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

Abstract

   The motion of relativistic electrons in the planar channeling mode in the accompanying reference system moving along the channel at a velocity equal to the longitudinal channel component of the electron velocity can be considered as an implementation of the one-dimensional atom model with parameters depending on both the kind and orientation of the crystal and the relativistic energy of the charged particle moving in the channel. The motion mode in the planar channel can remain stable even if the crystal and its planar channels of motion are curved if the bending angle is not too large. It is demonstrated that the energy quantization condition for one-dimensional channeled motion within the Bohr–Sommerfeld approach coincides with the calculated adiabatic invariant of this motion. Using the expression for the adiabatic invariant in planar channeling, the maximum bending angle of the single crystal at which stable motion in the channeling mode is still possible is estimated. It is noted that the maximum bending angle of a single crystal should not exceed the Lindhard critical channeling angle. Thus, the hypothetical possibility of using the bent single crystals to control the directions of propagation of the accelerated particles beams is limited to only small angles of deflection.