Splitting of the Transverse-Motion Energy Levels of Positrons during Channeling in the [100] Direction of a Silicon Crystal

Abstract

The motion of charged particles in the crystal can be both regular and chaotic. Within the quantum approach, chaos manifests itself in the statistical properties of the set of energy levels. The systems in which regions of regular motion are separated by that of chaotic motion in phase space are of special interest. The statistics of levels of these systems is greatly influenced by the possibility of tunneling between phase-space regions dynamically isolated from each other. Matrix elements for such tunneling transitions are estimated in the present paper. To do this, all transverse-motion energy levels of 20 GeV positrons moving in the axialchanneling mode along the Si crystal [100] direction, as well as the Hamiltonian eigenfunctions corresponding to these states, are calculated numerically. The superposition of these eigenfunctions that correspond to classical orbits localized in symmetric but dynamically isolated regions of phase space are found. The energylevel splitting makes it possible to estimate the tunneling-transition matrix elements.

On the Effect of Quantum Tunneling on the Energy Spectrum of the Transverse Motion of Channeled Positrons in a Silicon Crystal

“…Numerical methods are used to find all energy levels of the transverse motion of positrons and the wave functions corresponding to them in interval (9) near the upper edge of the potential well. The earlier developed methods [12,13,18] are used to find the relative contribution of regions of regular motion to the semiclassical average density of energy levels and the probability of tunneling transitions between such regions. According to the prediction published in [17], the presence of such transitions leads to the mutual repulsion of neighboring energy levels, which modifies the distribution of interlevel distances.…”

“…The technique used for estimating the amplitudes of tunneling transitions V ij is described in [18]. The found amplitudes also need to be normalized to the average interlevel distance.…”

“…The found amplitudes also need to be normalized to the average interlevel distance. However, given that tunneling transitions occur between the superpositions of states with a certain type of symmetry (A 1 , A 2 , B 1 , and B 2 ), as shown in [18] the V RC parameter in the Podolskiy-Narimanov distribution ( 5) should be set equal to the square root of the average of the absolute value of the found amplitudes, normalized to the unit interlevel distance, as follows: (11) For the set of arrays of energy levels of transverse motion in interval (9), this value was (12) = .…”

“…1 [17]. A theory that takes into account the effects of tunneling transitions on the statistics of the levels was proposed in [17]; this leads to the Podolskiy-Narimanov distribution (5) where (6) The technique for estimating the matrix elements of such transitions (from the values of which the V RC parameter is extracted) was described in [18]. In this study, we analyze the statistics of the interlevel distances of the energy of the transverse motion of highenergy positrons (20 and 40 GeV) for the case of channeling in the [100] direction of a silicon crystal.…”