A correspondence principle for the channelling of fast charged particles

The transmission of fast charged particles through thin crystals may be considered within the framework of either classical or quantum mechanics. The conditions under which the quantum mechanical diffraction approach attains correspondence with the more familiar classical analysis is discussed by considering only two waves, though a full quantum mechanical discussion requires a multiple beam consideration for the prediction of transmitted intensities. With increasing particle mass m for a fixed potential of interaction complementarity may be conveniently visualized in reciprocal space as being achieved when the angular width of the Bragg resonances exceeds the angular separation of the resonances themselves. Inelastic scattering produces a transmission pattern on the exit surface of the crystal of the kind first described by Kikuchi (1928). When correspondence occurs, and this may be determined by comparing the magnitude of a simple dimensionless parameter α with unity, angular widths of lines and bands in the transmission pattern are in a mass independent limit which is explicable on the basis of Newtonian mechanics and particle orbits. In both the classical and quantum mechanical treatments there is an essential division of the incident collimated beam into two distinctly separate components by the crystal, yielding a 'Pendellösung'behaviour which carries over from one analysis to the other. In the diffraction formulation this behaviour is associated with the extinction distance ξ h , and in the classical mechanical treatment with the orbit of wavelength λ.