Sine-squared Potential and Beam Extraction in Circulation Accelerator by Bent Crystal

In the classical mechanics frame and the dipole approximation the particle motion equation in bent crystal is reduced to the pendulum equation with a constant momentum by using the sine-squared potential.The phase planar properties are analysed by means of Jacobian elliptic function and the elliptic integral.The maximum deflected energy,the dechanneling factor and the dechanneled length are derived.The results show that the extrated efficiency is 70% for a proton beam with energy E=1.0 GeV moved in the channel on Si(110) if a radius of curvature R=1 m;and dechanneled length is of the order of μm for the particles with MeV energy;the dechanneled length is above 1 meter for the particle with TeV energy.