Focusing of channelled ions just below the crystal surface, known as superfocusing, was treated analytically as a combination of straight movement along the channel axis and perpendicular oscillations of an anharmonic oscillator. The anharmonic term in the channel potential is responsible for dispersion of the trajectories. It was shown that the anharmonic term vanishes for a specific oscillation plane along the channel, which would in principle allow a very sharp focusing point. However, the focusing pattern is blurred due to the multiple scattering by electrons. This effect, which was estimated from the projectile energy loss calculated for axially dependent electron density, can completely obscure the focusing effect from the second focusing point onwards.
Šmit et al. (Sat,) studied this question.