation index:

(a) gΩ = 0.1;(b) gΩ = 0.2;(c) gΩ = 0.3.



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



Slide. № 18

120 1600t

0 40 80

0.6

0.8

0.4

0.2

0.0

1.0|r|

a)

120 1600t

0 40 80

0.6

0.8

0.4

0.2

0.0

1.0|r|

b)

120 1600t

0 40 80

0.6

0.8

0.4

0.2

0.0

1.0|r| c)

120 1600t

0 40 80

0.6

0.8

0.4

0.2

0.0

1.0|r|

d)

39.5 400t

38 38.5 39 40.5

0.6

0.8

0.4

0.2

0.0

1.0|r| a)

40.10t

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


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.



Slide. № 20

The nonstationary modulation of the well parameters corresponds to a periodic spatial modulation of the well

parameters.

2max max max( ) (0) (1 cos / ) (0) (1 2 cos / )V z V g z v V g z v

/t z v

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.


Slide. № 21

US

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)

20 max2 / 8 / 2 (0)v v a M V


Slide. № 22

Using a typical crystal withVmax(0) = 20 eV

a0 = 1.5 × 10–8 cmProton beam with the energy

E = 100 keV v = 4.5 × 108 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.


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).

Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals M. Vysotskyy and.

Documents

strained crystals slide

particles momentum

positive particles positrons

channeling effect

nuclear processes

crystal nuclei

crystalling plane

major crystal direction