End effects in first-order theory of quadrupole lenses

Abstract Rapidly convergent series for the transverse position coordinates of a charged particle moving through an end zone of a quadrupole lens are derived by iterating integral equations obtained from the first-order differential equations. Not more than three terms of the series appear to be needed for lens strengths normally used. The series contain form-factors formed by integrating the field gradient along the axis with various weight functions. The focal lengths and the principal-plane coordinates of the whole lens are expressed in terms of numerically well behaved functions of the form-factors. Formulae are also derived for the effective lengths and strengths of the lens.