Proof of the existence of the axial-to-planar channeling transition in a simple crystal model

Abstract

Consider a fast nonrelativistic, positively charged particle (ion) traversing a crystal, assumed to be simple cubic and monatomic for simplicity. Let the Cartesian coordinates X1,X2,X3 be parallel to the crystal axes, and ηi be the initial value of the component of the particle’s momentum vector along Xi. If ∣η3∣ is sufficiently large compared to ∣η1∣,∣η2∣, then (under mild technical assumptions) the ion’s motion is well approximated for a long time by a solution of the equations of motion (EOM) of the axial-continuum Hamiltonian H¯, obtained from the ion’s nonrelativistic Hamiltonian H by replacing the potential V(X1,X2,X3), describing its interaction with the atoms of the crystal, by its average V¯(X1,X2) over X3. Furthermore, if ∣η2∣,∣η3∣ are sufficiently large compared with ∣η1∣,∣η2∣, respectively, then to a good approximation its motion is given, again for a long time, by a solution of the EOM of the planar continuum Hamiltonian H̿, obtained from H¯ by replacing V¯(X1,X2) by its average V̿(X1) over X2. We define motions of the first (respectively, second) type as axial (respectively, planar) channeling. In this paper, the transition from the first to the second kind of motion, occurring when the crystal is suitably rotated, is discussed in a mathematically rigorous way by using an improved version of first-order single-phase averaging theory.