THE NONLINEAR RESONANCE QUISTOR .1. POTENTIAL DISTRIBUTION IN HYPERBOLOIDAL QUISTORS

Non-linear resonance QUISTORS are generated by the superposition of multipole fields on the basic quadrupolar field. An easy way to superimpose even multiples is the variation of the hyperbolic angle of the classical Paul QUISTOR. In this first paper of a series, an analytical expression is deduced for the potential distribution in such boundaries, using the variational calculus method described by Ritz. The resulting complex potential formula can be reduced with sufficient accuracy to the simple form: u(r,z) = V0/r0(2)(r2 - 0z2) + c(2 - 0)V0[r0(4) - (r2 - theta-z2)2/r0(4)]exp(- theta-r2 + b-theta-z2)/r0(2) containing only three constants a, b, and c, which depend almost linearly on the parameter theta = r0(2)/z0(2) for the hyperbolic angle. The dependency of the constants is presented in graphical form. The method for the derivation of the potential is very general and can be used with other complex boundaries, even including space charge distributions. This potential distribution, among others, will be used in future papers to study the basic mass analysis principles of non-linear resonance QUISTORS.