Analytical Energy Eigenvalues of a Dirac Particle in Focusing Field of a Quadrupole Magnet

In the study, the motion of a charged spin-1/2 fermion is found out. It is assumed, in the system, that the free fermion is subjected to a linearly space-dependent magnetic field which can be supposed to be a focusing magnetic field of a quadrupole magnet in beam dynamics in the accelerator physics. In such an examination, two-component Dirac equation is solved via perturbation approximation of the Asymptotic Iteration Method, which has been widely used for the last decades. The results show that the fermion is bounded to the magnetic field for a certain condition of the strength of the field. For such a system, the analytical form of the energy eigenvalues are obtained. Moreover, to see whether this analytical expression works properly, the numerical eigenvalues are compared with the ones obtained by direct use of the AIM. We have an inspiration that the studies on beam dynamics and magnet design in particle accelerator physics may gain from this work.