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The motion of charged particles in a crystal during axial channeling can be both regular and chaotic [1]. The chaos in quantum case manifests itself in the statistical properties of the energy level set. These properties had been studied in [2] for the electrons channeling along [110] direction of the silicon crystal, in the case when the classical motion is completely chaotic. The case of channeling along [100] direction is of special interest because the classical motion here can be both regualr and chaotic for the same energy, depending on the initial conditions.
The semiclassical energy level density as well as its part that corresponds to the regular motion domains in the phase space is computed for the 10 GeV channeling electrons and positrons. It is demonstrated that the level spacing distribution for both electrons and positrons is better described by Berry-Robnik distribution [3] than by Wigner or Poisson distributions that are related to the completely chaotic and regular situations, respectrively (see, e.g. [4]).
[1] A.I. Akhiezer, N.F. Shul’ga, V.I. Truten’, A.A. Grinenko, V.V. Syshchenko // Physics-Uspekhi 38 (1995) 1119.
[2] N. F. Shul’ga, V. V. Syshchenko, A. I. Tarnovsky, and A. Yu. Isupov, Structure of the channeling electrons wave functions under dynamical chaos conditions // Nucl. Instrum. Methods Phys. Res. B 370 (2016) 1.
[3] Berry M.V., Robnik M. // J. Phys. A 17 (1984) 2413.
[4] H.-J. Stöckmann, Quantum Chaos. An Introduction, Cambridge University Press (2000).