1. High precision channeling experiments on the correlated dynamics of charged particles in nanothickness silicon crystals
V. Berec1, V. Guidi2, A. Mazzolari2, G. Germogli2, D. De Salvador3
1Institute of Nuclear Sciences Vinca, University of Belgrade
2INFN Ferrara and Ferrara University
3INFN LNL and Padova University
Rome, 1

2. Outline Planar channeling for positive ions Correlations
Low energy channeling apparatus
Experimental setup
Experimental results
Comparison with simulations
Summary and conclusions
Future outlook

3. Planar channeling
Beam impinges at an angle lower than θc -> planar channeling
Collisions with atoms are correlated
Oscillations between planes
Wavelength weakly depends on the impact parameter
-> coherence between the trajectories of the channeled particles
Multiple scattering on core electrons and atomic nuclei -> loss of coherence
Figure: Simulated trajectories of 2 MeV He++ ions between (110) planes of a Si crystal

4. Correlation
Arises when the particles traverse a sufficient distance in the crystal at constant velocity along a near-linear trajectory.
Three types of correlations are considered:
1. Superposition of Coulomb potential energy, waked by interaction of He++ ions with harmonic crystal potential, with exchange-correlation potential due to multiple scattering effect on electrons. Exchange-correlation potential becomes predominant as channeling trajectory increases its distance path from the entry face of the crystal.
2. Superposition of coherent radiation of the crystal lattice atoms itself induced by backscattered/channeled He++ ions, and radiation field secondary emitted by projectiles (He++ ions).
3. Interference effects exhibited in neighboring (110) Si planar stacking sequences between coherent He++ channeled trajectories.

5. Coherence and channeling
The effect of coherence arises during the interaction of charged particle with a crystal potential when the interaction amplitudes of the emitted/channeled particles are summing up in the phase with individual lattice atoms.
Coherence being the central part of the interference phenomena, produces phase summing, identifying an interferential multiplier in the scattering cross–section, which leads to appearance of coherent peaks when the transferred momentum coincides (it is in resonance) with one of the reciprocal lattice vectors.
This happens due to periodical arrangement of the crystal atoms, at selected parameters (beam energy, angles of incidence with respect to crystallographic axes or planes).
Channeling and coherence can occur simultaneously producing a joint effect.

6. Low energy channeling apparatus
Figure: Vacuum chamber and goniometer
Figure: Experimental apparatus
Channeling chamber at LNL-AN2000 accelerator:
α or p+ beams available
Energies: from 2 to 7 MeV
2 rotational degrees of freedom for the goniometer, 1 for the Si detector

7. Experimental setup
Figure: Scheme of the experimental apparatus.
2.0 MeV He++ ions delivered to a (100)-oriented Si wafer
To avoid <100> axial channeling, beam set to be a few degrees out of axis
3 different angles chosen for the detector (108°, 115°,170°)
Energy spectrum of backscattered particles recorded

8. Experimental results (I)
(b)
(c)
Figure: Measured energy spectra with the detector respectively at 108°(a), 115°(b), and 170°(c) with respect to the incoming α-beam.
Energy spectra recorded
End-point of the energy spectra depends on:
Beam energy (2 MeV)
Scattering angle (108°, 115°, 170° chosen)
Planar channelling oscillations recorded.

9. Experimental results (II)
(b)
(c)
Figure: Backscattering probability as a function of depth, estimated from the energy spectra recorded respectively at 108°(a), 115°(b), 170°(c).
Probability of backscattering as a function of the depth travelled into the crystal is estimated starting from energy spectra.
Planar channelling oscillations recorded.

10. Simulation input parameters
Comparison with simulations
Simulation input parameters
Beam: 2 MeV He++ particles.
Beam divergence: 0 degrees.
Target: Silicon crystal, (110) Planes.
Beam impinging parallely to atomic planes.
2 MeV He++ beam
(110) Planes
Inter- planar
distance 1.92 nm
1.47 nm

11. Coherence effect and correlation
Assuming the lattice periodicity within the of Bloch’s model we express coherence effect, induced by correlation of the ion soft collisions over electron density of states and the waked potential, in terms of a Fourier spectral decomposition:
In order to perform numerical simulation of channeled ion trajectories, coherent in spacetime with a lattice, we apply path integral formalism over a system ((100)Si crystal plane) whose degrees of freedom consist of effective potential field variable φx at each lattice site x.
A simulation model of a cubic lattice consists of the parameter points x = a (n0, n1,n2,n3) where a is the lattice spacing and n0, n1, n2, n3 are integers. Above slide equation defines Fourier transformed fields.

12. Correlated He++ trajectories
(I) Simulation results
Correlated He++ trajectories
interference effects exhibited between correlated channeled He++ trajectories
Moreover, comparison of simulation results (figure: top-right) shows for the fist time clear manifestation of interference effects exhibited between correlated channeled He++ trajectories in two neighboring planar stacking sequences.
Figure top left and bottom right:
(100) Si potential energy map during channeling He++ sequences, showing the crystal potential superposition maxima in region nm distant from the entry face of crystal. The effective potential intensity levels, which influence coherent ion motion, are shown in scaled gray/color tones. Coherence exhibits a cyclic behavior, i.e. it is evident again after 800 nm.
Effect of coherence, established over channeled He++ trajectories, becomes predominant at nm distances from the entry face of crystal. Coherent motion of channeled He++ ions is precisely guided through (110) planar stacking sequence of silicon atoms using simultaneous correlation, i.e. superposition of Coulomb potential energy caused by interaction of He++ ions with harmonic crystal potential, and exchange-correlation potential due to multiple scattering effect on electrons.

13. … the following differential transmission cross section is
Mapping of the quantum trajectories in phase space via two-stage process: in the coordination phase space, xy - the transverse position plane, and angular phase space,xy - the exit plane
..scattering for certain values ​​of the 2 MeV proton beam using the parameters: tilt angle and the 92 nm thick silicon nanocrystal.
Chosen value of variable  and the tilt angle of the crystal, , determine the Jakobian of the transformation:
represents the Jacobian of the components of the proton scattering angles 
… the following differential transmission cross section is

14. ..it was found out that beyond possibility to locate the atom (impurity) like in ordinary FPE, in the case of axial flux super-focusing one can probe interior subatomic structure of a target atom..
Demkov, Y. and J.D. Meyer, The European Physical Journal B - Condensed Matter and Complex Systems, (3): p
= 5.94
X 1013 Hz
The transversal proton yield projection for the proton peak at the superfocusing point Λ = 0.25 for ϕ = 0.
effective potential area as a function of the incident proton beam transverse energy
FWHM of the super-focused peak is much smaller than the thermal vibration amplitude
ρth = nm

15. Proton Beam Induction of Quantum Correlations
Quantum localization of single electron waive functions inside area of hydrogen ground state, induced by CP field. Chosen tilt angles are j = 0.05y, j = 0.01y and
j = 0.15y for L = nm Si n-c. The effect of quartic anharmonic terms in the exchange interaction initiates transition to triplet states.

16. The electron wavefunction frequency can be spatially distorted in order to coincides/shifts with applied CP field. A single spin excitation is then polarized along z axis coinciding with proton beam alignment.
Berec V PLosOne 2012

17. Possible schemes for ion-photon quantum correlation measurement

18. Summary and conclusions
We reported a detailed description of planar channeling oscillations of 2 MeV He++ particles channeled between (110) atomic planes of a silicon crystal.
Backscattering/transmission experiments with 2 MeV He++ ions were performed to study the exact correlation between the confined particles oscillating trajectories.
Regular patterns of channeled ion planar oscillations are shown to be dominated by the crystal harmonic-oscillator potential and multiple scattering effects. Quantitative estimation of channeling efficiency and electronic dechanneling length were performed.
For the first time it was shown that under the planar channeling conditions trajectories of positively charged particles exhibit observable correlation dynamics, including the interference effect.

19. Summary and conclusions
We have obtained new insights into phase space dynamics of 2 MeV protons oscillations, induced in axial channeling relative to <100> axis of a silicon nanocrystal.
Process of coupling of electron with ½ nuclear quantum spin states in silicon nanocrystal target, mediated by the polarized nuclear spin states of hyperchanneled protons through the quantum entanglement allowing the transfer of information originally deposited in the electrons to the spin state of the host 29Si.
An extremely fast transfer of quantum information in long-lived quantum state (polarization) of a nuclear spin, further addressable to a photon, with corresponding polarization/frequency.

20. Future outlook
Comparative analysis between 2 MeV protons and 2 MeV He++ particles, channeled between (110) atomic planes of a silicon crystal, considering quantitative and qualitative analysis of:
Electron dechanneling length;
Coherence effect and interference between neighboring planar stacking sequences;
Correlation due to multiple scattering effect on electrons..