Three-dimensional magnetic field distribution function beyond the bore radius of quadrupole magnets

Using the one-dimensional field gradient distribution along the symmetry axis of quadrupole magnets, the conventional three-dimensional field distribution function has been successfully describing the magnetic field within the bore radius R. In the present study, the conventional formalism was found to cause singularities beyond the radius. A new distribution function has been derived which is free from the singularities. The region of the reliable magnetic field distribution function has been extended from r<R of the conventional formalism to xy<R2/2 of the new one. By positive use of the extended region, the electric power consumption of quadrupole magnets can be reduced more than one order of magnitude.