Continuum-model planar channeling and the tangent-squared potential

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Abstract
The elements of continuum-model planar channeling are discussed using a general planar continuum potential with an emphasis on the depth-dependent phase-space density. This joint spatial-momentum density contains all the information concerning continuum-model planar channeled particles and allows a unified treatment of the depth-dependent and statistical equilibrium, spatial, and momentum densities for an arbitrary initial density. The Gaussian-beam-divergence case is discussed in some detail. A simple, two-parameter planar-continuum potential, the tangent-squared potential, is then introduced. We show that this potential is physically reasonable, and, for many calculations, easier to use than previously used planar continuum potentials such as the Lindhard, Moliere, and hyperbolic cosine. It simplifies many calculations because the channeled-particle wavelength function and the solutions of the associated equation of motion can be written in terms of elementary functions (specifically the square root, inverse sine, and trigonometric functions) and because the phase-space density has a simple analytic form.