Chaotic behavior of channeling particles

Channeling describes the collimated motion of energetic charged particles along the lattice plane or axis in a crystal. The energetic particles are steered through the channels formed by strings of atomic constituents in the lattice. In the case of planar channeling, the motion of a charged particle between the atomic planes can be periodic or quasiperiodic, such as a simple oscillatory motion in the transverse direction. In practice, however, the periodic motion of the channeling particles can be accompanied by an irregular, chaotic behavior. In this paper, the Moliere potential, which is considered as a good analytical approximation for the interaction of channeling particles with the rows of atoms in the lattice, is used to simulate the channeling behavior of positively charged particles in a tungsten (100) crystal plane. By appropriate selection of channeling parameters, such as the projectile energy E(0) and incident angle psi(0), the transition of channeling particles from regular to chaotic motion is demonstrated. It is argued that the fine structures that appear in the angular scan channeling experiments are due to the particles' chaotic motion.     

Dynamic chaos phenomenon and coherent radiation accompanying high energy particle motion through crystals

Abstract

A crystal has a regular structure, therefore every motion in such a structure seems to be regular. However, it is not actually so and even in perfect crystals the particle motion may be either regular or chaotic. Everything depends on the number of integrals of motion determining a particle trajectory.

The character of particle motion in a crystal, i.e. its regularity or chaoticity, affects many physical processes accompanying the particle motion through a crystal. In this paper we shall consider the effect of dynamic chaos on the coherent radiation of fast particles in a crystal.

We also consider the validity conditions of coherent radiation theory results, the role of the second and higher Born approximations in the radiation theory of fast particles in crystals, the continuous string approximation in this theory, the coherent radiation in the model of random strings, and the multiple scattering effect on the coherent radiation.     

Multiplicity distribution in proton-proton and proton-antiproton collisions at high energies

Multiplicity distributions of secondary hadrons produced in p $ \bar p $ \bar p and collisions are very different. There are three types of inelastic processes in p $ \bar p $ \bar p scattering. The first one is the production of a shower of secondary hadrons in gluon string decay. The second one is the shower produced from the decay of two quark strings, and the third one is the shower produced from the decay of three quark strings. At the same time, there are just two types of inelastic processes for pp scattering: the shower from the gluon string and the shower from two quark strings. The multiplicity distribution and the average multiplicity of charged hadrons for an energy of 14 TeV are predicted.     

Computer simulations of particle flux and energy peaking

A computer simulation of the channeling of 300 keV deuterons in a niobium lattice has been performed. The model calculations included: binary Thomas-Fermi interaction between projectiles and lattice atoms, thermal displacements of the lattice atoms in the transverse plane of particle motion, electron scattering of the deuterons dependent on the local electron density, and electronic energy loss dependent on the local electron density. Flux distributions and mean particle energy distributions of the deuterons were calculated in their development with penetration depth. Flux peaking and energy peaking was found to depend sensitively on the local variation of the electron density in the crystal.     

On azimuthal anisotropy in fragmentation of a classical relativistic string

A fragmenting relativistic string is widely used for modeling particle production via quark–gluon strings formed in hadronic and nuclear inelastic interactions at high energies. In this note we focus on motion and fragmentation of a relativistic string with non-zero transverse separation of its ends and study this scenario as a possible mechanism for bringing anisotropy into the azimuthal angle distribution of produced particles in inelastic interactions of hadrons.     

Immersing method for the multichannel scattering problem

We propose a method for solution of the multichannel scattering problem. In particular, the scattering problem is considered when a particle makes a finite motion in the transverse direction of the scattering. In this case the scattering becomes multichannel, which is connected with the presence of discrete energy levels of the transverse movement of the particle. For the case of two-channel scattering, the problem is formulated up to the end. A method for determination of scattering amplitudes for the potential V=V(x,y) is proposed.     

Dynamical chaos and stochastic mechanism of high-energy negatively charged particle deflection by bent crystals

Deflection of high-energy negatively charged particles in straight and bent crystals through multiple scattering by crystal atomic strings was considered for the case in which the initial angle between the particle momentum and one of the main crystallographic axes was approximately four critical angles of axial channeling. It was shown that in a bent crystal with a small crystal thickness, when the crystal bend was less than the beam incidence angle, the beam deflected in the direction opposite to the direction of the crystal bend. At larger crystal thicknesses, the large part of the beam starts to deflect in the direction of the crystal bend. In addition, there is a group of particles that follow the crystal axis bend in the angular region of approximately the critical angle of axial channeling with respect to the current direction of the crystal axis. It was shown that in all of these deflection processes, the periodicity of the location of atomic strings in the crystal does not influence the angular distributions of scattered particles. This fact is connected with the effect of dynamical chaos in particle motion in the periodical field of bent crystal atomic strings. It was also shown that observed in a recent CERN experiment effect of beam deflection, when the angle between the initial particle momentum and the crystal axis was approximately four critical angles of axial channeling, is due to peculiarities of the stochastic multiple scattering of particles by bent crystal atomic strings. These peculiarities are connected with the effect of dynamical chaos in particle motion in crystals.     

Superconducting strings

It is known that certain spontaneously broken gauge theories give rise to stable strings or vortex lines. In this paper it is shown that under certain conditions such strings behave like superconducting wires whose passage through astrophysical magnetic fields would generate a variety of striking and perhaps observable effects. The superconducting charge carriers may be either bosons (if a charged Higgs field has an expectation value in the core of the string) or fermions (if charged fermions are trapped in zero modes along the string, as is known to occur in certain circumstances). They might be observable as synchrotron sources or as sources of high-energy cosmic rays. If the charge carriers are ordinary quarks and leptons, the strings have important baryon number violating interactions with magnetic fields; such a string, traversing a galactic magnetic field of 10^?6 G, creates baryons (or antibaryons) at a rate of order 10¹² particles/cm of string per second.     

Computer simulation of proton channelling in beryllium

The channelling of protons through a thin beryllium crystal is simulated in a computer. The angular dependence of the momentum density is computed using particle trajectory approximation and is reported as the transmission spectra. In obtaining the spectra, the energy loss suffered by protons due to electron multiple scattering is considered and the effect of thermal vibrations treated separately. The spectra obtained are characteristic of the hcp structure of Be. Positions of the major dips in the transmission spectra are found to correlate well with the directions of neighbouring strings. Variations in the angle of incidence of the beam and in its initial azimuthal angle bring about modifications in the spectra depending on the transverse kinetic energy of the incident particles and the crystalline structure of Be. Thermal vibration of the lattice does not modify the spectra appreciably.     

Gravitational waves as string vacua II

Classical propagation of (super)strings through gravitational shock waves is analyzed. The exact classical solutions are used for quantization and for the identification of the exact quantumS-matrix describing string scattering by the wave. ThisS-matrix coincides with theS-matrix of the string-string scattering in theflat space-time for particular profile of the shock wave! This is interpreted as the generation of curved geometry from the flat space-time string theory. The quantum consistence of (super)string motion in gravitational plane wave backgrounds is then studied. It turns out that for the standard dimensionsD=26 (D=10) the vanishing of the Ricci tensor for the plane wave is sufficient condition for vanishing of the Weyl (superWeyl) anomaly. Thus, plane wave solutions of the Einstein equations are automatically the classical (super)string vacua. For particular plane waves the anomaly can be evaluated even nonperturbatively.This is the second part of the review based on the PhD thesis of the author defended in 1989 at SISSA, Trieste.     

Magnetized Bianchi Type III String Universe with Time Decaying Vacuum Energy Density Λ

Exact solution of Einstein’s field equations is obtained for massive string cosmological model of Bianchi III space-time using the technique given by Letelier (Phys. Rev. D 28:2414, 1983) in presence of perfect fluid and decaying vacuum energy density Λ. To get the deterministic solution of the field equations the expansion θ in the model is considered as proportional to the eigen value s² ₂\sigma^{2}_{~2} of the shear tensor s^j _i\sigma^{j}_{~i} and also the fluid obeys the barotropic equation of state. It is observed that the particle density and the tension density of the string are comparable at the two ends and they fall off asymptotically at similar rate. But in early stage as well as at the late time of the evolution of the universe we have two types of scenario (i) universe is dominated by massive strings and (ii) universe is dominated by strings depending on the nature of the two constants L and ℓ. The value of cosmological constant Λ for the model is found to be small and positive which is supported by the results from recent supernovae Ia observations. Some physical and geometric properties of the model are also discussed.     

Volume dependence of the energy spectrum in massive quantum field theories

The low-lying energy values associated to energy eigenstates describing two stable particles enclosed in a (space-like) box of sizeL are shown to be expandable in an asymptotic power series of 1/L. The coefficients in these expansions are related to the appropriate elastic scattering amplitude in a simple and apparently universal manner. At low energies, the scattering amplitude can thus be determined, if an accurate calculation of two-particle energy values is possible (by numerical simulation, for example).     

Chaotic phenomena of charged particles in crystal lattices

In this article, we have applied the methods of chaos theory to channeling phenomena of positive charged particles in crystal lattices. In particular, we studied the transition between two ordered types of motion; i.e., motion parallel to a crystal axis (axial channeling) and to a crystal plane (planar channeling), respectively. The transition between these two regimes turns out to occur through an angular range in which the particle motion is highly disordered and the region of phase space spanned by the particle is much larger than the one swept in the two ordered motions. We have evaluated the maximum Lyapunov exponent with the method put forward by Rosenstein et al. [Physica D 65, 117 (1993)] and by Kantz [Phys. Lett. A 185, 77 (1994)]. Moreover, we estimated the correlation dimension by using the Grassberger-Procaccia method. We found that at the transition the system exhibits a very complex behavior showing an exponential divergence of the trajectories corresponding to a positive Lyapunov exponent and a noninteger value of the correlation dimension. These results turn out to be linked to a physical interpretation. The Lyapunov exponents are in agreement with the model by Akhiezer et al. [Phys. Rep. 203, 289 (1991)], based on the equivalence between the ion motion along the crystal plane described as a "string of strings" and the "kicked" rotator. The nonintegral value of the correlation dimension can be explained by the nonconservation of transverse energy at the transition.     

Nonlinear Raman scattering of photons by a channeled particle

Channeled particles are characterized by the discrete spectrum of bound transverse motion. The interaction of photons with channeled particles in single crystals can be accompanied by energy transitions between the levels of transverse motion of the channeled particle. The Raman scattering of photons at a quasibound channeled particle leads to the appearance of a combination of frequencies: the incident radiation frequency ω₀ and the frequency Δω_{m, n}, i.e., ω = ω₀ ± Δω_m,n where Δω_m,n = 2Δε_m,nγ²; Δε_{m, n} is the energy of the transition between quantum states (m and n) of the transverse motion of the channeled particle; and γ = E/mc² is the Lorentz factor of the channeled particle. The appearance of a violet satellite (the anti-Stokes component) in the Raman scattering spectrum is analyzed. The three-photon Raman-type transition, which is the process of the simultaneous absorption of two photons with the frequency ω₀ with the emission of a photon with the frequency ωs = 2ω₀ ± 2Δε_m,nγ², is considered. The conditions for resonance observation during the formation of the second harmonic (ω = 2ω₀) are discussed.     

From independent particle towards collective motion for two polarized electrons on a square lattice

The two dimensional crossover from independent particle towards collective motion is studied using 2 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion in a L×L square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation Δr of the spacing r between the two particles is larger than the lattice spacing a, there is a scaling length L ₀ = π²(t/U) such that the relative fluctuation Δr/〈r〉 is a universal function of the dimensionless ratio L/L ₀, up to finite size corrections of order L^-2. L < L ₀ and L > L ₀ are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U ^*(L)/t, Δr becomes smaller than a, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of t/U become valid. A weak random potential reduces the scaling length and favors the correlated motion. Received 28 March 2002 Published online 19 November 2002     

Diffusion on random lattices

We study the motion of a point particle along the bonds of a two-dimensional random lattice, whose sites are randomly occupied with right and left rotators, which scatter the particle according to deterministic scattering rules. We consider both a Poisson (PRL) and a vectorized random lattice (VRL) and fixed as well as flipping scatterers. On both lattices, for fixed scatterers and equal concentrations of right and left rotators the same anomalous diffusion of the particle is obtained as before for the triangular lattice, where the mean square displacement is t, the diffusion process non-Gaussian, and the particle trajectories exhibit scaling behavior as at a percolation threshold. For unequal concentrations the particle is trapped exponentially rapidly. This system can be considered as an extreme case of the Lorentz lattice gases on regular lattices discussed before or as an example of the motion of a particle along cracks or (grain or cellular) boundaries on a two-dimensional surface.     

Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice II

In this second paper, using N = 3 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion on a two dimensional L×L square lattice with periodic boundary conditions and nearest neighbor hopping t, we show that a single unpaired fermion can co-exist with a correlated two particle Wigner molecule for intermediate values of the Coulomb energy to kinetic energy ratio r _s = UL/(2t ). This supports in an ultimate mesoscopic limit a possibility proposed by Andreev and Lifshitz for the thermodynamic limit: a quantum crystal may have delocalized defects without melting, the number of sites of the crystalline array being smaller than the total number of particles. When L = 6, the ground state exhibits four regimes as r_s increases: a Hartree-Fock regime, a first supersolid regime where a correlated pair co-exists with a third fully delocalized particle, a second supersolid regime where the third particle is partly delocalized, and eventually a correlated lattice regime. Received 22 October 2002 Published online 23 May 2003 RID="a" ID="a"e-mail: jpichard@cea.fr