The sine-squared potential and the band structure for channelling effects

Abstract

The interaction between the charged particle and the crystal is described by quantum mechanics. The schrdinger equation describing a paticle moving in the channel is reduced to Mathieu equation by using a sine-squared potential. The band structure of this system is discussed based on Bloch theorem. The instability zone and the stop-band width are found by the perturbation techniques. It is shown that the width of the 1-st order and 2-nd order instability zones are ΔE1=Kβ/2 and ΔE2=md2p(Kβ)2/16π22, respectively, and this system presents automatical the band structure, which is not present in Lindhard potential and Moliere potentials.