Classical and Quantum Description of the Channeling Effect as Mutually Complementary Approximations

Abstract—

This work continues a series of studies dedicated to various aspects of the channeling of relativistic electrons in single crystals. It is convenient to consider the motion of a charged particle in the channeling mode in the so-called comoving frame of reference moving along the channeling direction with a velocity equal to the longitudinal component of the channeled-particle velocity. In such a system, particle motion is finite and is similar to oscillatory motion in the case of a one-dimensional potential (in the case of planar channeling) or to two-dimensional finite motion in the central field (in the case of axial channeling). The motion of electrons with rather large (relativistic) energies can be considered in both the classical and quantum approaches. In the classical approach, it is possible to quite simply, analytically calculate the intensity of the emerging electromagnetic radiation, its spectral characteristics, even the characteristic lifetimes of quantum channeled states, and the probabilities of transitions between them, which can be done only numerically in the quantum approach. In this work, the method of simplified analytical consideration is applied to calculation of the spectral characteristics and intensity of radiation arising both in cases of the planar and axial channeling of electrons with ultrarelativistic energies (up to several gigaelectronvolts). It is shown that when passing through an oriented single-crystal target with a thickness of several millimeters, this radiation can lead to conversion of a significant portion of the electron-beam energy into high-energy gamma quanta.