Wave Functions of Positrons Channeling in [111] Direction of a Silicon Crystal

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Resumo

For a positively charged particle, the repulsive uniform potentials of the three neighboring [111] chains of the silicon crystal form a small potential well with the symmetry of an equilateral triangle is described by the C3v group. The motion of a quantum particle in such a well is of interest in terms of manifestations of quantum chaos. A previously developed procedure for numerically finding the energy levels and wave functions of stationary states, taking into account the symmetry of this problem, is used to study the transverse motion of the channeling positrons with energies of 5, 6 and 20 GeV. A classification of stationary states of transverse motion of a positron is given based on the theory of group representations. The wave functions of the stationary states in an axially symmetric potential well are also found, and it is shown how these functions are modified under the influence of a perturbation with the symmetry of an equilateral triangle. In the upper part of the triangular potential well, the classical motion is chaotic for the majority of initial conditions. The structure of the wave functions in this domain has the features predicted by the quantum chaos theory.

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Sobre autores

V. Syshchenko

Belgorod State University

Autor responsável pela correspondência
Email: syshch@yandex.ru
Rússia, Belgorod, 308015

A. Tarnovsky

Belgorod State University

Email: syshch@yandex.ru
Rússia, Belgorod, 308015

A. Parakhin

Belgorod State University

Email: syshch@yandex.ru
Rússia, Belgorod, 308015

A. Isupov

Laboratory of High Energy Physics, Joint Institute for Nuclear Research

Email: syshch@yandex.ru
Rússia, Dubna, 141980

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2. Fig. 1. Potential energy (1) of a positron moving near the direction [111] of a silicon crystal.

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3. Fig. 2. Graphs of eigenfunctions of the discrete spectrum of transverse motion of GeV positrons in the potential well (1), the lines mark the classical boundaries of motion. A diagram of the energy levels of the transverse motion of positrons is also shown, with a horizontal dotted line marking the height of the saddle point of potential (1) eV.

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4. Fig. 3. Comparison of the wave functions of states with eV in the potential well (1) and the wave functions of states with eV in the potential well (8). The lines mark the classical boundaries of motion. The irregularity of the black and white areas away from the center of the potential pit is due to numerical modeling errors.

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5. Fig. 4. The same as in Fig. 3 for states with eV in the potential well (1) and wave functions of states with eV in the potential well (8).

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6. Fig. 5. The same as in Fig. 3 for states with eV in the potential well (1) and wave functions of states with eV in the potential well (8).

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7. Fig. 6. The same as in Fig. 3 for states with eV and eV in the potential well (1) and wave functions of states with eV in the potential well (8).

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8. Fig. 7. The wave function of the GeV-type state for a positron in a potential well (1), corresponding to eV, (a) and a graph of the same wave function, on which the areas of positive values are shaded in white and negative values in black (b).

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