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

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Abstract

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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About the authors

V. V. Syshchenko

Belgorod State University

Author for correspondence.
Email: syshch@yandex.ru
Russian Federation, Belgorod, 308015

A. I. Tarnovsky

Belgorod State University

Email: syshch@yandex.ru
Russian Federation, Belgorod, 308015

A. S. Parakhin

Belgorod State University

Email: syshch@yandex.ru
Russian Federation, Belgorod, 308015

A. Y. Isupov

Laboratory of High Energy Physics, Joint Institute for Nuclear Research

Email: syshch@yandex.ru
Russian Federation, Dubna, 141980

References

  1. Ахиезер А.И., Шульга Н.Ф. Электродинамика высоких энергий в веществе. М.: Наука, 1993. 344 с.
  2. Ахиезер А.И., Шульга Н.Ф., Трутень В.И., Гриненко А.А., Сыщенко В.В. // УФН. 1995. Т. 165. № 10. С. 1165. https://doi.org/10.3367/UFNr.0165.199510c.1165
  3. Gemmel D.S. // Rev. Mod. Phys. 1974. V. 46. P. 129. https://doi.org/10.1103/RevModPhys.46.129
  4. Uggerhøj U.I. // Rev. Mod. Phys. 2005. V. 77. P. 1131. https://doi.org/10.1103/RevModPhys.77.1131
  5. Lindhard J. // Kongel. Dan. Vidensk. Selsk., Mat.-Fys. Medd. 1965. V. 34 (14). P. 1.
  6. Шульга Н.Ф., Сыщенко В.В., Тарновский А.И., Исупов А.Ю. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2015. № 7. С. 72. https://doi.org/10.7868/S0207352815070197
  7. Shul’ga N.F., Syshchenko V.V., Tarnovsky A.I., Isupov A.Yu. // Nucl. Instrum. Methods Phys. Res. B. 2016. V. 370. P. 1. https://doi.org/10.1016/j.nimb.2015.12.040
  8. Shul’ga N.F., Syshchenko V.V., Tarnovsky A.I., Isupov A.Yu. // J. Phys.: Conf. Ser. 2016. V. 732. P. 012028. https://doi.org/10.1088/1742-6596/732/1/012028
  9. Шульга Н.Ф., Сыщенко В.В., Тарновский А.И., Исупов А.Ю. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2016. № 4. С. 103. https://doi.org/10.7868/S0207352816040168
  10. Сыщенко В.В., Тарновский А.И. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2021. № 7. С. 84. https://doi.org/10.31857/S1028096021070207
  11. Сыщенко В.В., Тарновский А.И., Исупов А.Ю., Соловьев И.И. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2020. № 3. С. 103. https://doi.org/10.31857/S1028096020030188
  12. Shul’ga N.F., Syshchenko V.V., Tarnovsky A.I., Dronik V.I., Isupov A.Yu. // J. Instrum. 2019. V. 14. P. C12022. https://doi.org/10.1088/1748-0221/14/12/C12022
  13. Сыщенко В.В., Тарновский А.И., Дроник В.И, Исупов А.Ю. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2022. № 3. С. 79. https://doi.org/10.31857/S1028096022030207
  14. Сыщенко В.В., Тарновский А.И., Дроник В.И, Исупов А.Ю. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2023. № 6. С. 88. https://doi.org/10.31857/S1028096023060158
  15. Ландау Л.Д., Лифшиц Е.М. Теоретическая физика. Т. 3. Квантовая механика. Нерелятивистская теория. М.: Физматлит, 2016. 800 с.
  16. Gutzwiller M.C. Chaos in Classical and Quantum Mechanics. Springer, 1990. https://doi.org/10.1007/978-1-4612-0983-6
  17. Штокман Х.-Ю. Квантовый хаос. М.: Физматлит, 2004. 376 с.
  18. Райхл Л.Е. Переход к хаосу в консервативных классических и квантовых системах. М.–Ижевск: РХД, 2008. 756 с.
  19. Bolotin Y., Tur A., Yanovsky V. Chaos: Concepts, Control and Constructive Use. Springer International Publishing Switzerland, 2017. 281 p. https://doi.org/10.1007/978-3-319-42496-5
  20. Hénon M., Heiles C. // Astronom. J. 1964. V. 69. P. 73. https://doi.org/10.1086/109234
  21. Davis M.J., Heller E.J. // J. Chem. Phys. 1981. V. 75. P. 246. https://doi.org/10.1063/1.441832
  22. Syshchenko V.V., Tarnovsky A.I., Parakhin A.S., Isupov A.Yu. // J. Surf. Invest.: X-ray, Synchrotron Neutron Tech. 2024. V. 18. № 2. P. 274. https://doi.org/ 10.1134/S1027451024020186
  23. Feit M.D., Fleck J.A., Jr., Steiger A. // J. Comput. Phys. 1982. V. 47. P. 412. https://doi.org/10.1016/0021-9991(82)90091-2
  24. Шульга Н.Ф., Сыщенко В.В., Нерябова В.С. // Поверхность. Рентген., синхротр. и нейтрон. исслед. 2013. № 3. С. 91. https://doi.org/10.1134/S1027451013020183
  25. Shul’ga N.F., Syshchenko V.V., Neryabova V.S. // Nucl. Instrum. Methods Phys. Res. B. 2013. V. 309. P. 153. https://doi.org/10.1016/j.nimb.2013.01.022
  26. Галицкий В.М., Карнаков Б.М., Коган В.И. Задачи по квантовой механике. М.: Наука, 1981. 648 с.
  27. Шапиро Д.А. Представления групп и их применения в физике. Новосибирск: НГУ, 2005. 142 с.
  28. Исупов А.Ю., Сыщенко В.В., Парахин А.С. // Прикладная математика & физика. 2023. Т. 55. № 1. С. 49. https://doi.org/ 10.52575/2687-0959-2023-55-1-49-56
  29. Исупов А.Ю., Сыщенко В.В., Тарновский А.И., Парахин А.С. // Прикладная математика & физика. 2024. Т. 56, № 4. С. 320. https://doi.org/10.52575/2687-0959-2024-56-4-320-327

Supplementary files

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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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