Elsevier

Physics Letters A

Volume 374, Issue 46, 18 October 2010, Pages 4690-4694
Physics Letters A

Spectral method in quantum theory of channeling phenomena of fast charged particles in crystals

https://doi.org/10.1016/j.physleta.2010.09.025Get rights and content

Abstract

We show the possibility of using the spectral method to determine the quantum characteristics of interaction between fast charged particles and crystal when the channeling occurs. Basing on this method, we have developed the procedure for calculation of particles transversal motions eigenlevels for plane channeling. We considered origin of new levels with increasing of particle's energy and zone structure forming from these levels with increasing number of potential wells. We discuss also the possibility of using the spectral method in number of other problems of quantum theory of channeling.

Introduction

The motion of fast charged particle in a crystal near one of the crystalline axes or planes is mainly determined by continuous potential of crystalline lattice — by the potential of a crystal, that is averaged along the axis or plane, along which the motion occurs. The momentum component, parallel to axis or plane, is conserved in such field. Due to this the problem of particle's motion in a crystal transforms to the problem of it's motion in transversal plane [1], [2]. The continuous potential includes both potential wells, and potential barriers, in the field of which the particle's motion can be both finite (channeling) and infinite (above-barrier). For the particle's motion in such a field quantum effects in interaction can be significant. Potential wells in considered problem have quite complicated shape, that's why one needs to develop some methods that allow to determine transversal motion levels, corresponding wave functions and other quantum characteristics of interaction between the particle and the continuous potential of crystalline planes or axes. It is important that the number of transversal motion energy levels in such potential wells depends on the total energy of the particle.

The main goal of this article is to analyze the possibility of using the so-called spectral method to determine Hamilton operator's eigenvalues in problems of fast charged particle's channeling. Spectral method was developed and used in optics for calculation of resonant frequencies in waveguides and optical fibers [3]. In article [4] this method was successfully applied for the problems of nuclear physics, related to description of quantum effects in particle's motion in several-well potentials, where the quantum manifestations of dynamical chaos were considered. In the present Letter we consider the simplest case of this method's application for the problem of plane channeling of particle in separate potential well, created by continuous potential of a crystalline plane. On the basis of this method we develop the procedure for calculation of transversal motion levels' positions, when plane channeling occurs. We also considered dynamics of new levels appearing with increasing of particle's energy and dynamics of zone structure's forming from these levels with increasing potential wells' number. We discuss the possibility to apply this method for analysis of zone structure of particle's above-barrier states in periodic field of atoms' crystalline planes. The analysis of quantum manifestations of dynamical chaos when axial channeling of particle in complicated configuration potential wells occurs and in number of other problems of the channeling phenomena theory is also mentioned.

Section snippets

Plane channeling of fast charged particles

Let us consider the motion of fast charged particle in a crystal near one of crystalline planes. In this case the motion is mostly determined by continuous potential of atom's crystalline planes U(x), which can be written as the sum of potential of separate crystalline planes, that particle interacts with:U(x)=nUp(xnd), where d is the distance between the planes, Up(x) is the potential of the plane, x is the coordinate, which is orthogonal to atom's crystalline planes.

At first let's consider

The spectral method for solution of Schrödinger equation

Spectral method is destined for numerical calculation of energy levels for the particle in potential well, made up by external field U(x). In order to do that an auxiliary function is introduced [3]Pε=1T0Tdtexp(iεt)P(t)w(t). In the limit T it coincides (to within T1) with Fourier-component of correlation functionP(t)=+ψ0(x)ψ(x,t)dx, where ψ(x,t) is the solution of Eq. (4), that satisfies the initial condition ψ(x,t=0)=ψ0(x) and w(t)=1cos(2πt/T). An important property of the function P

Transversal energy levels of a particle in a field of crystalline planes

The potential U(x) is a many-well periodic function of x coordinate. The interaction of particle with this field is determined both by properties of its interaction with single potential well and by ones related to presence of large number of potential wells. At first let us consider the possibility to employ spectral method for calculation of energy levels for transversal motion of a particle in the field of separate potential well.

For negatively charged particles potential (3) forms a

Summary

When fast charged particles are channeled in an oriented crystal, one can satisfy the conditions, under which the quantum effects in interaction between the particles and the field of a crystalline lattice are significant. The quantum problems of particle interaction with continuous potential of a single crystalline plane is considered in this article. The possibility to employ the spectral method for this kind of problem was shown. On the basis of this method there was developed the procedure

References (8)

  • M.D. Feit et al.

    J. Comp. Phys.

    (1982)
  • V.P. Berezovoj et al.

    Phys. Lett. A

    (2004)
  • A.I. Akhiezer et al.

    Phys. Rep.

    (1991)
  • J. Lindhard

    Dansk. Vid. Selsk. Math.

    (1969)
There are more references available in the full text version of this article.

Cited by (10)

  • Geometrical optics method in the theory of channeling of high energy particles in crystals

    2019, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
    Citation Excerpt :

    In this case, there remains a possibility of manifestation of different coherent and interference effects in the interaction of particles with lattice atoms. Some of these effects that are connected with the study of angular distributions of the particles scattered by crystal were considered in the works [4,5], basing upon the operator method [6–8] of the wave function determination in crystal. In the present work we demonstrate the possibility of application of the geometrical optics method (see [9,10]) in the problem of particle scattering in crystal that allows one to express the wave function and the quantum scattering cross-section by the means of the family of classical particle trajectories in crystal.

  • On classical and quantum effects at scattering of ultrarelativistic electrons in ultrathin crystal

    2017, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms
    Citation Excerpt :

    With the crystal thickness increase the number of these peaks increases too. In the considered problem, the determination of the wave function at the electron’s emergence from crystal can be executed on the basis of numeric methods used in the so-called spectral method of determination of bound states levels in two-dimensional potential wells [5,9–11]. Note that the quantum peaks positions in common case do not coincide with the positions of peaks in the classical scattering theory connected with the rainbow scattering phenomenon.

  • On spectral method in the axial channeling theory

    2013, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms
    Citation Excerpt :

    Note that use of periodic crystal potential in [10–12] automatically leads to formation of band structure of the energy of transverse motion. In the paper [13] high resolution of the spectral method was used to trace out how the energy levels in the potential well formed by a single atomic plane are split due to possibility of tunneling between neighboring planes. For large number of such planes this leads to formation of the band structure.

View all citing articles on Scopus
View full text