Analytical determination of the critical angle and critical distance of approach of particles channeled along primitive and non primitive atomic rows

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Abstract
A parallel beam of charged projectiles, reflected by a static atomic row inevitably exhibits a spread in outgoing transverse energy. However, above a certain critical distance of approach, Pc' the dispersion of the transverse energy of the outgoing particles around Eψ2 in (transverse energy of entering particles) turns out to become exceedingly small. In the present investigation, analytical formulae to determine Pc and the corresponding ψc are derived for axial channeling along static primitive rows and rows with a basis consisting of two different equidistant atoms, two equal but not equidistant atoms, or two different and not equidistant atoms per periodic length. A plot for a convenient evaluation of ψc and Pc is given. Previous equations for ψc and Pc and the present results are examined and compared with computer calculations. The reliability of the new formulae is demonstrated over four orders of magnitude of particle energy. A comparison with almost all available experimental data on critical axial channeling angles exhibits the relevance of the present analytical solutions.