Adiabatic Invariants for Axial Channeling in a Bending Crystal (in the Accompanying Reference System)

A relativistic electron in the axial channeling regime in a single crystal moves along a helical trajectory around some regular chain of positive ions, forming an axial channel. In the accompanying reference frame moving along the channel axis at a velocity equal to the longitudinal component of the electron velocity, the channeling motion is finite and can be considered as an implementation of the two-dimensional relativistic atom model with controlled parameters. The depth and shape of the effective potential of the ionic chain depends on the chemical composition of the crystal, its crystallographic structure, and orientation, as well as on the relativistic energy of the particle moving in the channel. The motion in the axial channeling regime even in the bent crystal and its axial channels remains stable if the bending angle is not too large. The maximum possible bending angle is estimated using calculated adiabatic invariants of motion. It is demonstrated that the channeling motion regime can be stable only if the bending angle does not exceed the Lindhard critical channeling angle. Thus, the possibility of deflecting beams of accelerated charged particles by bent crystals is limited to just small angles of deflection.

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