Space-charge and third-order aberrations in quadrupole focusing systems☆

Abstract Our purpose is to consider third-order aberrations due to space charge. We derive the single-particle equations of transverse motion for a beam of elliptical cross section that propagates down a focusing channel. These equations include electric and magnetic fields caused by space charge and magnetic fields caused by quadrupoles and octupoles. We then calculate the third-order aberrations of a point-to-parallel system. We suggest that beam sizes should be minimized in order to control aberrations.