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We use the paraxial ray approximation equations to analysis the dynamics of particles in a tilted solenoidal focusing channel. In this case, the particles' initial canonical angular momentum is nonzero, so we need to add the term of centrifugal potential to the dynamics equation of particles. And in the dynamics equation this centrifugal potential term is nonlinear, which results in the emittance growth. In practice, we also need to consider the spherical aberration's effect on emittance growth and the linear part of the space-charge force of a Kapchinskij-Vladimirskij distribution beam in the dynamics equation of particles.
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